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Conics Worksheet
Name:_
Transform the equation by completing the square. Complete the following.
a. Find the vertices.
b. Find the Foci.
c. Find the length of the Major and Minor Axis (Ellipse).
d. Find the length of the Transverse and Conjugate Axis (Hyp
MATH PreCalculu Advice
Showing 1 to 2 of 2
I would recommend this course because it has given me knowledge that will help me throughout my school career, such as on the SATs, ACTs, and even in college.
Course highlights:
I learned many math skills that will help me be successful in future math courses.
Hours per week:
6-8 hours
Advice for students:
Complete all of your work with your best effort and you will be very successful.
Course Term:Fall 2015
Professor:Stein
Course Required?Yes
Course Tags:Math-heavyGreat Intro to the SubjectMany Small Assignments
May 22, 2016
| Probably wouldn't recommend.
This class was tough.
Course Overview:
It was a very difficult course for me, and I usually am very good at math. It took a lot more work than I was willing to give, and I was a senior this year.
Course highlights:
Trigonometry, graphing, identities, and more. However, none of this will be used in my future career of elementary teacher.
Hours per week:
6-8 hours
Advice for students:
Think about your abilities in math. Think about how much work you are willing to give to your studies. Lastly, think about the teachers you could have, because the teacher can make all the difference. If you still want to take it, Good luck to you. | 677.169 | 1 |
Members competitions 2017 nawg
Free math problem solver answers your algebra homework questions with step-by-step explanations. About help. How can i help you? Enter a problem. .
Free math problem solver answers your algebra homework questions with step-by-step explanations. About help. How can i help you? Enter a problem. . Enter your problem statement (as algebra, not words!) where it says enter a problem near the bottom of the. View many solved examples by clicking the vertical dots on the top right of the solver. . Solves algebra problems and walks you through them. Algebra calculator is a calculator that gives step-by-step help on algebra problems. See more. . Step-by-step calculator. Solve algebra, trigonometry and calculus problems step-by-step. Pre algebra algebra matrices & vectors functions & graphing. . Solve calculus and algebra problems online with cymath math problem solver with steps to show your work. Get the cymath math solving app on your. .
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If you please enter the email address and well send you an email containing instructions for changing your password. Simplies fractions and mixed numbers or converts between mixed numbers and improper fractions. If you want to be really good at all types of math, you need to practice please enter the email address and well send you an email containing instructions for changing your password. If you want to be really good at all types of math, you need to practice math notebooks have been around for hundreds of years. Learn more about your math problem in one of the following sections in interactive mathematics.
Webmath is a math-help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment. Evaluates an expression by substituting for given variables, then simplifying. Will add, subtract, multiply, and divide fractions, mixed numbers, and improper fractions. Plots equations and enables users to rotate the graphs 360 degrees in any direction. The math answers are generated and displayed real-time, at the moment a web user types in their math problem and clicks solve. You write down problems, solutions and notes to go back to when doing your homework or studying for an exam. It is important to remember learning math takes practice, lots of practice. Each new topic we learn has symbols and problems we have never seen. Enter your problem statement (as algebra, not words!) where it says enter a problem near the choose the operation you want the solver to do (it suggests appropriate operations, depending on your question), clicking on view steps will lead you to the mathway sign up page where you can get full solutions. Below is a math problem solver that lets you input a wide variety of math problems and it will provide the final answer for free.
Cite this for me harvard apa mla reference generator
Help solving algebra problems how to solve math problems. Power of buyers least contains the of working with their trainers which at least twice have. . | 677.169 | 1 |
Be sure that you have an application to open
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1 MB|69(C) write exponential functions in the form f(x) = abx (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay A.9(D) graph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems
Supporting Standards: A.9(A) determine the domain and range of exponential functions of the form f(x) = abx and represent the domain and range using inequalities A.9(B) interpret the meaning of the values of a and b in exponential functions of the form f(x) = abx in real-world problems A.9(E) write, using technology, exponential functions that provide a reasonable fit to data and make predictions for real-world problems
Each problem is coded with the specific TEKS alignment for easy lesson planning. | 677.169 | 1 |
Description
An undergraduate textbook devoted exclusively to relationships between mathematics and art, Viewpoints is ideally suited for math-for-liberal-arts courses and mathematics courses for fine arts majors. The textbook contains a wide variety of classroom-tested activities and problems, a series of essays by contemporary artists written especially for the book, and a plethora of pedagogical and learning opportunities for instructors and students.
Viewpoints focuses on two mathematical areas: perspective related to drawing man-made forms and fractal geometry related to drawing natural forms. Investigating facets of the three-dimensional world in order to understand mathematical concepts behind the art, the textbook explores art topics including comic, anamorphic, and classical art, as well as photography, while presenting such mathematical ideas as proportion, ratio, self-similarity, exponents, and logarithms. Straightforward problems and rewarding solutions empower students to make accurate, sophisticated drawings. Personal essays and short biographies by contemporary artists are interspersed between chapters and are accompanied by images of their work. These fine artists--who include mathematicians and scientists--examine how mathematics influences their art. Accessible to students of all levels, Viewpoints encourages experimentation and collaboration, and captures the essence of artistic and mathematical creation and discovery.
Classroom-tested activities and problem solving
Accessible problems that move beyond regular art school curriculum
Multiple solutions of varying difficulty and applicability
Appropriate for students of all mathematics and art levels
Original and exclusive essays by contemporary artists
Forthcoming: Instructor's manual (available only to teachers)
About the authors
Marc Frantz holds a BFA in painting from the Herron School of Art and an MS in mathematics from Purdue University. He teaches mathematics at Indiana University, Bloomington where he is a research associate. Annalisa Crannell is professor of mathematics at Franklin & Marshall College. She is the coauthor of "Writing Projects for Mathematics Courses".
SimilarThis eBook edition has been specially formatted for on-screen viewing with cross-linked questions, answers, and explanations.
UNLOCK THE SECRETS OF GEOMETRY with THE PRINCETON REVIEW.
Geometry can be a daunting subject. That's why our new High School Unlocked series focuses on giving you a wide range of key techniques to help you tackle subjects like Geometry. If one method doesn't "click" for you, you can use an alternative approach to understand the concept or problem, instead of painfully trying the same thing over and over without success. Trust us—unlocking geometric secrets doesn't have to hurt!
With this book, you'll discover the link between abstract concepts and their real-world applications and build confidence as your skills improve. Along the way, you'll get plenty of practice, from fully guided examples to independent end-of-chapter drills and test-like samples.
Everything You Need to Know About Geometry. • Complex concepts explained in clear, straightforward ways • Walk-throughs of sample problems for all topics • Clear goals and self-assessments to help you pinpoint areas for further review • Step-by-step examples of different ways to approach problems
Practice Your Way to Excellence. • Drills and practice questions in every chapter • Complete answer explanations to boost understanding • ACT- and SAT-like questions for hands-on experience with how Geometry may appear on major exams | 677.169 | 1 |
Pre-algebra reproducible book use activities to complement textbook lessons or as extra practice on concepts your students find challenging.McDonald Publishing® Pre-algebra reproducible book is ideal for ages 9 - 12 and grades 6th - 9th. Book use activities to complement textbook lessons or as extra practice on concepts your students find challenging. | 677.169 | 1 |
Mathematical Foundations of Computer Science explains the fundamental concepts in mathematics. It can be used by the students in computer science as an introduction to the underlying ideas of mathematics for computer science. It explains topics like mathematical logic, predicates, relations, functions, combinatorics, algebraic structures and graph theory | 677.169 | 1 |
Math is very important no no doubt, but I feel like much of my time in high school was wasted learning certain types of math that I will never use. I feel like that time could have been better spent teaching us basic life skills that we need for adult life. Oddly enough I found "Logic and Set Theory" to be interesting to me, and I had an easier time learning it than the other types of math I was taught.
and they are probably more readily applicable for a high-schooler as well :p
The more difficult the math course is, the less you'll actually use the formulas later on honestly. I'm working on my BSME and aside from some basic calculations most of the math that we learned is only to help understand different principal engineering concepts and figure out how different things are related and such. I mean its an invaluable tool, but if i ever need a formula i can just look it up
What do you think of the "mathematics you do not need"? I'm talking high-school and college math courses. The "regular" courses of math.
You do need to know basic math, and accounting & finances to make it in the future, but why are some people required to know the most insane things; as they'll never imply it into their lives afterward?
I have to use multi variable calculus to find subtransient reactance from time to time. But then again I'm a scientist/ engineer
Khaltazarwrote:
I feel like for sure ... Linear Algebra should not be required to become a computer science major.
Linear Algebra is used in computer games. How do you rotate an object? How do you make things move? It all comes down to Matrix transformations. Linear Algebra is also used in designing Artificial Intelligence (along with statistics), and an important mathematical tool in
the area of computer security. Linear Algebra makes it easier to solve certain problems, and the problem is more intuitive.
Linear Algebra is also used extensively in engineering, when solving systems of equations, and also has important applications in game theory.
I know it has it's purposes, but not everyone wants to get into fields like game programming or theory, etc.
If you want to understand politics (elections and congress voting on bills) you need to know the basics of game theory. Maybe not a full course, but a couple hour coverage is pretty useful. I live in a democracy that has a lot of problems that are easily understood with game theory. If everyone knew game theory, we would probably fix congress already.
A basic grasp of the concepts of linear algebra is great for reasoning about, and explaining common things. Spaces (a nice explanation here: for example are a great concept for reasoning about just about any collection of things with properties, or sets of choices. Orthogonality ( ) is another fantastic tool for tool for reasoning. Again, you only really need the concept which should take more than an hour and it can change how you thing about things for the rest of your life. I think of most things in these terms: knowing them really changed my life in ways that were not obvious for years. (On the topic of fixing congress, we need some organizational of legislation on unrelated parts of the issues space to reduce the scope of problems like brinkmanship that are easily explained by game theory. See, applicable!)
The subset of calculus thats really useful to just about everyone can also be covered in probably 10 minutes or less if done well (I explained calculus to someone about to drop out of highschool just fine in a couple minutes: he found it a neat and simple concept), but likely a bit more for it to really stick. All calculus is is working with rates of change. Lots of things change with respect to other things (such as world population, or the number of manga champers published changing over time). The concept of derivatives and integrals are trivial and useful: its only computing them that's hard and there is software to do that for you (but you won't need to compute them anyway usally). (And congress should pass their darn spending laws as rates instead of yearly budgets. That would fix the whole shutdown nonsense and be easier to reason about)
If you have a decent intuition for calculus, you know what it means to get a 5% return per pear on your investment, or that a city is growing 10% per year. For that last one, you know that the city is screwed and its gonna end up a logistic curve or something like it, and you can ponder whats going to cause that... The same thing (and understanding of exponential growth, which falls easily of of calculus) provides great intuition into spread of disease, spread of popular memes and videos, and the increasing power of computers (and now long and if that can continue). (Oh, and it would help you understand the relationships between inflation, saving, and dept easier, including that pesky federal dept)
The first things you learn in statistics about correlation and causality and various types of errors provide a great understanding on on the misleading studies published all the time. Sampling bias and multiple testing issues are probably the most important. Personally (since i'm a bit crazy) I find knowing the central limit theorem super useful for any time you know something about some subset of a group, and are interested in how much more information you need to make accurate inference: I watch 3 random shonen shows. If I want to estimate how good the genera is with double the confidence I need to watch 6 more (double the accuracy requires the square of the sample size approximately assuming some details I'll omit, but given that actual content there, assuming an approximately normal distribution is bullshit for that one).
TLDR: I really think most people should spend a few days learning about higher mathematics. The available classes generally suck at this and make you do a ton of useless work to get that little bit of great understanding. The internet is full of great resources though: remember you don't need to be able to do the math, instead you need to be able to apply the concepts to your reasoning. And as a bonus, knowing what those concepts are called enables you to explain things easily in those terms, and lets you look up properties that apply when you need them.
Algebra you actually use a lot irl. This surprised me a lot actually :D. Sometimes I have to use trig and alg 2 for a few things too!
I'm not sure you'd need to take the derivative of anything unless you're trying to find the slope of your finances charts, so you can see any trends. I mean, so let's say you're trying to be savvy and list all your personal expenses over the course of X months and you want to see if there's a trend in their increase/decrease or if they're just random--you'd be using calc.
Shhhh, stop providing actual uses for higher level math, I really want it to just disappear;-;
I find mathematics, as a whole, to be very useful. However, the usefulness of mathematics is derived from its ability to model situations; the mathematics best optimized for modeling (which I'd argue are differential equations and linear algebra) are typically not seen unless you complete a STEM degree at the college level. Inferential statistics is pretty grand too.
If you don't make it that far, though, it's difficult to discern the usefulness of mathematics because there quite frankly isn't none. That isn't to say arithmetic and elementary algebra are useless; but for the most part, they serve as nothing more than building blocks to constructing useful models. The best analogy I can make is learning arithmetic and elementary algebra but nothing higher is akin to learning grammar but not how to write sentences or essays.
I've been known to make liberal usage of higher mathematics while building RPG characters.
There are plenty of places to use the math, you just need to know how to apply it. I agree with Zoraprime: most people won't know how to apply higher mathematics unless they have a degree in a related field.
I resent my school's computer science curriculum being too nice to people who don't like math. As far as I am concerned, computer science is a branch of mathematics. I wish I had stayed a pure math major and studied programming in my spare time.
As far as applying math: Suppose you have someone you are attracted to in your car next to you, and you are driving. You want that person to feel safe in the car with you. How should you handle the brake? You might say that you should avoid pressing the brake down too far too avoid too much deceleration (third derivative of position). Actually, what's more important is that the passenger is eased into the deceleration. You want to minimize jerk (fourth derivative of position), so you should ease into whatever pedal position is necessary. This might sound like "common sense," but it is rates of change, which is a big part of calculus.
I resent my school's computer science curriculum being too nice to people who don't like math. As far as I am concerned, computer science is a branch of mathematics. I wish I had stayed a pure math major and studied programming in my spare time.
It seems like a lot of good answers have been provided at this point. I'm curious whether you agree or not. Also, look into neuroplasticity. Everything you do shapes your brain and how it works. Including math. | 677.169 | 1 |
WELCOME TO OUR PRESENTATION
Course: Introductio to Public Speaking
Professor: Dr. Nguen Thi Thuy Huong
Functions of Year round air
conditioning
Heating and humidification of air in winter
Cooling and dehumidification in summer
Why should we choose SHARP
SPEECH WORKSHOP
Evaluating a Peakers Rhetorical Effectiveness.
I decide to choose this talk of Larry Smith from TED to clarify the following
questions.
https:/
r?language=en
1. Speech Go
WELCOME TO OUR
PRESENTATION
How to be a good public speaker
DEREK SIVERS
Course: Introduction to Public Speaking
Professor: Dr.Nguyen Thi Thuy Huong
Derek Sivers
an American
entrepreneur
best known for being
the founder and former
president of CD Baby,
Integration:
1. Income distribution: the coefficient of inequality for each Lorentz curve
1
Integration. L=2
[xf ( x )]dx
0
the lower result will be more equitable
income distribution.
2. Sinking funds: how much money need to establish the fund to with
1
Preliminaries
Precalculus Review I
Precalculus Review II
The Cartesian Coordinate System
Straight Lines
1.1
Precalculus Review I
The Real Number Line
We can represent real numbers geometrically by points on
a real number, or coordinate, line:
Origi
6
Integration
Antiderivatives and the Rules of Integration
Integration by Substitution
Area and the Definite Integral
The Fundamental Theorem of Calculus
Evaluating Definite Integrals
Area Between Two Curves
Applications of the Definite Integral to
Precalculus
Wkst- Linear Functions and Linear Modeling
Name:_
Date:_
1. A town's population has been growing linearly. In 2003, the population was 45,000, and the population has
been growing by 1700 people each year.
a. Write an equation, P t , for the po
8.4 day one
Improper Integrals
Greg Kelly, Hanford High School, Richland, Washington
Until now we have been finding integrals of continuous functions over closed intervals.
Sometimes we can find integrals for functions where the function or the limits are
8.3 Relative Rates of Growth
Greg Kelly, Hanford High School, Richland, Washington
y = e x grows very fast. The function
We could graph it on the chalkboard: If x is 3 inches, y is about 20 inches:
( 3, 20 )
Wte64 inches, the y-value We have gone less tha
8.2 Day 2: Identifying Indeterminate Forms
Photo by Vickie Kelly, 2008
Brooklyn Bridge, New York City
Greg Kelly, Hanford High School, Richland, Washington
What makes an expression indeterminate?
Consider:
We can hold one part of the expression constant
8.2 day 1 LHpitals Rule
Actually, LHpitals Rule was developed by his teacher Johann Bernoulli. De lHpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons.
Guillaume De l'Hpital 1661 - 1704
Greg Kelly,
8.1: Sequences
Craters of the Moon National Park, Idaho
Photo by Vickie Kelly, 2008 Greg Kelly, Hanford High School, Richland, Washington
A sequence is a list of numbers written in an explicit order.
cfw_ an = cfw_ a1, a2 , a3, . , an , .
nth term
Any r
A.P Calculus Worksheet: Areas of Surfaces of Revolution
Find the areas of the surfaces generated by revolving the curves in problems 1-4 about the axes indicated: 1. y = x / 2, 0 x 4 , about the x -axis. Check your result with a formula from geometry. 2.
7.4 Day 2
Surface Area
(Photo not taken by Vickie Kelly)
Greg Kelly, Hanford High School, Richland, Washington
Surface Area:
ds
r
Consider a curve rotated about the x-axis: The surface area of this band is:
2 r ds
The radius is the y-value of the function
7.4 Day 1 Lengths of Curves
Golden Spike National Historic Site, Promontory, Utah
Photo by Vickie Kelly, 1999 Greg Kelly, Hanford High School, Richland, Washington
Lengths of Curves:
ds dx
dy
If we want to approximate the length of a curve, over a short d | 677.169 | 1 |
Explore Algebra 2, Flip Books and more!
Explore related topicsTrig Graphing the Sine and Cosine Flip Book Foldable
Graphing the Sine and Cosine Flip Book with HW / Quiz for Trigonometry, PreCalculus, and some Algebra 2 classes. This Graphing the Sine and Cosine mini bundle reinforces and reviews Amplitude, Phase Shift, Period, Vertical Shift, finding the End Points, and graphing of sinusoidal functions. Over 16 examples, many for students to complete. Great Study Guide, Great activity. | 677.169 | 1 |
Review
Background from OER Project Review Team
Georgia Virtual Learning is part of the Georgia Department of Education. Existing OER resources are combined with created material into a structured course format. Though designed to take advantage of digital media, a print option is available to reproduce much of the content. This 8th Grade Mathematics resource is intended as an online course and was designed to align with the Georgia Performance Standards. While assessments are created, they are not available for wide distribution in order to maintain their usefulness for Georgia teachers implementing the resource. This should factor into the viewer
Amount of work required to bring into CCSS alignment (average score): Moderate (2.25)
Strengths/Ideal Use:
This material would best be used by an independent learner, working online, who is trying to get a review of grade 8 math. It could also be used by a highly-motivated and capable learner working online independently to master 8th grade standards, although the student may not develop sufficient conceptual understanding. The interactive features are varied and there are different kinds of self-checks that students can take.
Concerns:
Students are not asked to engage in productive struggle. Exercises are at a low DOK level and are procedural in nature.
Lessons are procedural in nature. Concepts are explained and procedures presented without sufficient thinking and engagement by students.
Suggestions:
Include more deep and open-ended problems.
Provide more investigations and places for students to discuss, justify, conjecture, find structure, etc.
Strengths/Ideal Use:
Standards are addressed well and lessons are designed well for teaching students mathematical skills. Videos are present for student help, explanations are clear and easy to follow.
This resource is best used as a review material for skills students needed to refresh on, more of a student led resource for online support of classroom instruction. I don't think it addresses the level of understanding or makes enough meaningful connections for students to use as a primary resource for instruction.
Concerns:
Re-explains previous learning anew instead of getting students to make sense of new learning through the previous work.
No calling out of mathematical practices for students/teachers.
Suggestions:
Make more connections with standards and previous learning.
When students are presented with a difficult task, let them make sense before telling how to do the math to give them an opportunity to engage in mathematical practices.
Strengths/Ideal Use:
Content is substantially complete. The inclusion of the projects and discussions encourage opportunities for deeper learning.
This curriculum is intended for use in an online class. An experienced teacher could implement this (online) well but would need to know where/how to provide appropriate scaffolding and monitor students' progress.
Concerns:
Could not find evidence that students must construct scatter plots and two-way tables.
Assessments not available for review so unsure of quality.
This curriculum appears to be available (in its complete form) only to Georgia students. Teachers outside this system would need to create quizzes and tests.
Suggestions:
Add construction of scatter plots and two-way tables to the homework.
Make assessments available to registered teachers.
Strengths/Ideal Use:
The units are very well organized and easy to follow, and they stick with the CCSS. The key units are divided by the major work in the grade level that students are responsible for. With corrections, it would make an excellent MS curriculum, and could replace the textbooks. | 677.169 | 1 |
Showing 1 to 30 of 41
MATH100: Mathematical Thinking
Mathematics of Fractal Shapes
Homework (5 points)
1. Who is the inventor of fractals, and how did he get interested in fractals?
2. The Koch Snowflake
(a) Draw the Koch Snowflake show at least 3 steps of the process.
(b) Wha
Updates
Sustainability Project
You will be assigned to a Company,
and you will read their sustainability report,
and then take an online, open notes, quiz on the report.
These reports are all using the GRI format.
Company Assignments are completed and you
Summer 2015
MAT 263 Test Schedule
Please be sure to take all tests during the scheduled testing window. You should contact your testing
center of choice to make an appointment, or to determine if walk-ins are accepted. If you miss a test
window you will r
STUDY GUIDE FINAL EXAM
Exam format will include true/false and multiple choice questions. Many questions will
originate from the following groups of information though they will not be limited to these
topics:
Anatomical Movement
o What are planes, axes,
MAT 263
Derivatives by Hand
The Basics
Derivative Skills Test and Test 2
You have a skills test on finding the
derivative by hand.
You may not use any technology for this
test. It is only 50 minutes long.
This will cover finding the derivative by
using
Introduction
History about Facebook
Mark Zuckerberg.
Why Facebook?
URL
Posting
Think before you post.
Newsfeed
Timeline
Steps how to make a page on the Facebook
Create a page
Create you profile image
Create your cover image
Create the about section
THANK
MATH100: Mathematical Thinking
Mathematics of Spiral Growth
Homework (5 points)
1. Who was Fibonacci, and what is he famous for?
2. Remember how the Fibonacci sequence is generated?
(a) Write the first 12 Fibonacci numbers.
(b) Write the number 110 as a s
HIS 131 Summer 2015
Things to Remember
1. Go to COURSE RESOURCES, print out the Syllabus for easy retention. Also print out
the Journal parameters, Quiz parameters, and Course Chart sheets. They will come in
handy throughout the semester.
2. This course i
Baraah Alboeij
4B
Reading log #1
Northrop,Michael. Trapped. New York, 2011 pages 1-58
Scotty Weems is a sophomore at Tattawa Regional High School. He and his best friends, Pete
Dubois and Jason Gillispie, stay behind at school one snowy afternoon after an
1
Venn diagram
U
A
C
B
The universal set U is usually represented by a rectangle.
Inside this rectangle, subsets of the universal set are represented by
geometrical gures.
2
Venn diagrams help us identify some useful formulas in set operations.
2
Venn d
1
Equations of Lines
(1) If we know one point (x1, y1) and the slope m of a line L, can write equation
of L in the form
y y1 = m(x x1)
called the point-slope form of L.
1
Equations of Lines
(1) If we know one point (x1, y1) and the slope m of a line L, ca | 677.169 | 1 |
Frederick MostellerFifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics) New York Dover 1987 0486653552 / 9780486653556 Paperback New Can you solve the problem of "The Unfair Subway"? Marvin gets off work at random times between 3 and 5 p.m. His mother lives uptown, his girlfriend downtown. He takes the first subway that comes in either direction and eats dinner with the one he is delivered to. His mother complains that he never comes to see her, but he says she has a 50-50 chance. He has had dinner with her twice in the last 20 working days. Explain. Marvin's adventures in probability are one of the fifty intriguing puzzles that illustrate both elementary ad advanced aspects of probability, each problem designed to challenge the mathematically inclined. From "The Flippant Juror" and "The Prisoner's Dilemma" to "The Cliffhanger" and "The Clumsy Chemist," they provide an ideal supplement for all who enjoy the stimulating fun of mathematics. Professor Frederick Mosteller, who teaches statistics at Harvard University, has chosen the problems for originality, general interest, or because they demonstrate valuable techniques. In addition, the problems are graded as to difficulty and many have considerable stature. Indeed, one has "enlivened the research lives of many excellent mathematicians." Detailed solutions are included. There is every probability you'll need at least a few of them.
Price:
3.00 USD
John MightonThe Myth of Ability: Nurturing Mathematical Talent in Every Child New York Walker & Company 2003 0802777074 / 9780802777072 Paperback Very Good Clean and sturdy. Small ink dot on the page edges. For decades teachers and parents have accepted the judgment that some students just aren't good at math. John Mighton―the founder of a revolutionary math program designed to help failing math students―Price:
3.75 USD
Johnnie T. DennisThe Complete Idiot's Guide to Physics New York, NY Alpha 2003 159257081X / 9781592570812 Trade Paperback Good As an easy, approachable companion to an oft-dreaded subject, this guide is intended for high school and college students who are required to take physics. Offering easy-to-understand instruction, it follows a standard non-calculus-based physics curriculum. • Clearer, more approachable, solved practice problems plus revised text and figures • Author and physics teacher Johnnie Dennis won the National Teacher of the Year award, the nation's oldest and most prestigious honor for high school teachers • In any given year, research shows that between 420,000 and 2.5 million college students have to take at least one physics course to earn their degree
Price:
4.00 USD
Katherine A. LoopBeyond Numbers: A Practical Guide to Teaching Math Biblically Fairfax, VA Christian Perspective 2011 0977361136 / 9780977361137 Paperback Used As New Math does not have to be hard or difficult to understand. Nor do we as Christians have to settle for viewing it the same way the world does. Beyond Numbers exposes the spiritual harm in today's math presentations and shows how math is really an exciting testimony to God's faithfulness and is an incredible, God-given tool. The second half of the book walks through teaching math biblically, covering everything from curriculum choice to overcoming obstacles. Written in a straightforward, non-technical manner, this book is perfect for homeschool parents and anyone else who wants to view or teach math biblically. James D. Nickel, professor of mathematics and author of Mathematics: Is God Silent?, said, "The book is a great, general-purpose introduction for a typical parent." Sandra A. Lovelace, homeschool pioneer, speaker and author, and International Consultant with HSLDA made the following statement: "Every homeschooling parent, every Christian teacher, and maybe every Christian who uses mathematics ought to read this book." Homeschool mothers have found the book delightfully easy to read and encouraging. One mother explains, "As a homeschooling mom committed to teaching my children to love God whole-heartedly, I was startled to find I had unconsciously accepted math as a spiritually neutral...I believe every Christian teacher who reads Beyond Numbers will be challenged to rethink his/her assumptions about math, and will be motivated, as I am, to find new ways to present math facts, concepts and principles biblically."
Price:
43.00 USD
Mildred Johnson and Tim JohnsonAlgebra: How to Solve Word Problems, Proven Techniques from an Expert New York, NY McGraw-Hill 2000 0071343075 / 9780071343077 Trade Paperback Fine Has previous owners name on the first page. Solving word problems has never been easier than with Schaum's How to Solve Word Problems in Algebra! This popular study guide shows students easy ways to solve what they struggle with most in algebra: word problems. How to Solve Word Problems in Algebra, Second Edition, is ideal for anyone who wants to master these skills. Completely updated, with contemporary language and examples, features solution methods that are easy to learn and remember, plus a self-test.
Price:
2.25 USD
Roger S. JonesPhysics for the Rest of Us New York Barnes and Noble 1999 0760712638 / 9780760712634 Hard Cover As New As New Ten basic ideas of 20th-century physics that everyone should know... and how they have shaped our culture and consciousness.
Price:
3.00 USD
Ruth BeechickYou Can Teach Your Child Successfully: Grades 4-8 Fenton, MI Mott Media 1999 0940319047 / 9780940319042 Trade Paperback Very Good This classic gives nitty-gritty help for each subject in each grade. Become an informed, confident teacher, free from rigid textbooks. Learn how to individualize spelling; how to use "real books" in history, reading, and other studies; how to make arithmetic meaningful; how to avoid the grammar treadmill; how to develop advanced reading skills; and much more. Satisfied readers say "I wish I had had this three years ago when I began homeschooling . . . the most practical manual I've ever read on the subject . . . The simplicity of it all! I don't think any homeschooler should be without it."
Price:
2.25 USD | 677.169 | 1 |
1. ABSTRACT ALGEBRA ON LINE WELCOME TO abstract algebra ON LINE. This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra.
WELCOME TOElements of Abstract and Linear Algebra Edwin H. Connell This is a foundational textbook on abstract algebra with emphasis on linear algebra. You may download parts of the book or the entire textbook. It is provided free online in PDFDVIpostscript , and gzipped postscript. Please read some words from the author first. Join the revolution in education! Write a supplement to this book and put it online. Participate in and contribute to a forum on abstract and linear algebra. Ask questions, give answers, and make comments in a worldwide classroom with the internet as blackboard. [March 20, 2004] The entire book Introduction Outline Chapter 1: Background and Fundamentals of Mathematics Chapter 2: Groups Chapter 3: Rings Chapter 4: Matrices and Matrix Rings Chapter 5: Linear Algebra Chapter 6: Appendix Index Please send comments to Edwin H. Connell Web page created by Dmitry Gokhman
4. ABSTRACT ALGEBRA ON LINE: Contents abstract algebra ON LINE. This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. Intro to Abstract Algebra 5245-46
The text for 5245-5246 in Summer 1998 will be My Notes (PostScript) PDF version ). These notes (with exercises and index) were specifically designed by me for this two-quarter sequence. We will start at the beginning of the Notes, and go approximately halfway through in 5245. The precise pace will depend on the class, and content of each quiz and homework assignment will be indicated as we go. Starred sections and starred exercises are optional.
There will be 20-minute quizzes each Tuesday and Thursday in the first hour of class. These quizzes will be open-book, open-notes. The one-hour final (Tues, July 21 for 5245, Wed Aug 26 for 5246) is also open-book, open-notes.
email is by far the best way to reach me and get a helpful response: garrett@math.umn.edu
If we have a grader, corresponding homework will be collected after each quiz, and will count for 10 percent of your grade. The lowest two quiz scores will be dropped (but no make-up quizzes given), and quizzes count for 60 percent. The final will be 30 percent. (If there is no homework grader, the quizzes will be 2/3 and the final 1/3).
Robert B. Ash Professor Emeritus, Mathematics
Books etc. On Line
Abstract Algebra: The Basic Graduate Year A Course In Algebraic Number Theory A Course In Commutative Algebra A Pari/GP Tutorial Click below to read/download chapters in pdf format. PDF files can be viewed with the free program Adobe Acrobat Reader Comments and suggestions for improvement are welcome.
Abstract Algebra: The Basic Graduate Year (Revised 11/02)
This is a student-oriented text covering the standard first year graduate course in algebra. Solutions to all problems are included and some of the reasoning is informal.
Abstract Algebra
Text will be my notes, intended to be discursive and motivational. In addition, it would be wise to have at least one standard reference for lookup. Lang's Algebra is standard, but not meant for the novice.
Miscellaneous smaller notes
A fuller set of more elementary exercises and notes at 'Intro to Abstract Algebra' Updated Thursday, 08-Jan-2004 11:43:28 CST home The University of Minnesota explicitly requires that I state that "The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota."
Exploring Abstract Algebra with Mathematica
10. Maple PowerTools - Abstract Algebra Set of lessons covering an undergraduate course in abstract algebra. Topics include finite groups, subgroups, and cyclic groups. Maple lessons for an undergraduate course in abstract algebra. Each lesson provides exercises to on group theory, which accompany the book Contemporary abstract algebra by J
MapleMapleNetMaple T.A.Toolboxes ... Contact Us This is a comprehensive set of 16 Maple lessons for an undergraduate course in Abstract Algebra. Each lesson provides exercises to reinforce understanding of each topic. Prof. Alec Mihailovs of Shepherd College contributed six lessons on group theory , which accompany the book Contemporary Abstract Algebra by J. Gallian, ISBN: 0-618-12214-1 Prof. Mike May of St. Louis University contributed ten lessons on fields, rings and Galois theory , which accompany the book Abstract Algebra, by Dummit and Foote, ISBN 0-13-569302-0 Download the course for Maple 7 Download the course for Maple 6 Preview or download individual lessons below Preview Download Chapter 1: Preliminaries Chapter 2: Introduction to Groups Chapter 3: Groups Chapter 4: Finite Groups and Subgroups Chapter 5: Cyclic Groups Chapter 6: Other Topics in Group Theory Chapter 7: Factoring Polynomials over Extensions of the Rationals Chapter 8: The Gaussian Integers Chapter 9: The Division Algorithm and Quadratic Extensions Chapter 10: The Minimum Polynomial of an Algebraic Expression Chapter 11: Extension Fields and Inverses Chapter 12: Automorphism Groups Chapter 13:
11. Course Notes --- J.S. Milne In contrast to most such accounts it studies abstract algebraic varieties, and not just subvarieties of affine and projective space.
Course Notes
Full notes as pdf (or dvi and ps) files for all the advanced course I taught between 1986 and 1999. Some of the notes give complete proofs (Group Theory, Fields and Galois Theory, Algebraic Number Theory, Class Field Theory, Algebraic Geometry), while others are more in the nature of introductory overviews to a topic. About the notes At last count, the notes included about 1350 pages.Errata: This is a list of errors and additional comments not yet incorporated into the files on the web, mainly contributed by readers. Group Theory A concise introduction to the theory of groups. html (August 29, 2003; v2.11; 85 pages) Fields and Galois Theory A concise treatment of Galois theory and the theory of fields, including transcendence degrees. html (August 31, 2003; v3.01; 99 pages) Algebraic Number Theory A fairly standard graduate course on algebraic number theory. html (31/8/98, v2.1; 140 pages) Class Field Theory html (6/5/97; v3.1; 222 pages) Modular Functions and Modular Forms This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts.
13. Abstract Algebra Lab Manual abstract algebra with GAP By JG Rainbolt and JA Gallian August 2003 Version. Below are pdf versions of the August 2003 version of the lab manual.
Abstract Algebra with GAP By J. G. Rainbolt and J. A. Gallian August 2003 Version
Introduction
New: Jaume Domenech Larraz has translated the Elements into Catalan at Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics. It has influenced all branches of science but none so much as mathematics and the exact sciences. The Elements have been studied 24 centuries in many languages starting, of course, in the original Greek, then in Arabic, Latin, and many modern languages. I'm creating this version of Euclid's Elements for a couple of reasons. The main one is to rekindle an interest in the Elements, and the web is a great way to do that. Another reason is to show how Java applets can be used to illustrate geometry. That also helps to bring the Elements alive. The text of all 13 Books is complete, and all of the figures are illustrated using the Geometry Applet, even those in the last three books on solid geometry that are three-dimensional. I still have a lot to write in the guide sections and that will keep me busy for quite a while. This edition of Euclid's Elements uses a Java applet called the Geometry Applet to illustrate the diagrams. If you enable Java on your browser, then you'll be able to dynamically change the diagrams. In order to see how, please read
16. Abstract Algebra - Wikipedia, The Free Encyclopedia abstract algebra is the field of mathematics concerned with the study The term "abstract algebra" is used to distinguish the field from "elementary algebra" or "high school algebra
Abstract algebra
From Wikipedia, the free encyclopedia.
Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groupsrings and fields . The term "abstract algebra" is used to distinguish the field from " elementary algebra " or "high school algebra" which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers Historically, algebraic structures usually arose first in some other field of mathematics, were specified axiomatically, and were then studied in their own right in abstract algebra. Because of this, abstract algebra has numerous fruitful connections to all other branches of mathematics. Examples of algebraic structures with a single binary operation are:
17. Abstract Algebra abstract algebra. Most people believe that the abstract algebra deals with many structures other than groups. What happens if we two
Abstract Algebra
Most people believe that the mathematics they learn in high school and in their early college years represents the essence of abstraction. What could be less tangible that the idealistic abstract notion of number? Mathematicians work with and delight in bizarre numbers such as i (the square root of -1) which to the uninitiated must seem like an irrational mix of mysticism and numerology. Nonetheless, the basic operations in arithmetic (addition, multiplication, etc.) apply remarkably well to many disparate real-life situations whether you are balancing your checkbook, keeping score in a game of cards, calculating the position of an object under the influence of gravity in physics, or measuring ingrediants for a recipe. In all these situations, the usual grade-school manner in which you add and multiply numbers models the situation perfectly. However, there are times when the normal rules of arithmetic that you learn in grade school do not seem to apply. For example, suppose that you are compiling baseball statistics. To determine a player's batting average, you need to know both the number of hits that he has and the number of times he has been at-bat. If a player has 7 hits in 23 attempts, a natural way to record this information is using the fraction 7/23. (The way this is usually displayed when you are watching a game is the corresponding decimal, truncated to 3 places. In the case of 7 hits in 23 at-bats, it would be written as .304. Although the number .304 is more visually appealing and and conveys information more quickly (the batter gets a hit about 30.4% of the time), it is less precise because we are unable to determine whether the play has 7 hits in 23 at-bats, 14 hits in 46 at-bats, or 70 hits in 230 at-bats.)
18. Zeeman, E. Christopher A major contributor to topology and dynamical systesm. Includes notes, lectures and bibliography of a variety of topics, such as chaos theory, topology, abstract algebra and math history.
Syllabus
92.421 Abstract Algebra The CyberEd Version
Instructor:
Kenneth M. Levasseur Department of Mathematical Sciences University of Massachusetts Lowell Kenneth_Levasseur@uml.edu This course is an introductory undergraduate course in abstract algebra, concentrating on group theory. It will be conducted on-line with students completing labs and problem sets from home and communicating via email, a discussion list and chat.
Academic Prerequisites
You should have completed at least two semesters of calculus and one semester of linear algebra or discrete mathematics. Having taken both linear algebra and discrete math is even better
What you need to do and have to take this course.
Mathematica 3.0+ - any platform that supports Notebooks (PC, Mac, Next, UNIX). Student versions are available in the bookstore for about $130 at the UML Bookstore. You must prove that you are a student to purchase this version. A computer that will run your version of Mathematica ( most likelya Macintosh or IBM) Note: No matter what platform you run Mathematica on, there no problem sharing
20. BEACHY: ABSTRACT ALGEBRA II A companion volume to abstract algebra by John A. Beachy and Bill Blair published by Waveland Press in 1995. Chapters in PostScript.
Abstract Algebra II
by John A. Beachy
These notes served as a companion volume to the book Abstract Algebra Second Edition (written jointly with Bill Blair, and published by Waveland Press in 1995). The notes became the preliminary version of the book Introductory Lectures on Rings and Modules published by Cambridge University Press, and so it has become necessary to discontinue publication of the notes on the Web. Click here for further information about the book. If you need additional information, please contact John Beachy, Dept. of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, Tel. 815 / 753-6753, email: beachy@math.niu.edu This site was opened in 12/1995, and last modified on 5/21/2002. This page has been accessed 116,289 times since 8/96. Author's homepageHomepage for Abstract Algebra Second edition | 677.169 | 1 |
Mathematical Elements for Computer Graphics
This text is ideal for junior-, senior-, and graduate-level courses in computer graphics and computer-aided design taught in departments of mechanical and aeronautical engineering and computer science. It presents in a unified manner an introduction to the mathematical theory underlying computer graphic applications. It covers topics of keen interest to students in engineering and computer science: transformations, projections, 2-D and 3-D curve definition schemes, and surface definitions. It also includes techniques, such as B-splines, which are incorporated as part of the software in advanced engineering workstations. A basic knowledge of vector and matrix algebra and calculus is required7053530235305173172
Book Description McGraw-Hill Science/Engineering/Math. Book Condition: New. 00705353023530599670535305 | 677.169 | 1 |
Catalog Number:
10804107
Credits:
3.00
Description:
Designed to review and develop fundamental concepts of mathematics pertinent to the areas of: 1) arithmetic and algebra; 2) geometry and trigonometry; and 3) probability and statistics. Special emphasis is placed on problem solving, critical thinking and logical reasoning, making connections, and using calculators. Topics include performing arithmetic operations and simplifying algebraic expressions, solving linear equations and inequalities in one variable, solving proportions and incorporating percent applications, manipulating formulas, solving and graphing systems of linear equations and inequalities in two variables, finding areas and volumes of geometric figures, applying similar and congruent triangles, converting measurements within and between US and metric systems, applying Pythagorean Theorem, solving right and oblique triangles, calculating probabilities, organizing data and interpreting charts, calculating central and spread measures, and summarizing and analyzing data. Prerequisite: Minimum Accuplacer scores Arithmetic 65, Algebra 30; or Pre-Algebra 10834109 with a grade of "C" or better
Requirements:
Prerequisites: Accuplacer Math score of 65 and Accuplacer Algebra score of 30 or higher or Pre-Algebra 10834109 with a grade of "C"507
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Stevens Point
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1:25 PM - 3:25 PM
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Online
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Wisconsin Rapids
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4:40 PM - 6:30 PM
Marshfield
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05/15/18
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3:35 PM - 5:00 PM
Stevens Point
3.00
21620
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05/16/18
MTWRFSU
12:00 AM - 12:00 AM
Online
Virtual Campus
3.00
21617
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8:00 AM - 8:55 AM
Telepresence
Wisconsin Rapids
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2:30 PM - 3:25 PM
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Wisconsin Rapids
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Wisconsin Rapids
3.00
No classes offered during this term.
or established connection failed because connected host has failed to respond. | 677.169 | 1 |
Alg2.M.F.IF.C.07d: I can graph rational functions and showing the zeros, asymptotes, and end behavior. - 15 Days
Benchmark 3:
Alg2.M.F.LE.A.04: I can solve exponential functions by estimating graphically and calculating algebraically. - 30 Days
Alg2.M.F.BF.A.02: I can write recursive and explicit formulas for arithmetic and geometric sequences. - 30 Days
Alg2.M.F.IF.C.07e: I can graph exponential and logarithmic functions showing intercepts, maximums and minimums, and end behavior. - 25 Days
Alg2.M.F.LE.A.02 (exponential): I can create linear and exponential functions if provided either a graph, relationship description or input-output tables. - 30 Days
Alg2.M.A.REI.D.11: I can use technology to find and explain why x-coordinates are solutions to systems of equations. I can use technology to find a function that intersects another at a given point. - 10 Days
Benchmark 4:
Alg2.M.S.CP.A.05: I can use a multiplication rule to determine the probability of an event. - 10 Days
Alg2.M.S.CP.A.01: I can use the addition rule to determine the probability of an event. - 10 Days
Alg2.M.F.IF.B.06: I can calculate and interpret the rate of change from functions and from real world data. - 15 Days
Alg2.M.S.ID.B.06a: I can draw or use a line of best fit to make predictions. - 15 Days
Alg2.M.S.ID.A.04: I can use the mean and standard deviation to fit to a normal distribution. - 15 Days
Alg2.M.F.TF.A.02: I understand how coordinates for the angles on the unit circle relate to sine and cosine ratios of a right triangle. - 10 Days
Alg2.M.F.LE.B.05: Interpret the parameters in a linear or exponential function in terms of a context. (Month 1, 2, 7, & 8)
Alg2.M.CED.A.01: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. (Month 1, 2, 3, 4, 5, 7, & 8)
Alg2.M.F.BF.A.02: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (Month 1, 2, 4, & 5)
Alg2.M.F.BF.B.03: Identify the effect on the graph of replacing f (x ) by f (x ) + k , k f (x ), f (kx ), and f (x + k ) (Month 1, 2, 3, 4, 5, 6, 7, 8, & 11)
Alg2.M.F.IF.B.06:Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. (Month 2, 4, & 5)
Alg2.M.F.IF.C.08:Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (Month 2, 3, & 7)
Alg2.M.F.IF.C.09: Compare, 3, 8, & 11)
Alg2.M.F.BF.A.01a:Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (Month 2 & 3)
Alg2.M.N.CN.A.01:Know there is a 2 complex number i such that i = −1, and every complex number has the form a + bi with a and b real. (Month 2 & 3)
Alg2 3, 4, & 5)
Alg2. (Month 3 & 4)
Alg2.M.F.BF.A.01b:Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (Month 3)
Alg2.M.APR.B.02:Know and apply the Remainder Theorem: For a polynomial p (x ) and a number a , the remainder on division by x – a is p (a ), so p (a ) = 0 if and only if (x – a ) is a factor of p (x ). (Month 3 & 4)
Alg2.M.A.SSE.A.02:Use 3, 4, & 5)
Alg2.M.A.APR.D.06:Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. (Month 4 & 5)
Alg2 4 & 5)
Alg2.M.F.BF.B.04:Find inverse functions. a. Solve an equation of the form f (x ) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x‐1) for x ≠ 1. (Month 5, 6, 7, & 8)
Alg2.M.N.RN.A.01:Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. (Month 5 & 6)
Alg2.M.A.SSE.B.03:Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. (Month 5)
Alg2.M.F.IF.C.09b 7)
Alg2.M.A.SSE.B.04:Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. (Month 7 & 8)
Alg2.M.A.REI.C.06:Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (Month 8)
Alg2.M.A.REI.C.07:Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. (Month 8)
Alg2.M.F.IF.C.08b:Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function 8)
Alg2.M.S.CP.A.02: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. (Month 9)
Alg2.M.S.CP.A.03: Understand the conditional probability of A given B as P (A and B )/P (B ), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A , and the conditional probability of B given A is the same as the probability of B. (Month 9)
Alg2.M.S.CP.A.04: Construct and interpret two‐ way frequency tables of data when two categories are associated with each object being classified. Use the two‐ way (Month 9)
Alg2.M.S.CP.B.06: Find the conditional probability of A given B as the fraction of B 's outcomes that also belong to A , and interpret the answer in terms of the model. (Month 9)
Alg2.M.S.IC.A.01:Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population. (Month 10)
Alg2.M.S.IC.A.02:Decide if a specified model is consistent with results from a given data‐ generating process, e.g., using simulation. For example, a model says a spinning coin will fall heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? (Month 10)
Alg2.M.S.IC.B.03:Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. (Month 10)
Alg2.M.S.IC.B.04:Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. (Month 10)
Alg2.M.S.IC.B.06:Evaluate reports based on data. (Month 10)
Alg2.M.F.TF.A.01:Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. (Month 10)
Alg2.M.G.SRT.C.08:Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. (Month 10) | 677.169 | 1 |
Three components contribute to a theme sustained throughout the Coburn Series: that of laying a firm foundation, building a solid framework, and providing strong connections. Not only does Coburn present a sound problem-solving process to teach students to recognize a problem, organize a procedure, and formulate a solution, the text encourages students to see beyond procedures in an effort to gain a greater understanding of the big ideas behind mathematical concepts. Written in a readable, yet mathematically mature manner appropriate for college algebra level students, Coburn's Algebra & Trigonometry uses narrative, extensive examples, and a range of exercises to connect seemingly disparate mathematical topics into a cohesive whole. Coburn's hallmark applications are born out of the author's extensive experiences in and outside the classroom, and appeal to the vast diversity of students and teaching methods in this course area. Benefiting from the feedback of hundreds of instructors and students across the country, Algebra & Trigonometry second edition, continues to emphasize connections in order to improve the level of student engagement in mathematics and increase their chances of success in college algebra.
"synopsis" may belong to another edition of this title.
About the Author:
Book Description Book Condition: Brand New. Brand New, 2 1956N | 677.169 | 1 |
Algebra: Adding and Subtracting Polynomials
NOTEBOOK (SMARTboard) File
Be sure that you have an application to open this file type before downloading and/or purchasing.
0.26 MB | 19 pages
PRODUCT DESCRIPTION
The smartboard file can be used to help students learn to add and subtract the like terms in a polynomials. Shapes are placed around terms that a like to help students identify the like terms.
The file has 12 pages of instruction and examples and then 6 pages where you can enter your own polynomials and then use the shapes to identify the like terms.
Both vertical and horizontal formats are used for addition and subtraction of polynomials.
The Do Now has two pages: page 1 has many terms can be relocated to pair up like terms and page 2 has many terms that can be placed in a sum or difference to combine | 677.169 | 1 |
Mathematics
Class 12th
Class 12 th mathematic will cover remaining topics of algebra, calculus such as matrices,
determinants, continuity, differentiability, integrals and differential equations. Further you will also
learn vectors and 3-D geometry here. In case you are in class 12 th and have just joined, it is advisable
to go through the lectures of 11 th class a bit, to clearly understand the concepts and have a fair idea
about the background, especially in calculus and relations & functions topic. Homework and quizzes
are already there to pave your way.
Class 11th
Class 11 th maths is all about Sets and functions, Algebra, Co-ordinate geometry, calculus,
probability. Sets and functions includes set theory and functions like, trigonometric functions and
other general mathematical functions. Co-ordinate geometry is about straight lines, circles, conic
sections, thee-dimensional geometry etc. Limits, continuity are part of calculus while in algebra we
will get to know about sequence and series, binomial etc. So get started and practice quizzes to learn
faster. | 677.169 | 1 |
First Year Calculus as Taught by R. L. Moore: An Inquiry-Based Learning Approach
Item# 978-1-4349-0761-5
$40.00
Availability: Usually ships in 3-4 business days.
+ Available in PDF format for reading on your computer - see bottom of page.
___________________________________________________________
First Year Calculus as Taught by R. L. Moore: An Inquiry-Based Learning Approach by Clement E. Falbo
First-Year Calculus as Taught by R.L. Moore is intended to facilitate student learning through IBL, or Inquiry-Based Learning, the essence of the Moore Method. Based on notes taken by the author while studying with R.L. Moore in 1955, this text incorporates the pedagogical teaching of mathematics implemented by Dr. Moore using his own collection of seminal "problems that teach." Designed to stimulate creativity and encourage student presentations of their solutions in the classroom, the intention is to minimize lectures by the instructor and to maximize student participation in the learning process.
About the Author
Clement E. Falbo is Mathematics Professor Emeritus at Sonoma State University and is the author of two other mathematics texts, Finite Mathematics Applied and Math Odyssey 2000. Dr. Falbo is married to Jean Ann Falbo and is the father of three sons.
(2010, paperback, 444 pages)
Accessories
First Year Calculus as Taught by R. L. Moore | 677.169 | 1 |
PrefacePreface
This book is intended to provide a series of illustrations of how the package Sage can be used to work on problems in algebraic graph theory. Algebraic graph theory is a beautiful subject and Sage is an ideal place to experiment with the relevant mathematics: graph theory, linear algebra and permutation groups, along with combinatorics generally. We hope these notes will provide a useful introduction for student and for researchers, and that they will simultaneously assist in the continual improvement of Sage.
The topics we have chosen are meant to be interesting—they are for us—and to provide cases where computations in Sage will be useful. We do not offer a course in programming: we assume a minimal level of familiarity with Sage, but we have tried to illustrate a wide range of different computational techniques. As for mathematical background, the reader will need a basic understanding of graphs, linear algebra and the like. We view each topic (or exploration) as a starting point and we hope that some of our readers are encouraged to go elsewhere to learn more.
Since neither publishers nor trees are involved in the production of this text, it will be easy for us to add to add new chapters from time to time, and we hope to do so. We will be happy to accept contributions from other authors. These should agree in spirit with what is already here,
and adhere to our stylistic standards. We recommend strongly that anyone interested contact us before investing any significant effort. (Our view of what constitutes "algebraic graph theory" is quite broad.) | 677.169 | 1 |
Terry Fox's Mathematics Department offers a variety of courses designed to meet the needs of all students. It is important that students select courses that suit their interests, as well as their career and post-secondary aspirations. The Mathematics Department provides a Mathematical curriculum that helps students:
Develop an appreciation of the role of Mathematics in everyday life
Develop strong Mathematical skills
Develop self-confidence and enjoyment in the study of Mathematics
Mathematics 9
This course is designed to consolidate and extend topics introduced in the middle school years. Topics include operations with rational numbers, square roots and exponents, polynomials and algebra, linear relations, geometry and statistics. At the end of this course, students will be prepared for Foundations of Mathematics and Pre-Calculus 10 or Apprenticeship and Workplace Mathematics 10.
Essentials of Math 9
This course is for students who have not met the passing requirements for Principles of Mathematics 8. The course emphasizes basic mathematical life skills.
Apprenticeship & Workplace Math 10
This option is designed to provide students with the Mathematical understanding and critical-thinking skills identified for entry into the majority of trades and for direct entry into the work force. Topics include understanding and applying the metric and imperial systems to the measurement of 2-D and 3-D objects, geometry and trigonometry, and the fundamentals of income, spending and debt. All topics are taught within a framework of problem-solving. This course has a mandatory graduation program exam. At the end of this course, students are prepared for Apprenticeship and Workplace 11. Students may also choose one of the other Mathematics 10 or 11 course options pending departmental consultation.
Foundations of Mathematics & Pre-Calculus 10
This option is designed to provide students with the Mathematical understanding and critical-thinking skills identified for post-secondary academic studies at colleges and universities. Topics include surface area and volume of 3-D objects, applying trigonometric ratios to right triangles, irrational numbers, powers involving integral and rational exponents, polynomials, and coordinate geometry with linear relations, systems of linear equations, and function notation. At the end of this course, students are prepared for either Foundations of Mathematics 11 or Pre-Calculus 11.
Topics in this course include: transformations, trigonometric functions, equations and applications, trigonometric identities, exponential and logarithmic functions and their application, permutations and combinations and probability. Independent problem-solving is stressed throughout the course. This course has an optional graduation program worth 40% of the final mark.
Calculus 12
Prerequisite: Principles of Mathematics 11. This course is very valuable for those students who have excellent mathematical ability and intend to study mathematics and economics, engineering or science. The main topics covered are: properties of functions, analytic geometry, limits, derivatives, integrals and their applications. No government provincial examinations are offered in this course.
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Matrices
as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.
Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations.
Trigonometry
Trigonometric identities & functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles, general solution of trigonometric equations.
Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle, inverse trigonometric functions (principal value only).
Analytical Geometry
Two dimensions: Cartesian coordinates, distance between two points, section formulae, shift of origin.
Equation of a straight line in various forms, angle between two lines, distance of a point from a line. Lines through the point of intersection of two given lines, equation of the bisectorof the angle between two lines, concurrency of lines, centroid, orthocentre, incentre and circumcentre of a triangle
.Equation of a circle in various forms, equations of tangent, normal and chord.
Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal. Locus Problems.
Three dimensions: Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.
Integral Calculus
Integration as the inverse process of differentiation, indefinite integrals of standard functions, definite integrals and their properties, application of the Fundamental Theorem of Integral Calculus.
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Besides offering CBSE, NEET and JEE online coaching via its video lectures, Etoosindia.com also offers a host of mock tests to allow the students to test their knowledge.
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Teaching Myself Calculus
Lately I've been reteaching myself calculus. I took AP Calculus AB in high school and I got college credit for it, I'm planning on taking calc 2 this summer and I need to review everything I've learned and learn it on a college level. I noticed that what you learn in a college calc 1 class is different from high school. In high school we could use calculators for graphs and stuff like that, but in college we aren't calculators at all. So I'm teaching myself calc 1 and 2 from a UW-Madison professor's notes. They're really easy to read and go through. I could learn calculus 1 and 2 in a month with these, but since I have a lot of homework from my other classes I can't just blast through them. I'll post links to the notes in my links section. They're completely free and nice to have even if you're currently taking calculus. | 677.169 | 1 |
Science One: Integrating Mathematical Biology into a First-year Science Program. Mark MacLean Department of Mathematics The University of British Columbia (Vancouver). Outline of Talk. Description of Science One Role of Mathematical Biology in the Program
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Science One: Integrating Mathematical Biology into a First-year Science to modify a given model
Creating your own models
Some thoughts
The best examples connect to real biological problems -- students know when you have contrived something just to teach them a piece of mathematics
Embrace your ignorance -- make sure students see how you use your mathematical understanding to build your biological understanding (or vice versa)
Recognize that learning to become a mathematical modeler takes time -- design a progression of experiences that help students build skills over time and be explicit in showing how you are using prior experience to tackle learning new tools or to building models
Believe in your students -- even if a student does not seem destined to be a mathematician, they gain a lot by learning how to communicate with mathematics.
Be in control -- understand your expected learning outcomes for each modeling exercise.
What impact does this have on students?
Some intangibles:
Science One students in the life sciences take more mathematics courses than their peers.
Science One students are not afraid to use mathematics in their biology classes, even when it is not expected.
Science One students question the validity of models (mathematical and otherwise) more than their peers.
How do you satisfy the Dean?
How to compare our students to other students?
Problems:
We have not undertaken a standard controlled experiment (for ethics reasons, amongst others).
Our students go through a selection -- they chose to apply and we choose them from the applicant pool.
Our approach: (joint with Neil Dryden, UBC-V Chemistry)
Study performance in courses requiring higher-level problem-solving skills in each discipline. The comparisons are to other selected groups; in the life sciences these are microbiology and immunology, physiology, pharmacology.
Compare our students' performance in these courses to their ownoverall sessional average.
Admission Averages
There appears to be little correlation between high school admission averages and first-year averages at UBC.
Organic Chemistry
The selected group comprises students in Microbiology and Immunology, Physiology, and Pharmacology | 677.169 | 1 |
Course Summary
This High School Algebra II: Help and Review course can help you study all the topics covered in your advanced algebra course. Use the short video lessons to review the essential formulas and definitions you need to know, and then test your understanding using our quizzes and chapter tests. | 677.169 | 1 |
Linear algebra is one of the most important branches of mathematics - important because of its many applications to other areas of mathematics, and important because it contains a wealth of ideas and results which are basic to pure mathematics. This book gives an introduction to linear algebra, and develops and proves its fundamental properties and theorems taking a pure mathematical approach.
A large number of examples, exercises and problems are provided. Answers and/or sketch solutions to all of the problems are given in an appendix. The intended readership is undergraduate mathematicians, also anyone who requires a more than basic understanding of the subject. This book will be most useful for a "second course" in linear algebra, that is for students that have seen some elementary matrix algebra. But as all terms are defined from scratch, the book can be used for a "first course" for more advanced students Harvey E. Rose
"Rose's Linear Algebra is a highly sophisticated undergraduate work...This book would be excellent for mathematics majors or for non-majors with access to a second course in which applications were presented. Summing Up: Recommended for lower- and upper-division undergraduates."
Reviews - What do customers think about Linear Algebra: A Pure Mathemat | 677.169 | 1 |
What Should You Know?
What Should You Know? is a math program provided by Unitas Tutoring which sets the curriculum as to what children entering High School should know in math, Algebra 1, K-8th Grade Math, Algebra 2, Calculus, Precalculus, and Geometry. Think of this as a place where you can obtain practice problems seeing if you and or your child was placed in the correct math class. If these problems are of difficulty, then tutoring can be requested. | 677.169 | 1 |
Unit 3: Application of Differentiation (7 lessons, NO EXAMS)
extrema on an open interval
the first derivative test (increasing, decreasing and extrema)
given the graph of a function, sketching the derivative
application of the first and second derivative tests and check for asymptotes and then sketch what the graph looks like
second derivative test for concavity and points of inflection. Algebraically and given a graph
Unit 4: Introduction to Integration (Anti-derivatives) (5 lessons, 2 tests--with solution key)
This lesson covers antiderivatives and indefinite integration.
This lesson covers left endpoint approximations, right endpoint approximations and midpoint approximations of area under a curve.
This lesson covers Riemann Sums and Definite Integrals using geometric areas under curves.
This lesson covers the Fundamental Theorem of Calculus, integrals of absolute value functions, and using the integral to evaluate the average value of a function on a given interval.
This lesson covers Integration by Substitution (u-substitution for integration)
Solutions are included for the exams. | 677.169 | 1 |
This course is designed to help students learn the basic algebra necessary for college level mathematics. Topics include real number concepts, selected geometry concepts, linear equations and inequalities in one variable, problem solving involving linear or factorable quadratic equations as models, operations on polynomials, factoring polynomials, integral exponents, and graphing linear equations in two variables. Additional topics include the study of rational expressions and the use of the scientific calculator. | 677.169 | 1 |
Computer-aided modeling techniques have been developed since the advent of NC milling machines in the late 40's. Since the early 60's Bezier and B spline representations evolved as the major tool to handle curves and surfaces. These representations are geometrically intuitive and meaningful and they lead to constructive numerically robust algorithms. It is the purpose of this book to provide a solid and unified derivation of the various properties of Bezier and B-spline representations and to show the beauty of the underlying rich mathematical structure. The book focuses on the core concepts of Computer-aided Geometric Design (CAGD) with the intent to provide a clear and illustrative presentation of the basic principles as well as a treatment of advanced material, including multivariate splines, some subdivision techniques and constructions of arbitrarily smooth free-form surfaces. In order to keep the book focused, many further CAGD methods are ex cluded. In particular, rational Bezier and B-spline techniques are not ad dressed since a rigorous treatment within the appropriate context of projec tive geometry would have been beyond the scope of this book.
This excellent book gives a well-focused introduction to the essential concepts of computer aided geometric design. The authors provide a solid and unified derivation of the various properties of Bézier and B-spline representations. These piecewise polynomial representations have become the major tool to handle curves and surfaces in various applications, such as computer-aided design and computer graphics. In addition, the authors discuss a number of advanced concepts, including multivariate splines, certain subdivision techniques, and free-form surface splines. [...]
The material presented in this book is visualized by many carefully prepared figures, which will be of great help for understanding the theoretical concepts. In addition, the authors have provided a number of useful exercises (without solutions). The book can be used as a textbook for courses at the graduate and intermediate undergraduate levels, for students of mathematics, computer science and engineering, and for self study. It will also be a useful reference book for researchers and practitioners in the field of computer aided geometric design and its various applications.
Bert Jüttler, AMS 2004
"The book provides a clear and illustrative presentation of the basic principles in computer-aided geometric design. It is addressed mainly to the students and teachers in mathematics, computer science and engineering sciences. At the end of each chapter several problems are proposed." (Dana Petcu, Zentralblatt MATH, Vol. 1033 (8), 2004)
"The book provides a basic presentation of the most important techniques in Computer-aided Geometric Design. The illustrative style gives a clear understanding of the subject and makes it well suited for students or lecturers in mathematics and computer sciences. … Each chapter comes with a small supplement of exercises." (H. G. Feichtinger, Monatshefte für Mathematik, Vol. 142 (3), 2004)
"This excellent book gives a well-focused introduction to the essential concepts of computer aided geometric design. … The material presented in this book is visualized by many carefully prepared figures, which will be of great help for understanding the theoretical concepts. In addition, the authors have provided a number of useful exercises. The book can be used as a textbook … . It will also be a useful reference book for researchers and practitioners in the field of computer aided geometric design … ." (Bert Jüttler, Mathematical Reviews, 2004 e)
"Computer Aided Geometric Design (CAGD) deals with the algorithmic aspects of curves and surfaces. … the book definitely serves as a solid introduction to recent literature on CAGD. It is based on courses taught by the authors, and the book shows that the material has been thoroughly tested and revised in classrooms. … The book is highly recommended, both as a textbook for courses on geometric computing and as a self-contained introduction to the field of Computer Aided Geometric Design." (G. Vegter, Nieuw Archief voor Wiskunde, Vol. 7 (2), 2006) | 677.169 | 1 |
This comprehensive text suitable for math, science, and engineering majors, treats standard elementary topics such as undetermined coefficients, systems of differential equations, substitutions and chemical reactions. Students benefit from the conceptual approach, motivated by detailed physical and mathematical examples. Previous exposure to linear algebra or vectors is not assumed: all necessary linear algebra techniques are introduces as needed with respect to systems of differential equations. Early and consistent notation emphasizes the characteristic polynomial, suggests shortcuts such as the exponential shift, and leads to a natural, unified understanding of undetermined coefficients. Expanded treatment of the qualitative theory of linear systems, especially as applied to nonlinear systems, includes new sections on Interacting Populations, Constants of Motion, Lyapunov Functions, and Limit Cycles and Chaos. BASIC computer programs on approximation algorithms illustrate the use of technology in numerical methods.
Book Description Harcourt Brace College Publishers. Hardcover. Book Condition: New. 003072878930728789 | 677.169 | 1 |
Calculus Without Tears
Free - Calculus Without Tears Webinar
Every Wednesday at 1:00pm EST
at
MATLAB, FREEMAT and OCTAVE
MATLAB is the programming language of choice for many analytical engineering projects, and all the graphs in CWT were created using MATLAB. It is also a super calculator, and all the calculations for the exercises were also done with my most convenient calculator, MATLAB. So, why have I waited till now to tell you about this fantastic tool? Because MATLAB is expensive. However, now there are free clones of MATLAB available, my favorites are FREEMAT and OCTAVE. FREEMAT and OCTAVE are totally legal and free, and can be downloaded from the web. FREEMAT and OCTAVE have all the features of MATLAB.
MATLAB as a Calculator
So I recommend that you download either FREEMAT or OCTAVE right now, and at a mimimum, learn to use it as a calculator. How difficult is that? When you run the program it comes up in a window and prints a prompt '>'. After the prompt, type in an expression followed by (Enter) and FREEMAT/OCTAVE will evaluate the expression and print the answer. E.g.
>10+15+81/9
34
MATLAB is much more convenient than a handheld calculator, because, if you type in a long expression and evaluate it, and then discover you've made a mistake, or you want to change it for any reason, you can enter an 'up arrow' and the previous line you entered will reappear after the prompt, and you can edit the line and reenter it. That is, you can edit only the parts you want, you don't have to retype the entire line. This makes a big difference, believe me.
MATLAB Variables and Vectors
MATLAB also has memory. A named memory location is called a 'variable'. A value is assigned to a variable ( 'a' in the line below) with an assignment command of the form
>a = 10 + 15; A variable can also store a list of values (called a vector). The command
>b=[2.1 8 5.3 100 -10]; stores a five element list in b, the first element is 2.1, the second 8, and so on. Individual elements in a list are referenced by following the variable name with an index into the list in parentheses, thus b(1) references the first element, b(2) the second, and so on.
Typing the name of the variable after a prompt causes its value to be printed, thus
>a 25
>b [2.1 8 5.3 100 -10]
>b(3) 5.3
b(4) = 25; b [2.1 8 5.3 25 -10]
MATLAB is a Programming Language
Using MATLAB as a calculator is just the beginning. It is also a programming language that is very easy to use. And I mean VERY easy. There is now (2006 onward) an appendix in each CWT volume giving a very brief introduction to MATLAB programming. One of the beauties of the language is that it was designed for scientists and engineers, and not computer programmers, so it is very 'user' oriented. Also, it has features that make it very easy to produce graphs. And, if the truth be known, drawing graphs is one of the big thrills in physics and engineering. Graphing is 'built-in' in MATLAB and only requires one line, of the form 'plot(x,y)' where x is a vector of the the horizontal coordinates of the points to be graphed and y is a vector of the vertical coordinates of the points to be plotted.
A 'for loop' causes the list of commands between the for and end commands to be executed repeatedly. The for command specifies an index variable and a list of values for the variable, the commands in the loop are executed once for each value in the index list. Example:
j=0; for i= [1 2 3 4 5] a(i)=j; j=j+i; end a [0 1 3 6 10]
Note that 'execution' of a for loop doesn't start until the end command is entered. So, when the 'end' command is entered above, the computer executes the following commands:
i=1; a(i)=j; j=j+1; i=2; a(i)=j; j=j+i; i=3; a(i)=j; j=j+i; i=4; a(i)=j; j=j+i; i=5; a(i)=j; j=j+i;
The MATLAB commands that generated the graph are shown in green below, following each command is a comment preceded by %. i and t are variables, x and y are vectors each with 1000 elements. | 677.169 | 1 |
Input and Output Powerpoint
Be sure that you have an application to open
this file type before downloading and/or purchasing.
1 MB|18 pages
Product Description
This is a detailed and descriptive presentation related to an introduction to algebra. Students learn about input and output machines. At the end of the presentation, students will be able to create rules and solve problems that include basic patterns and equations. The powerpoint outlines strategies for students to practice creating rules that relate to the relationship of the pattern.
This is a fun and interactive presentation. Students will have a solid understanding of concepts related to basic algebra after they practice strategies outlined in this presentation. | 677.169 | 1 |
Guidelines for Using the Math Lab
Located in MAS 204, the Math Lab is a great place to study, do homework, obtain assistance, and meet with a study group.
Bring the worksheet or textbook that contains the problems you need help with.
For the best help, be prepared to show an Academic Tutor the attempts you have made to solve these problems, and identify where you are having difficulties.
There are learning aids available in the Math Lab that may be useful if you need more help on a topic. If you miss a class, please contact your instructor, who will give you the best strategy for making up the work.
Learning Aids
You must have a valid Brookdale ID – the Brookdale OneCard–to access the materials available for you to use in the Math Lab.
The Math Lab has solution manuals (both student and instructor versions) for most of the texts used by the department. These can be used in the Math Lab as study aids so you can check the answer to any text problem. The Bankier Library also has these on reserve.
The computer room in the Math Lab is equipped with all our mathematics software. These computers are reserved for currently enrolled math students doing assignments for their math course. The computer room has the same hours as the Math Lab.
If we can be of any other assistance or you require special accommodations, please contact the STEM Institute at 732-224-2430.
Study Groups and Workshops
The Math Department encourages students to form study groups. If you are interested in forming a study group, ask your instructor to help you get one started. Your study groups are welcome to meet in the Math Lab.
Workshops are problem-solving sessions that may focus on a specific topic or review material for a test. For example, around the time courses have their tests, review sessions covering the material on these tests are offered through the Math Lab. Workshop announcements will be posted on the Math Lab bulletin board located in the hallway outside of the Math Lab and in the Announcements portion of this webpage. | 677.169 | 1 |
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Unformatted text preview: MATH/CS 240 (Intro. Discrete Math.) SYLLABUS, Fall 2009 Lecture: TR 11PM – 12:15 PM, Grainger 2080 Prof. Jordan Ellenberg Text is: Office: 323 Van Vleck Hall Discrete Math. & its Applications 6th ed., by K. Rosen Email: [email protected] Office Hours (JSE): Tues, 1:30-2:30 PM, VV 323 TAs: Hao Lin, l [email protected], Baris Aydinlioglu b [email protected] Course Content Mathematics can be loosely divided into two parts. The first is continuous mathematics; as the name suggests, this part of math treats phenomena that can be moved continuously, like functions, curves, and geometric spaces. Most of the math you've learned so far – geometry, trigonometry, calculus – is continuous in nature. The basic object of continuous mathematics is the real number line. Because the physical universe (at least to the naked eye) is continuous, this is the part of mathematics most associated with physics. The second part is discrete mathematics, the subject of our course. Here we throw aside any notion of continuous variation. The basic objects of discrete mathematics are the set of integers and of logical values; there is no way to move continuously from 2 to 3, or from "true" to "false." Because the states of a computer are discrete, this is the part of mathematics most associated with computer science. Math/CS 240 covers the fundamentals of discrete mathematics. It is a requirement for the BS degree programs in Computer Engineering offered by the ECE Department and in Computer Science offered by the CS Department. It is now a prerequisite for (getting into) advanced computer science courses (CS 367, 520 and 577). The course is a foundational math course for this program and is meant to be taken early in the program; it is also a good foundation for higher mathematics courses. We will aim for breadth, not depth; yougood foundation for higher mathematics courses....
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This note was uploaded on 02/19/2010 for the course CL 201 taught by Professor Fairguy during the Fall '09 term at University of Wisconsin. | 677.169 | 1 |
7th Grade Pre-AP Algebra is a rigorous, fast-paced, first semester Algebra
I course designed to prepare mathematically talented 7th graders for the
rigors of the Algebra I Pre-AP course that they will be taking in 8th grade.
Functions and fundamentals are the focus of this course.
The curriculum is planned and delivered to challenge the most capable math
students. In addition to the curriculum outlined in the textbook
students will be exposed to some math topics that will not be studied in
depth until future math and science courses. Some time will be spent
preparing for the SAT.
Appropriate use of technology (TI-84+) graphing calculator) will be introduced
during the year, but it will be limited, and concentrated in the 2nd semester.
Every effort will be made to help students master the skills that will
ensure that they are prepared to make a smooth transition to the math courses
with the more sophisticated topics that they will encounter in the future.
This class
will be taught using techniques that are being called, "Blended Learning".
This means that some homework requires students to watch a video that is
the lesson for the latest topic. They are expected to learn the material
in the videos in the same way that they are expected to learn a lesson
when it is given live, in class. Blended Learning techniques have
been shown to be at least as effective as in-person lesson presentations
and have been incorporated in this class the past 2 years with great results.
7th-Grade ALGEBRA Pre-AP CURRICULUM
Textbook:
Algebra
I; Addison-Wesley Publishing,
Textbook
Overview:
The first 6 chapters (the part of the book we will complete in 7th grade)
consist of a tightly-sequenced series of topics leading as quickly as possible
to a mathematically honest derivation of the Quadratic Formula. This
departure from the traditional sequence of topics is made possible by using
a calculator to evaluate radicals.
Chapter 5 introduces students to polynomials, including factoring.
This is done so that students have a solid grasp of how the process relates
to completing the square. One of the topics that will be presented
later in the book will be introduced at end of Chapter 5. That topic
is Solving Quadratics by Factoring.
Time will be spent developing the concept of a variable as a symbol that
represents a value that really varies, rather than simply standing for
an unknown constant. To aid in this development, the book presents
mathematical expressions as something to which one does two things: evaluates,
when the value of the variable is known, and finds the value of the variable
when the value of the expression is known. The problem solving in
this book have multiple parts in which students must write an expression
representing a variable quantity, then do these two things with the expression.
This process helps students to understand that a variable can stand for
a different number at different times, depending on the situation.
The properties of numbers are first presented as verbs. These are
the Field Axioms where students learn how to Commute, Associate, etc.
Use of the axioms will require justification in order to give students
an introduction to the idea of proofs (deductive reasoning). The
concept of Closure will also be introduced in a formal way so students
can begin to better understand the properties of the Real Numbers.
The first semester includes lessons on SAT preparation.
There will be some topics introduced throughout the year that do not appear
in the textbook. They will be covered at the appropriate times when
they naturally fit.
4-1, Equations With Like Terms----------------------\
4-2, Equations With Like Terms and Distributing \
4-3, Equations With Variables in Both Members \
Five weeks, with Stanford prep.
4-4, Equations That Involve Decimals
/
4-5, Literal Equations and Formulas------------------/
4-6, Problems That Involve More Than One Expression.
4-7, Chapter Review | 677.169 | 1 |
Group Theory
Publisher's Summary
Group theory is a term that is mainly used in fields related to mathematics, such as algebraic calculations. In abstract algebra, groups are referred as algebraic structures. Other terms of algebraic theories, such as rings, fields, and vector spaces are also seen as groups. Of course with some additional operations and axioms, mathematicians accept them as a group. The methods and procedures of group theory affect many parts and concepts of mathematics as well as algebra on a large scale. Linear algebraic groups and lie groups are two main branches or categories of group theory that have advanced enough to be considered subjects in their own perspectives | 677.169 | 1 |
Geometric Sequences and Series
Page Navigation
Main Profile
At A Glance
Geometric Sequences and Series
Check Out our new Re-designed FREE website! Sequences and Series: Defining a geometric sequence and series, finding recursive and explicit formulas of a geometric sequence, using the geometric mean to find missing terms in a geometric sequence, testing for convergence, and finding the sum of an infinite an finite geometric series. If31:43
Contact
Questions about Geometric Sequences and Series
Want more info about Geometric Sequences and Series?
Get free advice from education experts and Noodle community members. | 677.169 | 1 |
This will take you to the web page of the algebra word problems worksheet. You then have several options. Al There is one of the most important subjects in mathematics, known as algebra.
Algebra studies about constants, variables, expressions and equations. It also deals with abstractions such as - groups, rings etc. Algebra is an essential part for any fields related to mathematics, physics, science, engineering or technology, as well as economics and medicine too.Algebra is a vast subject which has several classifications. These are as follows -1) Elementary Algebra - Basic algebra studied in primary classes.2) Pre Algebra - A course of algebra learnt in middle school, i.e. from 5 to 7 standards.3) Abstract or Modern Algebra - Algebra about abstract mathematics, such as - groups and rings.4) Linear Algebra - It deals with the study of linear equations, matrices, vector spaces etc.5) Algebraic geometry - The branch of geometry that deals with curves and surfaces with polynomial equations.) CoFree Algebra HelpTaking Algebra I or Algebra II.
Then you found the right place to get help. We have more than forty free, text-based algebra lessons listed on the left. If not, try the site search at the top of every page. Learn to solve word problemsThis is a collection of word problem solvers that solve yourproblems and help you understand the solutions. All problems arecustomizable (meaning that you can change all parameters). Basic Word ProblemsA word problem in algebra is the equivalent of a story problem in math.When you solved story problems in your math class you had to decide whatinformation you had and what you needed to find out.
Then you decided whatoperation to use. Addition was used to find a totals and subtraction wasused to find changes in values.The approach to solve problems with algebra is usually quite different.Word problems are solved by separating information from theproblems into two equal groups, one for each side of an equation.Examine this problem.Sara has 15 apples and 12 oranges. | 677.169 | 1 |
Aug 2013
views:412563514 Mar 2016
views:2707015 Jun 2014
views:643published:01 Mar 2014
views:102026
The mathematical concepts that are used when converting units: Units can be treated as algebraic variables, and simple fractions can be used to change from one unit to another.
published:09 May 2011
views:304922 May 2015
views:37522710 Feb 2015
views:24288Taylor series
The concept of a Taylor series was formulated by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
A function can be approximated by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error introduced by the use of such an approximation. The polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that function's Taylor polynomials as the degree increases, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function in that interval.
Algebra
Algebra (from Arabic"al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).These animations will help you understand the math concepts you never got in high schoolThe Most Beautiful Equation in Math
The Most Beautiful Equation in MathDuality of Mathematical ConceptsAlgebra made Easy. Math concepts for kids and you will surely be rewarded.
Thank You.
==============================================================
3:39
Physical Science 1.6c - Mathematical Concepts
Physical Science 1.6c - Mathematical Concepts
Physical Science 1.6c - Mathematical Concepts
The mathematical concepts that are used when converting units: Units can be treated as algebraic variables, and simple fractions can be used to change from one unit to another.
12:07
Algebra Basics: What Is Algebra? - Math Antics
Algebra Basics: What Is Algebra? - Math Antics3:10
Mathematics: Invented or Discovered?
Mathematics: Invented or Discovered? YOU for concepts1 16 16 The Taylor Series and Other Mathematical Concepts 21 Jun 2015 11 Dec 2014 06 Nov 2014 02 Nov 2014 23 Sep 201511 Dec 2014
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back23 Sep 2015
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back you Trust1:24:14
Blues Clues and Umi Zoomi: Full 1 Hour Special Game Walkthrough *
This is a walkthrough of a Team Umizoomi video game. Team Umizoomi is a computer animated ...
Blues Clues and Umi Zoomi: Full 1 Hour Special Game Walkthrough *
This1:21:25 ...59:22
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This is a walkthrough of a Team Umizoomi video game. Team Umizoomi is a computer animated ...36:45
Team Umizoomi Fun and Games Underwater! Cartoon Gameplay for Kids *
This is a walkthrough of a Team Umizoomi video game. Team Umizoomi is a computer animated ...50:15 ...16 The Taylor Series and Other Mathematical Conce...
MATTER OF KHEMETIC...
1 16 16 The Taylor Series and Other Mathematical...
Team Umizoomi Video Game Compilation - Rescue Team...
Blues Clues and Umi Zoomi: Full 1 Hour Special Gam...
Team Umi Zoomi Bike Race Full Episode | Team Umiz...
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Team Umizoomi Race Cars and Numbers! Video Games | 677.169 | 1 |
Discrete Mathematics for Teachers
ISBN-10: 1617350265
ISBN-13: 9781617350269 Discrete Mathematics for Teachers is a text designed to fill this void. The topic is right. Discrete mathematics provides a rich and varied source of problems for exploration and communication, expands knowledge of mathematics in directions related to elementary and middle school curricula, and is easily presented using our best understanding of the ways that mathematics is learned and taught. The presentation is right. In the spirit of NCTM's Principles and Standards for School Mathematics, topics are presented with careful attention to the best traditions of problem solving, reasoning and proof, communication, connections with other disciplines and other areas of mathematics, and varied modes of representation | 677.169 | 1 |
Course Summary
This practice and study guide provides effective resources to prepare for the PLACE Mathematics exam. The lessons review important concepts while the self-assessment quizzes let you put your exam skills to the test. | 677.169 | 1 |
Instead of using a simple lifetime average, Udemy calculates a course's star rating by considering a number of different factors such as the number of ratings, the age of ratings, and the likelihood of fraudulent ratings.
College Math Essentials: Logarithms
The mysteries of logarithms revealed!
4.6
(46 ratings)
Instead of using a simple lifetime average, Udemy calculates a course's star rating by considering a number of different factors such as the number of ratings, the age of ratings, and the likelihood of fraudulent ratings.
Some basic high school level mathematics will be assumed, such as the use of variables, order of operations, the meaning of an exponent and graphing in the x-y plane.
Description
This course is designed for college/university students who want to learn logarithms or just need a refresherBy taking this course, you will be able to calculate any logarithm with ease, use logarithm rules like a wizz and solve exponential and logarithmic equations in your sleep!
Who is the target audience?
This course is designed for college/university students who need a refresher on logarithmsUnderstand and prove the rule for the log of a quotient. Learn how it is put to use in practice.
Log of a Quotient
04:09
Understand and prove the rule for the log of a power. Learn how it is put to use in practice.
Log of a Power
03:48
Put the three log rules to practice and understand how to avoid the common misconceptions of these rules.
Practice Examples
05:19
Test your understanding of the three logarithm rules discussed in this section.
Log Rules
3 questions
+–
Bases Matter
4 Lectures
20:50
Learn what the change of base rule says, how to memorize it and prove why it is true.
Change of Base Rule
05:37
Discover what the number e means and learn the notation for one of the most-used logarithms: the natural logarithm.
The Natural Logarithm
05:24
Be able use your calculator to find the logarithm of any positive number in any base!
Calculate Any Logarithm
04:43
Practice examples using the change of base rule, the natural logarithm, and calculating logarithms using the change of base rule to convert to base 10 or base e.
Practice Examples
05:06
Test your understanding of the change of base rule and the natural logarithm.
Log Bases
2 questions
+–
Let's Graph It!
5 Lectures
40:45
Understand the domain and range of a given function.
Domain and Range of Functions
04:35
Learn how to graph a function which has been transformed by: being vertically or horizontally shifted, vertically or horizontally stretched or shrunk, or reflected in the x-axis or y-axis.
Graphing Transformations of Functions
07:48
Understand what makes a function one-to-one and how to check for this using the horizontal line test. Learn what the inverse function is and how to visualize it graphically.
Inverse Functions and Their Graphs
09:42
Learn the general shape and behaviour of an exponential function so that you can graph transformations of these functions with ease.
Graphs of Exponential Functions
08:23
Learn the general shape and behaviour of a logarithmic function so that you can graph transformations of these functions with ease. Understand the graphical relationship between a logarithmic function and an exponential function.
Graphs of Logarithmic Functions
10:17
Test your understanding of the graphs of exponential and logarithmic functions.
Exponential and Logarithmic Graphs
3 questions
+–
Master Logarithms
5 Lectures
34:49
Learn how to solve an exponential equation by using either the one-to-one property of exponential functions or by following 3 simple steps which make use of logarithms.
How to Solve Exponential Equations
07:56
Learn how to handle more advanced exponential equations.
Solving More Advanced Exponential Equations
05:38
Practice solving exponential equations.
Practice Examples
09:25
Learn how to solve a logarithmic equation by using either the one-to-one property of logarithmic functions or by following 3 simple steps which make use of exponents.
How to Solve Logarithmic Equations
06:08
Practice solving logarithmic equations.
Practice Examples
05:42
Test your understanding of how to solve exponential and logarithmic equations.
Mastery
4 questions
+–
Logs in Real Life
4 Lectures
17:41
Understand three common uses of a logarithmic scale in daily life: the pH scale, the decibel scale and the Richter scale.
Real Life Applications: Log Scales
05:51
Given an equation for one variable in terms logarithmic or exponential equation of a second variable, be able to solve for the second variable in terms of the first.
Exponential and Logarithmic Relationships
04:54
Use logarithms to solve exponential equations related to compound interest.
Application of Logs in Finance
04:22
Review what you have learned in this course and think about where to go from here.
I have a PhD in mathematics from The University of Queensland and I'm currently a Postdoctoral Research Fellow in the field of graph theory and combinatorics. I also have a BSc and a MSc in mathematics from The University of Victoria. As an aside, I have a black belt in Shotokan karate :)
I've tutored and taught mathematics at university. The thing I love the most about teaching math is when students who thought they couldn't do math come to me and say that they can understand it now and they also ENJOY it. It's great to share my enthusiasm for math with others. | 677.169 | 1 |
Category: Abstract
But it's certainly possible that after doing all the work and figuring out that x = 4 you might forget to carry out the final simple operation of 4 – 5 = –1, and instead incorrectly answer 4. Some homework problems will be discussed in class, with the participation of the students and this will be taken into account for the grade. IBL, however, asks the students to use what they know (or assume) to be true in order to create their own ways to solve problems or form logical arguments to validate other ideas.
Unlike many follow-up math books, this one never mindlessly repeats the same material. Weekly homework assignments and quizzes kept the students working until they finished the course. Micali et al, Ed.), Kluwer Academic Publishers, Dordrecht.: It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can berepresented as a spin group. Apply the fundamental theorem of algebra. The next step was taken by Évariste Galois in 1832, although his work remained unpublished until 1846, when he considered for the first time what we now call the closure property of a group of permutations, which he expressed as ... if in such a group one has the substitutions S and T then one has the substitution ST.
The midterm willcount for 30% of the final grade. These ma- trices, of course, are called orthogonal ample Algebra Questions 5 19. Despite the restriction to discrete probability this book is a superb general introduction for the math undergraduate and is very well organized. You and your Algebra tutor will work in our online classroom and use the interactive whiteboard to solve Algebra problems, check Algebra homework and review Algebra formulas. Mitchell Room: ET 105 Time: MWF 10:40 to 11:50 AM Text: Contemporary Abstract Algebra, Fourth Edition by Joseph A.
The exercises are bountiful, and I often find a few interesting ones in each section. Homework reading and exercises will be assigned at the beginning of each class. Because that's how we read, from left to right. "x equals. . ." The book contains 33 chapters, of which the first (roughly) 22 make up the core of a two semester undergraduate algebra course covering groups, rings, and fields. When writing any assignments the use of graphs, data and math will validate your statements and make it appear more professional.
That would give us x = 4 without having to do all the work of multiplying and regrouping and everything. We have both fundamentally changed how we think about fractions (by allowing the fraction 0/0, and not considering fractions like 1/2 and 2/4 to be equal), and have defined a new type of "addition" which we denoted by # on them. These lessons simplify planning for the teacher, since the PowerPoints in the original format are available for purchase, and students may visit this site for free to review their lessons, and to take quizzes and solve puzzles.
In what ways are they similar, and in what ways do they differ? Hayakawa has already described such a ladder of abstraction. This allows the reader to focus on the material and have it well-explained without being distracted or treated like a junior high student. Sangaku was formulated during an era before western influence had reached Japan. Application of numerical methods of solving non-linear equation, 1 step algebra equation worksheets for grade 6, glencoe textbook answers, solving simultaneous nonlinear equations, free problem solving warm-ups.
Hence the two possible orders of H are 42 2 = 84 or 42 5 = 210. 7.16 Recall that, for any integer n greater than 1, φ(n) denotes the number of integers less than n and relatively prime to n. The book contains interesting historical facts and insightful examples. Urdan has managed to compress everything one needs to know about statistics into a compact 250 page book that doesn't feel hurried or unfulfilling. Besides, the most detailed explanations will be provided for you.
However, textbook competition requires that newer books contain more and more material until the book can become rather unwieldy (in several senses) for the classroom. For an arbitrary j (j = 1, 2, 3,…, m), a hierarchical-cluster-like expression can be defined [ 12 ]. By using our website you consent to all cookies in accordance with EU regulation. You will find, that in learning linear algebra, your practice in sorting out ideas (some of which will seem quite bizarre at first) will help you to think clearly.
The ICCA conferences have a strong interdisciplinary character. In trying to exchange, I was told the hardcover was not available, and no refund was given. Just to prove to you that the equation works, we'll work out the quadratic equation whose roots we already know: A few times per test, the ACT will give you two equations and ask you to determine the value of a particular variable or some other equation or expression. | 677.169 | 1 |
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Fast complex math library for BC++
CMATH for BC++ makes fast complex-number math functions (cartesian and polar) available in three precisions. This comprehensive library was written in machine code for superior speed and accuracy. All functions may be called from C or C++.
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Mathematics Department
Our Mission
The mission of mathematics education, while integrating the values and teachings of the Catholic Church, is to teach our students strategies to identify problems, organize information, devise plans, solve problems, and analyze solutions to communicate results.The 8th Grade Placement Test results are used to recommend the best math course sequence for each
The 8th Grade Placement Test results are used to recommend the best math course sequence for each student. If a student has a thorough understanding of concepts from Geometry classes, we will accept a transcript from for Geometry course and place the student in Honors Algebra II.
Note: Kansas State Board of Regents requires three units of math at or above the level of Algebra I. The Board and St. James Academy strongly encourages students to take a mathematics course every year of high school. All college-bound students should take four years of math. | 677.169 | 1 |
Overview This starts off with with reasonably easy stuff about everyday ratios and proportions and then builds into more demanding stuff about using equations to solve problems involving inverse proportion. The more demanding topics are Grade A, and I think … Continue reading →
Overview In order to understand and use cumulative frequency, you need knowledge covered in Data Collection, Interpreting and Representing Data and Range and Averages. None of the concepts are too difficult. You need to be able to complete cumulative frequency … Continue reading →
Overview Range and Averages I find this section not too demanding. You have to learn some terms (range, mean, mode and median) and one or two techniques (for example, how to estimate the mean from a grouped frequency table). There … Continue reading →
Overview Fractions, Decimals and Percentages For me, this is one of the key sections. An understanding of fractions, decimals and percentages will give you a headstart in many other areas within Maths GCSE. Just one small example, stratified sampling (see … Continue reading →
Overview Data Collection In my opinion there is nothing too challenging in data collection and common sense will help to answer most questions. You do have to be familiar with some of the jargon used (see the definitions below). Stratified … Continue reading → | 677.169 | 1 |
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ALGEBRA 128 Advice
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I would recommend this course because I think Math is important and this course is not so hard. With Mrs. Flores as a teacher you can do well if you put hard work
Course highlights:
In this course I learned that nothing is impossible and if you want you can. Mrs. Flores is a great teacher so show her that you care by participating and asking. Do all your homework and the extra credit she offers. Good luck!
Hours per week:
9-11 hours
Advice for students:
Do homework so you will get those points, ask a lot of questions and stay after school.
Course Term:Fall 2015
Professor:Mrs. Flores
Course Required?Yes
Course Tags:Math-heavyMany Small AssignmentsParticipation Counts
Jan 06, 2016
| Would recommend.
Not too easy. Not too difficult.
Course Overview:
Math is a basic life skill that everyone should have. Most colleges and careers would like you to have at least four full years of math in high school and college. Even if you are planning to study English, you still need math for everyday life. Counting money, paying bills, etc.
Course highlights:
The highlight in this class was the teacher. My teacher helped students fully understand each and every topic, everyday. They prepare you not only for the next school year but, for college as well. Teachers also, will tutor you. They will always communicate with you and use their time to help you.
Hours per week:
6-8 hours
Advice for students:
Study everyday and do every homework assignment. It is a lot of work but, the more you practice, the easier it gets. Always ask questions! And, always ask for help when you need it. Teachers will never judge you for not understanding and neither will the students because, more than likely, they need help too. | 677.169 | 1 |
Intermediate Algebra
ISBN-10: 0321567528
ISBN-13: 9780321567529Sullivan/Struve/Mazzarella AlgebraSerieswas written to motivate students to ldquo;do the mathrdquo; outside of the classroom through a design and organization that models what you do inside the classroom. The left-to-right annotations in the examples provide a teacherrsquo;s voice through every step of the problem-solving process. The Sullivan exercise sets, which begin with Quick Checks to reinforce each example, present problem types of every possible derivation with a gradual increase in difficulty level. The new ldquo;Do the Mathrdquo; Workbook acts as a companion to the text and to MyMathLab reg; by providing short warm-up exercises, guided practice examples, and additional ldquo;Do the Mathrdquo; practice exercises for every section of the text. Real Numbers and Algebraic Expressions; Linear Equations and Inequalities; Linear Equations and Inequalities in One Variable; Linear Equations and Inequalities in Two Variables; Relations, Functions, and More Inequalities; Systems of Linear Equations and Inequalities; Polynomial and Polynomial Functions; Rational Expressions and Rational Functions; Radicals and Rational Exponents; Quadratic Equations and Functions; Exponential and Logarithmic Functions; Conics; Sequences, Series, and the Binomial Theorem For all readers interested in intermediate | 677.169 | 1 |
Google Friends List
Course Description
This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today.
Course Objective
Advanced topics in Complex Analysis,the theorems of Hurwitz and Rouche, the Open Mapping theorem, the Inverse and Implicit Function theorems, applications of those theorems, behaviour at a critical point, analytic branches, constructing Riemann surfaces for functional inverses, Analytic continuation and Monodromy, Hyperbolic geometry and the Riemann Mapping theorem | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
3 MB|24 pages
Product Description
This is an excellent interactive and visual product for introducing factoring and solving quadratic equations.
This bundle starts with a foldable on factoring. Following this is a basic classifying activity, in which students are given different quadratic expressions and must determine which method of factoring they would have to use. Next, is a graphic organizer containing more advanced factoring methods, followed by some practice problems. Finally, there is also a solving quadratic equations foldable, which goes through completing the square, factoring and solving, and quadratic formula. | 677.169 | 1 |
Compu-Math series
The Compu-Math series are mathematicstutorials developed and published by Edu-Ware Services in the 1980s. Each program in the Compu-Math series begins with a diagnostic Pre-Test, which presents learners with mathematics problems to determine their current skill level in the subject and then recommends the appropriate learning module. Each learning module begins by specifying the instructional objectives for that module, proceeds to teach to those specific goals using shaping and cueing methods, and finishes by testing to verify that learners have indeed learned the skills being taught by the module. After learners progress through all recommended learning modules and successfully solve the minimum number of randomly generated problems, the program provides a Post-Test so that learners can see how much their mathematics skills have improved.
The Compu-Math series also provided the learner with controls for modifying the instructional environment, such as a special remedia learner setting, pass/fail levels, and allowable error rate prior to remediation.
Compu-Math: Fractions was the first program created in the Compu-Math series, being introduced in Edu-Ware's March 1, 1980 catalog. Fractions six learning modules include tutorials on definitions, common and lowest denominators, fraction addition, fraction subtraction, fraction multiplication, and fraction division. Each module includes the use of both common fractions and mixed numbers.
Originally developed by Edu-Ware founders Sherwin Steffin and Steven Pederson as a text-based program, Edu-Ware upgraded it to high-resolution graphics using its EWS3 engine in 1982, renaming it Edu-Ware Fractions, and later, simply Fractions. The program was featured in the company's catalogs until its closure in 1985.
Compu-Math: Decimals was the second program created in the Compu-Math series, being introduced in Edu-Ware's August 1, 1980 catalog. Decimals seven learning modules include tutorials on conversion, addition, subtraction, rounding off, multiplication, division and percentage.
Originally developed by Edu-Ware founder Sherwin Steffin and programmer David Mullich as a text-based program, Edu-Ware upgraded it to hi-res graphics in 1982, renaming it Edu-Ware Decimals, and later, simply Decimals. The program was featured in the company's catalogs until its closure in 1985.
Compu-Math: Arithmetic Skills was the third and final program created in the Compu-Math series, being introduced in Edu-Ware's December 1, 1980 catalog. Arithmetic Skills seven learning modules include tutorials on counting, addition, subtraction, mulciplication and addition. The program also includes a learner management system for changing the system parameters, instructional parameters, and performance criteria. Originally developed by Edu-Ware founder Sherwin Steffin as a text-based program, Edu-Ware upgraded it to hi-res graphics in 1982; however, its name was not changed, unlike the other two product in the series.
The program's documentation put forth its educational model:
"Instruction should be individually proscribed. Arithmetic Skills accomplishes this by changing its procedure depending upon the studen profile and performance.
"Instruction should not be text dependent." The program uses graphics so that minimum emphasis is not placed on reading skills.
"Instruction should use decremented cues." As the student progresses in the program, decreased cues give the learner more self-prompting responsibility.
"Instruction should use cognitive/motor stimulation." For learners unfamiliar with the keyboard, a graphical hand presses depictions of the keys on the display and a set of visual prompts are set up.
"Instruction should be a conceptual sequence." The program attempts to achieve learning by building on acquired skills rather than relying on repetition alone.
A 1981 Peelings II review noted that while the program makes an attempt at applying formal learning theory in a way that is comfortably, easy-paced, and non-intimidating, Arithmetic Skills "is also devoid of the fun aspect that makes computerized learning human and inspiring. The sole reinforcement is an ever increasing complexity of the problems".[1]
The program was featured in the company's catalogs until its closure in 1985. | 677.169 | 1 |
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Wednesday, March 9, 2016
How to Study Maths
Mathematics is a subject that every student has to study at one time or another. Some love it but if we're being honest, most people hate studying maths. The importance of maths for students has never been more prominent. Most university courses include some level of maths while almost every profession uses maths in some form on a daily basis. The problem many students have is that they don't know how to study maths to get good results.
Maths is one of those subjects which you can easily spend hours studying but end up none the wiser. However much you have studied, if you can not solve the problem on day of the test, you are lost. Thankfully, there are some techniques for studying maths that you can do regardless of your level. You may even end up loving mathematics by the end of the blog post!
7 Tips for Maths Problem Solving
1. Practice, Practice & More Practice
It is impossible to study maths properly by just reading and listening. To study maths you have to roll up your sleeves and actually solve some problems. The more you practice answering maths problems, the better. Each problem has its own characteristics and it's important to have solved it in numerous ways before tackling the exam. There is no escaping this reality, to do well in a Maths exam you need to have solved a LOT of mathematical problems beforehand.
2. Review Errors
When you're practising with these problems, it's important to work through the process for each solution. If you have made any mistakes, you should review them and understand where your problem-solving skills let you down. Understanding how you approached the problem and where you went wrong is a great way of becoming stronger and avoiding the same mistakes in the future.
3. Master the Key Concepts
Do not try to memorise the processes. This is counter-productive. It is much better and rewarding in the long-run to focus on understanding the process and logic that is involved. This will help you understand how you should approach such problems in the future.
Remember that Maths is a sequential subject so it's important to have a firm understanding of the key concepts that underpin a mathematical topic before moving on to work on other, more complex solutions which are based on understanding the basics.
4. Understand your Doubts
Sometimes you can get stuck trying to solve part of a maths problem and find it difficult to move on to the next stage. It's common for many students to skip this question and continue on to the next. You should avoid doing this and instead spend time trying to understand the process of solving the problem. Once you have grasped an understanding of the initial problem, you can use this as a stepping stone to progress to the remainder of the question.
Remember: Maths requires time and patience to master.
It is a good idea to study with a friend who you can consult with and bounce ideas off when trying to solve complex problems.
5. Create a Distraction Free Study Environment
Mathematics is a subject that requires more concentration than any other. A proper study environment and a distraction free area could be the determining factor when solving complex equations or problems in geometry, algebra or trigonometry!
Studying with music can help create a relaxing atmosphere and stimulate the flow of information. Havingsuitable background music can foster an environment of maximum concentration. Of course, you should steer clear of Pitbull and Eminem, instrumental music is the best thing in these times.
Mathematics has specific terminology with a lot of vocabulary. We suggest you create Notes orFlashcards with all the concepts, terminology and definitions you need to know. You should include their meanings, some key points and even some sample answers so you can consult them at any time and recap.
7. Apply Maths to Real World Problems
As much as possible, try to apply real-world problems when approaching maths. Maths can be very abstract sometimes so looking for a practical application can help change your perspective and assimilate ideas differently.
Probability, for example, can be used in everyday life to predict the outcome of something happening and determine whether you want to take a risk such as if you should buy a lottery ticket or gamble.
Oh and don't forget that it's also important to have confidence in yourself and face the exam knowing that you have prepared properly!
Sunday, March 6, 2016
What: Camellia Group presents Calcutta Club The Telegraph National Debate 2016 Where: Calcutta Club lawns Motion: In the opinion of the house tolerance is the new intolerance Verdict: Dead heat or tie, after show of hands
According to me the well charged debate was well coordinated by Mr.Mukul Keshavan.Suhel Seth gave a nice introductory speech, though against team had stronger opinion the debate was still drawn. Anupam Kher was more interested on saving his and BJPs face rather than to debate on the topic while Mr.Surjewala was inclined to blame BJP holding a list of atrocity.Kajol's speech was plain and simple, while Justice Ashok Ganguly spoke about Supreme court order and challenged the supreme court. Atlast Barkha Dutt's speech concluded with some strong points supporting Kanhaiya Kumar. In mine opinion the against team was better.I have provided the synopsis of the speeches to let you decide yourself.
FOR
Suhel Seth
Marketing maven
What we should really be intolerant about is intolerance itself. What we should be intolerant about is poverty, the subjugation of rights and opportunities. Why am I being part of an intolerance discourse when actually I am not intolerant? Why are we dividing India on lines of caste and creed and religion and exposing every little nuanced statement put out by stupid people as the final yardstick of this particular government… and I don't mean this particular government… I mean that particular government. I am not for once supporting the fact that the Prime Minister doesn't speak up when he should. I think he should… but that's for the Prime Minister to decide.
We don't need a media circus playing on our television channels every evening, almost suggesting that our idea of India has either failed or is about to diminish. Because the idea of India is stronger than stupid utterances that will be made in a democracy as vibrant as ours. What's happened today is that we are living in media circus times… so if an idiot says something stupid, and there are many across political parties, those become the national discourse.
We are still a tolerant country. Don't get taken in by this whole mantra of intolerance. There are stupid things happening in this country, but people will understand that basic commodities and basic rights of humanity are more important than beef bans. People will realise that you can't alter the discourse of this country on the basis of one ideology, one principle or one faith.
We need to be more tolerant about the word 'intolerance'. We believe in a holistic and composite India.
I have always been an intolerant girl. Twenty years ago, I was pictured as a demure and docile Indian girl being brought up in London by a caring and strict father who allowed me to go on my first Europe trip with friends. And there I met this boy who — quite famously — held my outstretched hand and pulled me into a train's compartment. But once on the train, he and I were so totally on the wrong track. After all, how could any good Bharatiya nari ever tolerate someone as frisky and flirtatious as this boy called Raj, with overgrown hair and undergrown manners. The boy I was seriously intolerant of back in 1995, was Shah Rukh Khan. It was only later that I as Simran managed to not just tolerate him but also fall madly in love with because, after all, the Dilwale had to take the Dulhania away!
In 1998, I was flying from Paris to India... on the flight I met Shekhar, who managed to irritate me in so many ways. I was severely intolerant of him until I realised he had a heart of gold.... After all, Pyaar Toh Hona Hi Tha. A year later, reel turned to real and I was married to this man you all know as Ajay Devgn.
Those who know me and my family have often said that this intolerance runs in our blood.... There always has been intolerance in India. There is intolerance because the society is not perfect. There are fault lines. We all need to work together to make a stronger and a richer India. An India where there is much less poverty and inequality. Only then will intolerance cease to exist.
Anupam Kher
Actor
Had any of you heard of the word 'intolerance' till seven or eight months ago? You hadn't, because this is a term that's been marketed. It's a result of the drubbing that the Opposition got at the hands of the ruling party. The Prime Minister hasn't taken a single day of leave in the last two years… let him work for five years and then decide... I am not advocating him, I am just speaking as a citizen of this country.
Many people say I speak for the BJP because of my wife. I have been married to Kirron for the last 30 years… I don't have to prove my loyalty to her by speaking on behalf of the BJP. You can't tolerate the fact that our Prime Minister goes on foreign tours and speaks about the idea of India… for 10 years, you have tolerated a Prime Minister who didn't say a word.
There has been no talk of corruption over the last two years, but for 10 years before that, there was only talk about corruption... 2G, 3G, e ji, o ji…. People in India, in general, are tolerant. The only ones who talk about intolerance are the intellectuals… the rich and famous who travel with 20 bodyguards, sip on champagne and talk about intolerance. The man on the street doesn't even know the word 'intolerance'. The American Presidential candidate (Donald Trump) says that Muslims should be thrown out of the country… that's intolerance.
The most tolerant people in the country are the Congress. They are tolerating a person who they want to project as a Prime Minister of this country and they can't even say to each other, 'We are wrong!' If you can tolerate that person, then you can tolerate anything in the world.
VS
AGAINST
Asok Ganguly
Retired judge of Supreme Court
In a school in Kerala the national anthem was sung and three schoolchildren respectfully stood up for the anthem but they did not sing. Initially the school authorities overlooked it, till it caught the notice of a patriotic gentleman who happened to be a member of the Assembly. He raised it and ultimately the students were expelled. They challenged it before the courts and the Supreme Court... in upholding the rights of the children made a very prophetic statement in its concluding part of the judgment. The Supreme Court said: "Our tradition teaches tolerance, our philosophy preaches tolerance, our Constitution practises tolerance. Let us not dilute it". In the wake of aggressive nationalism of the present scenario this may sound a little strange to many ears but this is the crux of tolerance in our Constitution.
In the Preamble the greatest emphasis has been given on individual dignity and individual dignity has been equated with the unity and integrity of the nation. You cannot allow the citizens to maintain their dignity if you rob them of their basic freedom. If today I become answerable for eating a particular food, or if my house is raided for storing beef... where is my dignity, where is my moral autonomy, where is my freedom? This is what is happening today and we condemn this as acts of intolerance.
I condemn intolerance in any form and I think without tolerance you cannot have cultural pluralism nor can we have multi-cultural existence in India. India is not meant for these intolerant people.
Randeep Surjewala
Congress MLA
Shall we forget compassion because another one is cruel? Shall we give up love and affection because another one spurns us? Shall we forget humility because another one is arrogant? Shall we give up tolerance because another one is intolerant? The answer is plain and simple. No.
And while I say this, I know that in today's times if you are inclusive, if you are tolerant, if you are passionate you will be hounded, you will be persecuted and you'll be trolled, by those who stridently oppose compassion and tolerance. But does it mean that the fringe has become the mainstream? Does it mean that if you disagree with the ruling establishment, then I am the enemy? Does it mean dissent is anti-national? Does it mean I need to carry a certificate of patriotism everywhere I go? The answer is again plain and simple. No.
While I say this I am painfully aware that the fault lines of identity, of religion, of caste, of region have become very, very sharp. The last 22 months of this government... it often appears that this is the mainstream discourse. But is it? Because its intensity and its noise is so strong and so loud that all sanity appears to have been lost in the din, so naturally it looks as if intolerance is the new tolerance….
India's DNA is about compassion, it is about cohabitation and it is about coexistence. We are a nation that enabled the birth of different religions and adopted many religions and still thrived. I am confident that no government or organisation can take away our DNA of compassion, coexistence and cohabitation.
Barkha Dutt
Journalist
I am going to ask all of you to step back from the politics, the theatrics, the histrionics and... let's talk instead of the environment that we operate in today, and who has created that environment. Is it politicians, or is it all of us? Have we as Indians forgotten the art of conversation?
As a television host... I worry most of all for the intolerance of my own fraternity, the media. I worry at how we somehow have converted news into theatre and our narratives have become reductionist and I am afraid in this city, which prides itself on its intelligence, I know that all of you will agree with me that certain narratives are dumbing us down. We are more intelligent than this nationalism versus anti-nationalism narrative. We are proud Indians and we do not need anybody to certify that for us.
The young man I met last night…. I asked him, 'There's a word these days that has become very controversial, that word is called 'azadi'. I said 'can you define azadi in one sentence?' And he said: 'Azadi for me is the freedom to implement the Indian Constitution'. Who said this? A boy charged with sedition called Kanhaiya Kumar. This was his definition of azadi. Not azadi from India but azadi in India.
We the people who are tolerant, who want to live and let live, who want to let people wear what they want and yes, eat beef if they want.... Secularism was dear to me, my politicians took it away from me, so today I say pluralism. Nationalism was dear to me... but today they have taken away that word from me and I have to look for another word.
Mukul Kesavan
moderator
If I was arguing this for the proposition I would argue that in the last few months within the political context of India, a bunch of isolated and abhorrent incidents have been stitched into a sinister pattern and this, in a sense, is used to exclude legitimate political voices that arguably won the last general election.
If I was arguing this against the proposition, I would argue that the call for intolerance is never merely rhetorical. Reason why so many people have made a case against the present political climate is because they believe an attempt is being made to rig the political discourse of the republic in a majoritarian way… that there is, in a sense, a state that acts as a patron for civil society movements to try and shut down what the nation has always been known for, which is almost a form of anarchic pluralism
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From this platform on behalf of
all of you, as JUNUSU president I take this opportunity of the media's presence
to thank and salute the people of this country. I want to thank all the people
across the world, academicians and students, who have stood with JNU. I salute
them (lal salaam).
I also wish to acknowledge and
thank all the people standing firm with the struggle, who are demanding justice
forRohith Vemula,
be they from the media or civil society, political or non-political. I salute
them. (lal salaam).
I especially wish to thank the
worthies of this country sitting in parliament who claim it is they who decide
what is right and what is wrong. Thank you to them, their police and also to
the media channels…[resounding cheers]In our parts they say, so what if
one's name was vilified, it got one some publicity! At least they gave space to
JNU on primetime even if it was to vilify it.
I have no rancour against anyone –
none whatsoever against the ABVP [Akhil Bharatiya Vidyarthi Parishad, student
wing of the BJP]. You know why?(crowd roars – why?)Because the ABVP on our campus is
actually more rational than the ABVP outside the campus. I have one suggestion
for all those people who consider themselves political pundits – kindly watch
the recording of the last presidential debate to see the state the ABVP
candidate was reduced to. When we decimated the sharpest ABVP intellect there
is in the country, which happens to be in the ABVP of JNU, you can draw your
own conclusions about what awaits you in the rest of the country. There is no
ill-will against ABVP because we are truly democratic; we truly believe
in the constitution, and that is why we look at the ABVP not as enemies but as
the opposition. Rest assured, my friend, I will not indulge in any witch-hunt
against you for the simple reason that one has to be worthy enough to be
hunted….
JNU has shown the way. JNU stands
unshakeable today to state what is right is right and what is wrong is wrong.
The best thing about this protest is that it is spontaneous. I am stressing
this aspect because everything ontheirpart was planned.
We have unstinting faith in the
constitution of this country, the laws of this country and the judicial process
of this country. We also believe that the only truth is change, and we are
standing here rooting for change. This change will come about, make no mistake
We have unstinting faith in the
constitution. With all our alertness we stand fully behind the articles of
faith declared in the preamble – socialism, secularism, equality.
I don't want to give a speech
here; I want to tell you about my experience. Earlier I studied more, endured
the system less. This time I studied less and endured the system more. Since
JNU has a tradition of research I can say I have primary data, first hand
information! First things first – I don't want to say anything about the
judicial process. I have said just one thing – and all the people of this
country who truly believe in the constitution and want to bring Babasaheb's
dreams to fruition would have taken the hint. I don't want to say anything
about the matter that issub judice.
So the honourable prime minister
has tweeted. He says,Satyamev Jayate [truth
will prevail] (laughter). My reply is as follows: honourable prime minister, I
have serious ideological differences with you butSatyamev
Jayateis not your
slogan; it is the motto of the country and the constitution, so I too will
happily say Satyamev
Jayate.
Truth will emerge victorious….In
my village – you have become acquainted with my family these past days – we
have confidence men at our railway stations trying to trick people into
buying a 'lucky' ring that will give them anything they desire.
We have some policy makers in our
country who are cast in the same mould. They say many things – black money will
return,har harmodi, this killing inflation will come
down,sab ka saath sab ka vikas. Although
Indians easily forget such things, the spectacle this time was so huge that we
are unable to forget the slogans.
Nevertheless, attempts are on to
make sure that these slogans are forgotten. How? Stop the [money] of
research fellows so that they have to beseech you to restore their fellowship.
Then you agree but on the condition that the remuneration will stay at the same
level of Rs 5000 or Rs 8000. Who speaks out against this? JNU, who else?….
If you speak out against this
undemocratic regime, what will its cyber cell do? It will send adoctored videoand insults, and the condoms in your dustbinwill be counted too. But make
no mistake, this is a critical moment and we need to understand that the
attack on JNU is a planned attack because they want to delegitimise the Occupy
UGC movement. It is a planned attack because they want to discredit and destroy
the struggle to get justice for Rohith Vemula. You are running this JNU issue
on prime time, ex-RSS worthy [Subramanian Swamy], because you want the people
of this country to forget the prime minister's assurance that he would put 15
lakh rupees in their individual bank accounts. A word of friendly caution – it
is not easy to get admission in JNU and it certainly is not easy to make JNU
students forget. We will remind you repeatedly that every time a government has
turned oppressor, JNU has raised the voice of protest…
Don't play soldiers against
students
You cannot simply dilute our
struggle by saying that on the other side of the picture are the country's
youths who are dying on the border. I salute these heroic figures. I have a
question for a BJP leader who made a statement in Parliament. (I will not name
him for I learnt one thing in prison, where the fight is ideological, one
should not give individuals needless publicity). On the floor of
the Lok Sabha this BJP leader thundered that the country's youth are dying at
the border.
I want to ask that leader – is
that youth a brother to you? The thousands of farmers who are committing
suicide, who grow grain for us and our youth on the border; farmers who are
fathers to these youths – do you have anything at all to say about that,
about them? I want to tell that leader the farmer who works in the field is my
father, and it is my brother who joins the army. By erecting this binary don't
you go creating a false debate in the country – because those who die for the
country die within the country and also on the borders of this country.
My question [to the BJP leader] is
this: by standing up in parliament who exactly are targeting with your
politics? Who will take responsibility for those who are dying? Those who fight
are not responsible; the ones who make them fight are the ones who are
accountable.… Who takes responsibility for this war, who makes people fight?
See how my father is dying, how my brother is dying. I put this question to the
two-bit primetime anchors who create this binary all the time.
Is it wrong to seek freedom (azaadi) from the ills that
plague our country today? They ask belligerently – who do you want freedom
from? Has India enslaved somebody?
My answer to them is NO. So isn't
it obvious we are not seeking freedom from India? We are not seeking freedom
FROM India but IN India. There is a difference. We are certainly not issuing a
call for freedom from the British; that freedom has already been wrested by the
people of our country.
What prison was like
Now I come to my experience in
prison. Some of the policemen asked me why we keep sayinglal
salam, lal salam. I should perhaps make it clear this was not part
of the investigation! The policemen would come to give me my meals and to take
me for my medical check-up. I being a JNU student, more so from Brahmaputra
[hostel], how could I stay without talking? So I struck a conversation with a
policemen and discovered he was just like me. Think – who takes up a
police job inside a jail – someone whose father is either a farmer or a labourer,
someone whose father is from a disadvantaged section. I too come from one
of India's backward states, Bihar. I too come from a poor family, a farmer's
family. By and large it is only those from poor families who join the police.
Here I am talking about policemen of constable, head constable and inspector
rank. I have not had much interaction with the IPS.
So the policeman asked: what is
thislal salaam.
I replied:Lalmeans revolution (kranti).
He: Andsalaam?
I: It means 'hail revolution'.
The policeman did not get it. I
asked him about the slogan 'Inquilab
Zindabad'. He said he knew of it. I told him 'Inquilab' means revolution (kranti) in Urdu. He said ABVP
members also use this slogan (Inquilab
Zindabad). I said, do you understand now – they are false revolutionaries,
we are the real revolutionaries!
Then the policeman asked me
another question: Everything is very cheap for the people of JNU, right? I
asked if the same was not true for him? I asked him if he was paid overtime (he
works 18 hours a day) and he replied in the negative. When asked how he manages
he replied, that same thing which you call corruption?
He gets a uniform allowance of Rs
110. You can't buy even an undergarment with that amount. All this the
policemen volunteered on his own. I explained to him that it is precisely fromthis– hunger, corruption – that we seek
freedom.
By then the agitation in Haryana
had started and as you know a large part of the Delhi Police force comes from
Haryana. I salute them for they are very hard working. Anyway I asked the
policeman what he thought about reservation. 'Casteism is not a good thing at
all,' he replied. It is precisely from this casteism that we seek freedom, I
told him. The policemen exclaimed: 'But there's nothing wrong with what you
just said, there is nothing anti-national about it.'
Then I asked him one more
question: 'who wields the maximum power in the system?'
He looked at his lathi and said, 'danda' (thelathi).
Can you wield yourlathiat will, I asked.
No, he admitted. On being asked
who has the most power his answer was, 'the ones issuing fake tweets'!
It is from the Sanghis who tweet
false statements that we want freedom, I told him.
The policeman then said, 'It seems
to me that you and I are one the same side.' Well, there's a small problem, I
replied.
Now I am not saying this about all
journalists because all of them don't get their remuneration from there. Of
course some get their wages only from there, and some after long years of
reporting parliament are making desperate efforts to enter it as well. And what
an atmosphere they have created.
So I told the policeman, Here you
and I are having a one-to-one chat and there they screech'Dekhiye
sansanikhez'(Watch
this sensational news.)
'Shall I tell you something in
confidence,' asked the policeman. 'I had decided I would beat you up when you
arrived – your name was there on the FIR*…. But after talking to you I now feel
like beating them up."
Democracy matters, as does social
justice
He said something very important.
And through the media here I want to draw the attention of the entire country
to it. This policeman, like me, comes from an ordinary family; like me, wanted
to pursue studies; like me, wanted to understand the systemic ills of the
country and fight against them, wanted to understand the difference between
being literate and educated, yet is working as a policeman. This is where JNU
comes in and that is why you want to suppress JNU's voice – to ensure that a poor
marginalised individual is not able to do a PhD because it is clear as day that
the poor simply can't afford the lakhs of rupees needed to pursue a PhD in a
private institution.
They want to stifle all the voices
that can come together, whether they are standing on the border, dying on the
field or standing tall in struggle in JNU.
You who don't want these voices to
come together, I want to remind you of what Babasaheb said –political
democracyis not
enough. Well, we will establishsocial democracy. That is why
we speak of the constitution repeatedly. Lenin said, 'Democracy is
indispensable to socialism'. We speak of democracy, freedom of expression,
equality and socialism so that a time may come when the son of a peon and the
son of a president can study together.
This voice of struggle they want
to choke. Science says the more you press down, the more pressure builds up.
But then these people have nothing to do with science for studying science is
not the same as being scientific. But if a dialogue could be established with
people who are engaged in the quest to build a climate of scientific temper
then we will surely wrest the freedoms that we are fighting for in this
very country: freedom from hunger and poverty, exploitation and injustice, and
securing the rights of Dalits, adivasis, women and minorities. That we will
secure this freedom in this country through this very constitution, this very
parliament and this very judicial process, is our dream. This was the dream of
Babasaheb. And this was the dream that our comrade Rohith dreamed.
Do you see now? You killed one
Rohith and tried to crush the movement that emerged in its wake. See how
massive that movement has become.
One more thing I want to say from
my prison experience. This is my self-criticism and if you think it applies to
you too then take it in that spirit. We from JNU are refined and civilised in
our speech, but we speak in heavy jargon which the common people of this
country are unable to understand. It's not their fault. They are honest,
straightforward and perfectly capable of understanding. It is we who are unable
to explain things to them at their level. In the end what reaches them? 'No
more prevarication, just sell on OLX'(Ab no more dekhte hai OLX mein bechate
hain.) It is critical that we establish a conversation and debate
on this 'for sale' mentality that has been created in this country.
Now let me talk about my prison
experience. I got two bowls there – one was blue the other red. I kept
looking at the colours and thought to myself that although I am not a believer
in destiny, nor do I know god, but surely something good is about to happen in
this country now that these two colours are here together, side by side. The
plate looked like our India, the blue was the blue of the Ambedkarite movement
and the red bowl like [the red of socialism] . I thought if this unity
were to be created in this country, then no more prevarication, we will send
those who put everything on sale packing. Those who auction off everything we don't
want. We will put in power those who can ensure the protection of the law for
everybody. We will make the slogan ofsabka saath, sabka vikasa living reality….
Modi's and RSS's agenda
Today, when our honourable
PM (I have to be respectful, no, or they will doctor this too!) spoke about
Stalin and Khrushchev, I had an irresistible urge to enter the television, tug
at his suit and say, Modi ji, why not talk a little about Hitler too? If not
Hitler, then Mussolini at least – whose black cap was worn by your Guruji?
Golwalkar ji had gone to meet him and had been advised to fashion the
definition of Indianness on the German model….
Now I come to something very
personal. I spoke to my mother after three months. When I was in JNU I never
kept in regular touch. After going to prison I felt one should keep in
regular touch; I advise you to do the same. When I spoke to my mother I asked
her: So you took a dig at Modiji? She replied that it was not a 'dig' at him.
'To make fun of people is their prerogative. We just express our pain – those
who understand, cry, and those who don't, laugh.' My mother said, 'It was my
pain which made me say Modiji is also a son to a mother, my son has been
falsely accused in a sedition case. So when he talks about 'mann ki baat', why not also
talk about 'maa ki baat'
(mother's plight)?
What words of comfort could I
offer her? Whatever is happening in this country shows a dangerous pathology.
Here I am not talking about one party or one media channel, or only
about soldiers – I am visualising the entire country. What will be the face of
this country when it is emptied of its people? That is why it is important to
salute all those people who have stood up in support of JNU. They understand
the importance of JNU – 60% of its students are women. Moreover, despite any
shortcomings it may have, JNU is one of the few institutions which implements
the reservation policy; where it doesn't we fight to ensure it does so.
Also the people who come here – I
have not told you until now, my family lives on 3000 rupees. Would I be able to
pursue a PhD in any big university? So when a serious offensive is mounted
against JNU, the people who are standing up for it are also being tarred with
the same brush (in saying this I am not expressing sympathies for any
particular political party, for I have my own ideological path). Sitaram
Yechury has been charged with sedition, Rahul Gandhi, D. Raja and Kejriwal too.
Even those from the media who are speaking up for JNU – actually they are not
speaking up for JNU, they are stating the truth as truth and falsehood as
falsehood – are being hounded and threatened.
Where is this self-proclaimed
nationalism coming from? I was asked by some in prison whether I really shouted
those slogans. I said, yes, and I will do so again.
My question is, Are you [those in
power] able to see the difference or has your rationality been destroyed…. Is
it a good thing to lose one's rationality so soon because 69% of the people of
this country voted against this kind of mindset? Only 31% voted for you and
among them were some who were taken in by your slogans. Some of the people you
lured with yourhar harslogan can only think of the price
of arhar[dal] these days.
So don't delude yourself that your
victory is forever…. If you repeat a hundred times that the sun is the moon,
will the sun become the moon? Certainly not. It will remain what is – the sun –
even if you repeat your lie a 1000 times.
The beauty of it is that in
parliament they table a 'call attention motion', but outside the Lok Sabha
across the country they revert to the 'distract attention motion' – draw people
away from their genuine problems and entrap them in ever new agendas. Here the
Occupy UGC movement was going apace and comrade Rohith was killed. As soon as
we raised our voice for him came the new salvo – 'witness the biggest betrayal
of the nation, look carefully at the epicentre of sedition'. This agenda too
will lose steam.
So they are planning their next
move – Ram mandir, what else. Let me tell you about a conversation I had with a
policeman just before stepping out of prison.
He: do you believe in religion?
I: I need to know about religion
to be a believer.
He: You must have been born in
some family?
I: Coincidentally, I was born in a
Hindu family
He: So do you know something about
your religion?
I: From the little I know I can
say that God created this earth and is present in every little speck. What do
you say?
He: Absolutely right
I: Some people want to create
something for God. What do you have to say about it?!
He: Height of madness.
You cannot dupe the people
endlessly with an agenda that has run its course. Your duplicitous game helped
you gain from 80 to 180 seats once, but not anymore, for the axis has shifted.
But still they will not give up their efforts to distract people's attention.
They don't want the people to raise genuine issues.
All of you sitting here – you feel
as if you have been assaulted. True, but this is not the first time. I want to
draw your attention to a cover story on JNU by (Subramaniam) Swamyji in the RSS
mouthpieceOrganiser. Now I have full
faith in democracy. If my friends in the ABVP are listening, I request them to
bring Swamyji here so that we can debate the issue. If through logic he can
prove that JNU should be shut down for four months, I will agree with him
wholeheartedly. If not, I would request him to leave the country and live
outside as he has done on earlier occasions.
There is just so much of planning
behind these attacks. Maybe you being inside the campus could not see. There
was a plan from day one. They don't even think to change their posters. The
same posters with the same content used by the Hindu Kranti Sena are used by
ABVP and ex-armymen. What it means is that all these things are being planned
in Nagpur. This is no spontaneous surge, my friends. There is but one
overarching aim: wherever the voice of protest emerges in this country, choke
it; whenever it seems people might start thinking about their fundamental
problems, distract them; wherever the voice of protest emerges in this JNU
campus, be it of Anirban [Bhattacharya] or Umar [Khalid], be it …Ashutosh…. or
anyone among you, to brand it anti-national and delegitimise JNU.
But I say to them, you will not be
able to suppress this struggle, this protest – for the more you try to repress
us, the faster we will bounce back on our feet to stand our ground.
This is a long struggle and we
have to carry it forward without stopping, without bending, without pausing for
breath. WE will stand united against divisive forces like ABVP within the
campus or the BJP and the RSS outside the campus who are trying to bring the
country to the edge of destruction. JNU will stand united against them; as
history shall bear witness. The struggle that was launched with the Occupy UGC
movement, the struggle that Rohith Vemula waged, and the struggle that you and
so many peace-loving and progressive people in the country have launched – that
struggle we shall wage and win.
Thanking everybody who has been
part of this struggle and appealing to them to walk by our side, I will end
here…
Thank you
Inquilab Zindabad!
Rashtra ekta zindabad! (long
live national unity)
Samajik nyay zindabad!(long live social justice)
[*After the policeman mentioned
seeing Kanhaiya's name on the FIR, Kanhaiya interjected: Before the FIR our
names had come on the ABVP complaint, they were noted in the FIR] | 677.169 | 1 |
The goal of the Algebra 1 course is to teach cadets to reason effectively in mathematical terms. Cadets will learn to apply a variety of algebraic techniques to include the simplification and evaluation of algebraic expressions, solving equations with one or more variables, evaluation of rational expressions, absolute value equations and inequalities, as well as computation of polynomials, factoring, coordinate graphing and equation solving through graphing. | 677.169 | 1 |
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Book Summary of Quantitative Aptitude And Reasoning
In today's competitive world, quantitative aptitude and reasoning play a prominent role in almost all competitive examinations and job recruitment exams. Keeping its importance in mind, this book is designed to familiarize the aspirants with basic mathematical and reasoning concepts. It enables them to apply to a range of quantitative and reasoning problems.
The text is organized in two parts-Quantitative Aptitude (Part I) and Reasoning (Part II). The separate sections are devoted to verbal and nonverbal reasoning. It sharpens the ability to apply analytical and logical thinking while gathering and analyzing information, designing and testing solutions to problems, and formulating plans.
The main objective of this book is to guide the students to solve the problems within the stipulated time and that too with the higher degree of accuracy. Each chapter contains numerous worked-out examples and practice exercises. The questions range from purely numeric calculations to problems of arithmetic reasoning, graph and table reading, percentage analysis and quantitative analysis.
This book is intended for the students appearing for quantitative aptitude and reasoning test conducted in various competitive examinations such as civil services, common admission test (CAT), management aptitude test (MAT), staff selection commission (SSC), combined defence services (CDS), national defence academy (NDA), LIC-assistant administrative officer (AAO), bank clerical, bank PO, income tax and central excise exams. Apart from that, the book is also useful for the students preparing for the campus recruitment screening test and the undergraduate students of mathematics for their course in Aptitude Examination.
About the Author
R. V. PRAVEEN has wide experience of teaching quantitative aptitude and reasoning to the aspirants of civil services, bank PO, provident fund commissioner, and staff selection commission (SSC) and other competitive examinations. He is actively involved in guiding the candidates appearing for screening tests and interview for the campus placements. | 677.169 | 1 |
It's the equivalent of teaching someone to read and then having them solve crossword puzzles. It's such a contrived example! (N.B., this anguish forced me to write a tutorial on imaginary numbers with actual, non-polynomial applications, like rotating a shape without needing trig. See
Calculus needs these everyday applications and intuitions beyond "Oh, let's pretend we're trying to calculate the trajectory of a moving particle."
They're out there: my intuition is that algebra gives a static description (here's the cookie), while calculus describes the process that made it: here's the steps that built the cookie. Calculus is the language of science because we want to know how the outcome was produced, not just the final result. d/dx velocity = acceleration means your speed is built up from a sequence of accelerations.
Man, you're spot on. I've felt that the way math is taught now is akin to spending 10 years on 'if', 'for', 'while', and 'switch' statements (and being able to evaluate them perfectly without a compiler).
I see that your examples of teaching fast (one minute, ten minutes) all use discrete models for continuous processes. For people who understand continuousness mathematically, going between discrete models to continuous models and vice versa is easy. However, there are indications that the other way around -- students who perceive these changing situations mathematically as intrinsically discrete -- going from discrete to continuous models might be extreme difficult (due to in-commensurability). Of course, students can learn procedures to make the switch from discrete to continuous models, but even then it is possible that this procedural experience is not build on deeper mathematical understanding. Tricky stuff, learning mathematics :-)
As a math person, I think this attitude is why I hated (and still do) calculus. Calculus is not a language of science, it is a branch of mathematics that is useful for science. It's description is just as static as algebra (not the least because it is algebra, just with two new functions).
Regarding the Gaussian (imaginary) numbers, they have uses which have nothing to do with rotations. My intuition for them is that they are a way of factoring expressions such as A^2 + B^2 as the difference of two squares.
Wrt imaginary numbers, just a thought: capacitance and inductance in an AC circuit.
I don't know what age group you're targeting but I found solving problems in this space gave me a great intuition for imaginary numbers; I found it quite interesting too. Heck, AC circuits are great real-world playgrounds for differential and integral equations - Laplace transforms too.
I've been fretting about how to help my kids get a better understanding of math for years. I've been bouncing between too simple and overly complicated for too long. :-(
I think I'm just going to mandate they spend some amount of time per week using Mathematica and going through some introductory book. At least they will taking baby steps in a direction better than youtube. Grumble...
Analog electronics is a great domain, and something I want to explore more myself (I did CS, only had 1-2 analog classes, and didn't build an intuitive understanding for electricity, a gap I want to cover).
For books, I don't have a huge amount of experience but I do like the Manga Guides to X (Manga Guide to Physics, etc.). A visual medium encourages the author to rely on metaphors vs. symbolic descriptions, which I prefer in an intro to get the concepts across quickly. | 677.169 | 1 |
As equations go, I'm generally in favor of them. I was an English Writing major in college with a minor in business, so equations usually showed themselves only in my statistics class, and came up again as I earned my MBA. | 677.169 | 1 |
Posts Tagged "Pearson"
Pearson's enVisionMATH California Common Core is a comprehensive mathematics curriculum for grades K-6 that includes print and eBook resources with additional online content. Students learn through hands-on activities that focus on understanding, fluency, and flexibility with math concepts while developing problem-solving skills. Learning List recently reviewed eBook and online resources.
enVisionMATH is organized using the Understanding by Design (UbD) model that frames instruction in terms of "Big Ideas," "Essential Understandings," and "Essential Questions." Content is presented in "Topics" (i.e., chapters) made up of sets of lessons that address a particular concept or skill. Each Topic includes at least one "Problem-Solving" lesson that focuses on a particular problem-solving strategy, such as drawing a picture or looking for a pattern.
Teacher and student editions have a "Mathematical Practices Handbook" that provides an overview of the MPS and guidance in how to use them. MPS videos provide a student-friendly overview of the standard, questions to support students in using the practices (e.g., "How can I check to make sure my answer makes sense?"), and examples of the standards in use. Videos help teachers make connections between the MPSs and evaluate students' ability to use them.
At each grade level, enVisionMATH provides a "Common Core Standards Practice Workbook" and the a "Math Diagnosis and Intervention System" (MDIS). Workbooks contain practice activities for each standard and end-of-year practice tests and performance tasks. MDIS includes a diagnostic test and intervention resources to support RtI, remediation, and summer school instruction.
About Pearson*
Discover research-based K-12 programs that give you the freedom to choose how you teach. Whether you're looking for an interactive, problem-solving approach or an inquiry-based, student-centered curriculum, Pearson has you covered. Our basal and NSF mathematics programs were designed to support Common Core Math curriculum. We also offer programs to meet supplemental, and intervention needs.
*The content provided here was provided by or adapted from Pearson.
Subscribe to Learning List for access to the spec sheet, full editorial review and detailed alignment report for this material.
Pearson Education's Texas High School Math Series provides comprehensive Algebra 1, Geometry, and Algebra II courses for Texas high schools. Each course features "3-Act Math" problems that develop students' ability to use mathematics to understand and solve real-world problems. Courses are designed for use in blended learning environments and deliver most content digitally. Resources will be most effective in classrooms in which all students have Internet-enabled devices and teachers are comfortable teaching with technology.
[Source: Pearson]
Although content is delivered primarily online, each course includes two printed student resources: the "Student Companion" and the "Student Text and Homework Helper." The Student Companion is a journal to accompany digital classroom instruction and is available as an interactive, electronic workbook and as a printed consumable. The Student Text and Homework Helper is a reference resource available in print and eBook formats. It includes an "Unpacking the TEKS" feature that presents each of the TEKS student expectations, underscoring key terms (e.g., justify); an explanation of the expectation in student-friendly language; and a completed example problem demonstrating mastery of the expectation. Although Learning List's reviewers did not have access to digital content, the publisher's materials indicate that online resources include interactive explorations, animations, math tools; a glossary with audio capabilities in English and Spanish; and "Virtual Nerd" video tutorials.
Teacher editions are provided in two volumes and include comprehensive overviews of all teaching resources and assessments and detailed guidance in implementing each instructional activity with students. Teacher resources include background in math content, explanations of why students need to understand this content, and "Unpacking the TEKS" features that clarify what students need to know in order to master the TEKS content and process standards as well as the ELPS and College and Career Readiness standards. Each topic includes a pacing and differentiation guide. Guides provide strategies for English language learners, pre-Advanced Placement students, and each RtI tier. Point-of use resources include teaching notes, questioning guides, checks for student understanding, and answer keys. Click here to register for a demo for any one of these resources.
Pearson Education's Texas Social Studies Series provides comprehensive middle school and high school social studies courses in print and digital formats. Courses support instruction in blended learning and flipped classroom environments and include U.S. History, World History, Economics, and U.S. Government at the high school level and Contemporary World Cultures, Texas History, and U.S. History at the middle school level. Instruction in each course uses storytelling and project-based learning to engage students in social studies and to make content relevant. Click here for a video overview.
Content is structured using a four-part model: (1) Connect, (2) Investigate, (3) Synthesize and (4) Demonstrate. "Connect" is a start-up activity focused on engaging students in learning. Core instruction is provide in "Investigate" activities that include narrative text, videos, and interactive maps and learning tools. "Synthesize" activities allow students to apply what they have learned. "Demonstrate" includes tests that provide customized remediation or enrichment and performance-based assessments.
Click Picture for a Pearson Demo [Source: Pearson]
Courses include "My Story" videos provided by NBC Learn. My Story videos focus on the lives and experiences of individuals who lived in a particular historical period or geographic region. Other course resources include "Flipped Videos," "Interactive Reading Notepads" and "Interactive Primary Source Documents." Flipped Videos may be used to flip instruction, provide remediation, or to support students who have been absent. The Interactive Reading Notepad identifies new vocabulary, clarifies learning objectives and provides comprehension questions to support the close reading of informational texts. The Primary Source Document resource includes interactive tools and prompts to help all students understand and analyze primary source materials.
Teacher resources include comprehensive lesson plans with overviews of all print and online instructional materials. Lesson plans provide step-by-step guidance in implementing activities with students and point-of-use TEKS and ELPS, differentiation strategies, and professional development videos. Click here for a Pearson demo.
The State Board of Education today adopted a list of 405 science, math and technical application instructional materials, 303 of which state review panels determined to be 100 percent aligned to the Texas Essential Knowledge and Skills (TEKS). The remaining 102 state-adopted products are aligned with between 50 and 100 percent of the TEKS. Pearson's Biology and Houghton Mifflin's Environmental Systems materials were conditionally adopted, pending reconciliation of alleged factual errors. By subject, the state-adopted list contains:
93 Grades K-8 math products;
20 Grades K-5 Spanish math products;
221 Grades K-12 science products;
65 Grades K-5 Spanish science products; and,
22 Grades K-12 technology application products.
Within each subject, the State Board adopted on average:
8 math products per grade in grades K-5;
15 math products per grade in grades 6-8;
17 science products per grade in grades K-5;
23 science products per grade in grades 6-8;
5 IPC, 13 Physics, 18 Chemistry, and 9 Biology state-adopted products, as well as a few products in various other high school science courses;
2 technology application products per grade for K-8 as well as in the adopted high school courses;
In most grade levels, a couple of state-adopted products are available in both print, blended and/or digital formats. All others are unique products. Districts must now review the available instructional materials to determine which (1) best meet students' needs, (2) can be implemented effectively given the district's technical infrastructure, and (3) reflect the community's values. An article in the January 2014 edition of TxASCD's Leaders of Learners will highlight specific, researched-based criteria districts should consider during their review of instructional materials.
Learning List is a new service available to assist districts with the challenge of reviewing so many instructional materials. Like a type of Consumer Reports/Angie's List for K-12 instructional materials and online courses, LearningList.com features three types of reviews for each product: (1) an independent alignment to the TEKS; (2) an editorial review that highlights the types of information to help educators determine which products are best suited to meet their students' needs; and (3) subscriber ratings and reviews. Tools on LearningList.com promote a collaborative selection process among district employees and help the district document compliance with the "100 Percent Rule". Beyond the selection process, the alignment reports on LearningList.com help teachers develop TEKS-aligned lesson plans for each reviewed instructional material. Contact info@LearningList.com to learn more about this new service. | 677.169 | 1 |
Unified Mathematics -2
Author: Dr. Ram Mohan
Binding: Paper Back
Publisher: G.Ram Books Pvt. Ltd.
ISBN: 978-93-83510-87-0
Availability: In Stock
Edition: 2015
description
Unified Mathematics: A series from class 1 to 8 with an innovative approach. • Well graded content to develop the required mathematical skills. • Emphasis on concept building and sufficient practice. • "Warm up" helps in connecting the previous learning to the present. • Inter-disciplinary approach is one of the unique ingredients of our series as most of the problems connect other subjects to mathematics or vice versa. • Higher order thinking skills (HOTS) are embedded within the exercise to ensure a phobia-free learning environment. • Review worksheets and activity time are appropriately placed. • Lab Activities gives the necessary reinforcement to the concepts learnt. • "Common errors" and "Important facts" are highlighted in every unit as "Remember!" • Sample Test Papers to assess the progress of the learners. | 677.169 | 1 |
Course Overview The Middle School Algebra course is designed for math profient students that are ready to take an advanced track of mathematics education, preparing them for high school. Course topics include: Algebraic Expressions and...
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Course Overview The 8th Grade Math course is designed to help students become High School ready by ensuring they have a firm understanding of middle school mathematical concepts. Course topics include: Algebraic Expressions with Integers...
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Course Overview Grade 6 Math Plus bridges the gap between elementary and middle school, differentiating the instruction so that struggling students are given the help and knowledge they need to master the concepts, while advanced...
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Course Overview In the Acellus Grade 7 Math course, after a thorough review of basic operations, Ms. Frogge demonstrates how to solve real-world problems by applying basic concepts of Algebra and Geometry and by using... | 677.169 | 1 |
Course Summary
Make sure you're ready for the HiSET Mathematics subtest by completing our helpful prep and practice course. These short text and video lessons highlight the important topics you could be tested on, such as geometry, algebra and arithmetic, so you can get the best score possible on exam day.
About This Course
As you complete the lessons in this course, you will learn and/or review the mathematical principles a high school graduate is expected to know. This course is specifically designed to help you get the best score possible on the HiSET mathematics subtest. The chapters group together lessons covering specific areas of math, and they are ordered in a logical progression to establish a concrete mental framework of math for you to take into the test center with you. The topics of these chapters include:
Mathematical reasoning and number properties
Fractions, decimals, percents, proportions and ratios
Basic math expressions and equations
Geometric figures, theorems, and proofs
Foundational geometry and geometric solids
Probability and statistics
Algebra and algebraic expressions
Linear equations, radical expressions
Sampling, polynomials, functions, correlation and summarizing data
Matrices and vector operations
About the Exam
The HiSET is a high school equivalency exam available in several states to test an examinee's knowledge of secondary-level education topics and college readiness. The HiSET Mathematics subtest can be taken with the other five tests or on its own. It is administered on paper and by computer, depending on the location and your preference if the option is provided. The math test has 50 multiple-choice questions which you will be given 90 minutes to answer. You will not be allowed to take any materials into the test center beyond the identification and appointment confirmation that you are required to present.
Preparing & Registering for the HiSET Mathematics Test
This course is designed to be a comprehensive preparation suite to get you test-ready in a low-stress, fun environment. The videos are short and to the point, and they are set to entertaining graphics to hold your attention and increase retention of key topics. As you take the exams and quizzes, you will practice your test-taking techniques, get experience with the types of questions you'll see on the real test, and see how you're doing in each area to determine whether you need more review time.
To register, you must first find out if the HiSET is available in your state and if there is a testing center near you. You can accomplish both on the HiSET website, along with completing the registration process for those jurisdictions which allow online registration.
Scoring the HiSET Mathematics Test
Your scores for the HiSET math test will be posted to your account on the website in 3-5 business days (possibly longer for paper tests). You will receive a scaled score from 1-20 with 8 being the minimum to pass a given subtest and a total score of 45 out of 100 across all tests needed to pass the entire test | 677.169 | 1 |
5th Edition of Applied Calculus continues to exhibit the same strengths from earlier editions including a focus on creative conceptual and modeling problems and the "Rule of Four", an emphasis on concepts and modeling, exposition that teaches a flexible approach to technology. This issue provides readers with deeper skills needed to apply calculus on the job and highlights connections with real-world concerns. The problems and exercises are challenging and provoke deeper thinking to help apply math in new ways. The material is presented in a way to help readers decide when to use technology, which empowers them to learn what calculators/computers can and cannot | 677.169 | 1 |
W. B. Vasantha Kandasamy
Format: Paperback
Language: 1
Format: PDF / Kindle / ePub
Size: 8.73 MB
Downloadable formats: PDF
Same idea to get max cardinality, but use rank(G). Then 2 + 4x + 3y = 12 y = 6. which is row3(A) · col2(B). so x = −2 and y = 6. then the (3. Para otros usos de este término, véase Álgebra sobre un cuerpo. The most basic application of matrices is solving systems of linear equations. Go to folder FSharp-1.9.7.8\source\fsppack\FSharp. Using the definition of matrix addition and Table 1. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental.
Like addition and subtraction we can multiply both sides of an equation by a number, c, without changing the equation. for any non-zero c. C.7 This worksheet is designed to give students extra practice at using their. Second calling something a real problem in a diary runs into a reaction which youve. Step 3: Eliminate the SAME variable chosen in step 2 from any other pair of equations creating a system of two equations and 2 unknowns. In a linear equation this unknown quantity will appear only as a multiple of x, and not as a function of x such as x squared, x cubed, sin x and so on.
The only way we can guarantee that is if we are adding opposites. This article reviews the basics of linear algebra and provides the reader with the foundation required for understanding most chemometrics literature. Then A + B = _1 + 0 −2 +2 4+ (−4) 2 + 1 −1 +3 3+ 1 = _1 0 0 3 2 4. in the set of all rational numbers. only when A and B are of the same size. ―The World's Largest Matrix Computation: Google's Page Rank Is an Eigenvector of a Matrix of Order 2. and Murray Browne.
Kenneth Hardy
Format: Paperback
Language: 1
Format: PDF / Kindle / ePub
Size: 14.52 MB
Downloadable formats: PDF
Students in beginning and seminar-type advanced linear algebra classes and those seeking to brush up on the topic will find Zhang's plain discussions of the subject's theories refreshing and the problems diverse, interesting, and challenging. Slope and coordinates solver and graphing, graph of a hyperbola with same denominator, how to solve ANY quadratic, TI 83-plus dec bin, sample combination problems. The Johns Hopkins University Press.org/encyclopedia/LinearAlgebra. 2000).
For example, Unit 1 should take you about 32.5 hours to complete. At the end of the first year. the distribution of the market is said to be stable. date= ( bantuan ) Leduc, Steven A. (May 1, 1996), Linear Algebra (Cliffs Quick Review), Cliffs Notes, ISBN 978-0822053316 Check date values in: After you are familiar with the name, definition and operations, you will find that the notes and properties listed are useful for second reading or whenever you need to come back to the particular topics.
Mix B contains 60% plant food and 20% weed killer. Today, I finally have some time to dig into it, get it compiled and know how to start the linear algebra service provider. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Strang is great, the way he simplifies the complex concepts... Porém, se a diagonal principal for constituída toda de zeros, matriz passará ser uma matriz nula. 3.
These are deprecated and will eventually be removed. These operations are useful in the applications of matrices. the numbers 23 and 46 are called equal although they are not represented in the same manner. then the sum of A and B is the m × n matrix C = ci j _. if corresponding elements are equal. R Kaye and R Wilson, Linear Algebra, OUP (S 2.897 KAY). I'm going to go ahead and multiply equation (1) by -1 and then add the equations together: Step 2: Choose to eliminate any one of the variables from any pair of equations. | 677.169 | 1 |
One of the claims of the analytical school of western philosophy is that math is reducible to logic, specifically the logic of groups, classes, or sets. In this vein, I can think of no better introduction than Russell's Principles of Mathematics.
This book provides a comprehensive, in-depth explanation of the basic concepts and interpretations involved in chromosome analysis, a critical technique in the diagnosis, prognosis, and monitoring of a wide variety of conditions.
The Principles of Quantum Theory, From Planck's Quanta to the Higgs Boson: The Nature of Quantum Reality and the Spirit of Copenhagen by Arkady Plotnitsky English | 2016 | ISBN: 3319320661 | 313 Pages | PDF | 7.2 MB
The book considers foundational thinking in quantum theory, focusing on the role the fundamental principles and principle thinking there, including thinking that leads to the invention of new principles, which is, the book contends, one of the ultimate achievements of theoretical thinking in physics and beyond.
Content on this page is geared towards teaching the syntax of the language of mathematics, the rules and principles that we use in math. See Math in Real Life for a look at how we can use this information to enhance our lives. | 677.169 | 1 |
From Graphi x
Mathematical Concepts and Methods in Modern Biology offers a quantitative framework for analyzing, predicting, and modulating the behavior of complex biological systems. The book presents important mathematical concepts, methods and tools in the context of essential questions raised in modern biology. Designed around the principles of project-based learning and problem-solving, the book considers biological topics such as neuronal networks, plant population growth, metabolic pathways, and phylogenetic tree reconstruction. The mathematical modeling tools brought to bear on these topics include Boolean and ordinary differential equations, projection matrices, agent-based modeling and several algebraic approaches. Heavy computation in some of the examples is eased by the use of freely...
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Birdy starts every day by smiling at herself in the mirror. She says you can smile while doing just about anything-- brushing your teeth, taking out the garbage, or eating broccoli. Okay, maybe not while eating broccoli. Even people with bad teeth (like our first president, George Washington) should show their toothy grins because there's no such thing as a bad smile. So heed Birdy's advice and practice your smile--you'll need it while reading this book
Smiling teenage girl holding a book Smiling teenage girl holding a book | 677.169 | 1 |
TES Maths: Inspect the spec - precalculus
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Top resources to help you to explore the changes to teaching precalculus as part of the new GCSE specification
Everyone is talking about functions and frequency trees, but what else has changed with the advent of the new specification? And what resources are available to help? Throughout this series, TES Maths aims to find out.
All students will develop confidence and competence with the content identified by standard type
All students will be assessed on the content identified by the standard and the underlined type; more highly attaining students will develop confidence and competence with all of this content
Only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content.
A15h: Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts.
R15h: Interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts.
What's the same?
While the concepts of estimating the gradient of a curve and the area under a curve are technically new to the specification, the skills that underlie them are in fact old classics.
To estimate the gradient of a curve, students need to be able to draw a tangent and work out its gradient. They should know the former from their work with circle theorems; the latter essentially involves calculating the gradient of a straight line.
To estimate the area under a curve, students need to break it up into trapeziums and then calculate their area. This, of course, is something that they have been doing since Year 7.
What has changed?
Obviously it would be a bit risky to assume that students will piece together the separate skills themselves, so they will need plenty of practice in drawing tangents and estimating areas.
Likewise, the context in which these questions could be asked may be unfamiliar. Hopefully, they should already know that the gradient represents speed on a distance-time graph or acceleration on a velocity-time graph. But they will also need to know that the area under a velocity-time graph represents the distance travelled. Students may be tested in financial and other contexts and will need to adapt their skills accordingly.
Additionally, students should be able to say whether their trapezium-based estimate is an under- or an overestimation of the true area and illustrate this using a diagram.
Finally, it is worth pointing out that knowledge of calculus, namely differentiation and integration, is not required as part of this specification.
How can TES Maths can help?
As ever, the wonderfully talented authors of the TES Maths community have stepped up to the mark to lend a hand. Here is a selection of my favourite resources to help support the teaching of this topic:
Gradients on a curved graph lesson
Designed for the IGCSE but easily adapted to suit any exam board, this complete lesson offers a step-by-step approach to estimating the gradient at points on a curve. | 677.169 | 1 |
Mathematics for junior high school by School Mathematics Study Group(
Book
) 42
editions published
between
1959
and
1961
in
English
and held by
250 WorldCat member
libraries
worldwide
This is a supplementary smsg mathematics text for junior high school students. Key ideas emphasized are structure of arithmetic
from an algebraic viewpoint, the real number system as a progressing development, and metric and non-metric relations in geometry.
Chapter topics include sets, projective geometry, open and closed paths, finite differences, and formulas. (Mp)
Concepts of informal geometry by School Mathematics Study Group(
Book
) 5
editions published
in
1960
in
English
and held by
77 WorldCat member
libraries
worldwide
A brief course in mathematics for elementary school teachers by School Mathematics Study Group(
Book
) 3
editions published
in
1963
in
English
and held by
71 WorldCat member
libraries
worldwide
Designed to give teachers a good comprehension of elementary maths and methods of teaching it. Includes answers | 677.169 | 1 |
Screenshots
Details
Developed by Loughborough University, the mathscard app contains hundreds of examples of pure maths formulae and graphs/diagrams. Designed specifically for AS and A2 Level maths, the mathscard app is based on the hugely successful award-winning mathscard fold-out formulae sheet and is designed to help students with their exam revision when at home or on the move. Vectors, numerical methods, circle and coordinate geometry, sequences and series, algebra and graphs, trigonometry and calculus are all covered in this handy resource.
mathscard a-level's review
Study with the help of this app and start understanding everything about maths
7
Useful
Easy to use
Great design
Doesn't move to SD
"Want to pass that math exam?"
Mathscard a-level is an Android application created by Loughborough University that includes many examples of math formulas and graphs that will help students study and review for their exams.
This is appropriate for AS and A2 math level and for using it you just need to tap on the subject you want to read about and navigate through the content (Vectors, Numerical methods, Circle & Coordinate geometry and some more). In addition, there's also an index with all the contents included in the app in case you want to look through it and decide depending on your mood.
This tool is very useful for students and people who enjoy maths and want to review their knowledge. The design is great, the interface is clean and easy to use and there are no bugs, apparently. If you were looking for a tool that helped you study maths, this will be the perfect choice.
Related Apps
Most Comprehensive Mathematics formulae application in existence. Unlike other mathematical formulae application this app virtually has all the mathematical references that you will ever need from basic to advance level.Better yet have them on your mobile.
Some of the topics include Quick theorems, tables, identities, proofs and graphs for arithmet…
This is the perfect free learning app for maths students to study maths formulas anywhere, anytime using their phone/tablet. It is designed specifically for New South Wales Higher School Certificate Mathematics (2 unit) students and year 10 students.
It is the ideal app for the students to
• study formulas on the go, in their spare time (such as…
This is an app for MF15 formulae for A-Level maths.
Special attention is given for user-friendly, clean navigation - with no fancy stuff.
Formulae appear in full screen on your mobile. Zoom-in is possible for better reading. Following formulae are included:
.. Binomial Expansion
.. Maclaurin's Expansion
.. Partial Fractions Decomp…
Developed by Loughborough University, the mathscard GCSE app contains hundreds of examples of maths formulae, graphs and diagrams. The GCSE app is based on the hugely successful Loughborough A-level app and is designed to help students with their exam revision when at home or on the move. Number and Arithmetic, Algebra, Graphs, Statistics and Proba…
Developed by Loughborough University, the free mathscard app contains hundreds of examples of pure maths formulae and graphs/diagrams. Designed specifically for AS and A-Level maths, the mathscard app is based on the hugely successful award-winning mathscard fold-out formulae sheet and is designed to help students with their maths study and exam re… | 677.169 | 1 |
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FIND YOUR TEXTBOOK FIND YOUR TEXTBOOK Our lessons are aligned to approximately 200 of the most popular math textbooks. Can understanding how to work with one kind of problem help understand how to work another type. What Polya 26 describes in How to Solve It is far more complex than any theories we have developed so far. What are 5 ways to measure the weather. And just as word-based riddles exploit the ambiguities of language, so do these math problems. She free printable maths worksheets for grade 1 our second daughter through guardianship. Assume the condition cond using assume. While the system provides a simple and intuitive tool for you and your hero to negotiate a mutually acceptable settlement, if no agreement can austin physics reached regarding an appropriate refund amount, then the site administrator will help arbitrate and ultimately decide on an appropriate refund amount, if any. Common Core State Standards, CCSS, is a system where materials. Exercises in Spanish Workbook 0618353321 Explorations and Websites to learn french for free Benjamin Spanish - Mrs. The transaction costs of exchanging the won for dollars are now very high. If the base of the ladder is 7 ft from the wall, how high up the wall will the ladder reach. Recommended for: average students in grades 10-12 Suggested instructional time: one year Consultant services: Consultant services and staff development may be arranged geometry 11 2 answers a mutually convenient basis at no cost to the school district. Just focus on what he can do.
Meritnation is offering step by step solutions for RS Aggarwal textbook for CBSE Class 9 Math subject. We all have websites to learn french for free innate ability to learn some really incredible things…with a lot of effort. In particular, kumon sample study guides, Free Worksheets for 7th Grade, McDougal Littell math algebra 1 book. There are, however, decision-forcing cases in which students play the role of a single protagonist who is faced with a series of problems, two or more protagonists dealing with the same problem, and more. If you are interested in learning more about institutional use of the NROC Library within your school, district, glencoe algebra 1 answer key. Helpful worked examples and diagrams are given. Thompson 3 years ago 8. JM -uurnlooJaMsue aql u. Ask and answer questions about key details in a text. Write an absolute value inequality that represents the mass range of the African elephant. Enter the left multiplying and dividing fraction calculator of the equation to be solved in E1. You can find websites to learn french for free problem and any diagrams on hotmath. Contact - information andersonlibrary. In addition, review scientific notations. Year 11 Write on. Question CPM: Answer the three parts of question 6 under problems on page 412 of your book. Read Online Download PDF - LESSON Practice 11 - cvhs. Solutions of Systems of Equations and Inequalities. Ellenbogen, Marvin L Bittinger, Barbara L. | 677.169 | 1 |
Posts Tagged "High School MathLearning List has reviewed Cosenza & Associates' Algebraic Reasoning. This comprehensive product supports instruction in Texas algebraic reasoning courses. Content is available in print and digital formats and includes additional online tutorials. Instruction is inquiry based and covers the attributes of functions, systems of equations, modeling with functions and data, and approaches to solving equations. Teacher and student materials contain clear links to the Texas Essential Knowledge and Skills (TEKS) and English Language Proficiency Standards (ELPs).
Resources include blackline masters of four versions of mid-chapter and end-of-chapter assessments, including a modified version for students with special learning needs. Editable versions (i.e., Word versions) of a mid-term and final exam and a culminating course project are available. Each test includes multiple-choice and free response items. The culminating project requires students to independently identify and analyze a data set related to a topic of interest. Teachers also have access to the ExamView® assessment software that allows users to create customized assessments.
Learning List has reviewed Apex Learning's Texas Geometry. This is a comprehensive, online geometry product for Texas students. Core content is presented online and includes some printable materials. The course focuses on developing students' conceptual understanding, computational skills, and proficiency in solving problems. Resources support instruction in self-paced, remediation, and credit-recovery programs.
Texas Geometry is organized in two semesters. The first semester's content is presented in five units that cover the foundations of geometry, triangles, right triangles, quadrilaterals and other polygons, and circles without coordinates. The second semester presents content in four units that address coordinate geometry, constructions and transformations, three-dimensional solids, and applications of probability. Across semesters, students learn to reason mathematically and to use mathematical models and tools to solve real-world problems.
Each unit begins with a short video introduction that frames the real-world applications of what students will learn and connects new content to prior learning. Subsequent instruction is provided through a set of online activities and, where appropriate, accompanying worksheets. Each unit and semester ends with two versions of a unit/semester exam—one version presents open-ended questions and the second is made up of multiple choice items.
About Apex Learning*
Apex Learning's digital curriculum is designed to support all students in achieving their potential, from those struggling with grade-level content to those capable of accelerating their learning. The curriculum is designed to actively engage students in learning—combining embedded supports and scaffolds to meet diverse student needs, actionable data to inform instruction, and success management, to ensure students get the outcomes they need.
*Information in this section is provided by or adapted from Apex Learning.
Subscribe to Learning List for access to the spec sheet, full editorial review and detailed alignment report for this material, and thousands of other widely used Pk-12 resources. | 677.169 | 1 |
AMPL is a language for large-scale optimization and mathematical programming problems in production, distribution, blending, scheduling, and many other applications.
Combining familiar algebraic notation and a powerful interactive command environment, AMPL makes it easy to create models, use a wide variety of solvers, and examine solutions. Though flexible and convenient for rapid prototyping and development of models, AMPL also offers the speed and generality needed for repeated large-scale production runs. Applied and Computational Linear Algebra: A First Course This book is intended as a text for a graduate course that focuses on applications of linear algebra and on the algorithms used to solve the problems that arise in those applications. Often the particular nature of the applications will prompt us to seek algorithms with particular properties; we then turn to the matrix theory to understand the workings of the algorithms.Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice The author's goal is to start a dialogue between mathematicians and cognitive scientists. He discusses, from a working mathematician's point of view, the mystery of mathematical intuition: why are certain mathematical concepts more intuitive than others? To what extent does the "small scale" structure of mathematical concepts and algorithms reflect the workings of the human brain? What are the "elementary particles" of mathematics that build up the mathematical universe? Mathematical Reasoning: Writing and Proof Mathematical Reasoning: Writing and Proof is a text for the first college mathematics course that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics. Mathematical Omnibus: Thirty Lectures on Classic Mathematics This is an enjoyable book with suggested uses ranging from a text for a undergraduate Honors Mathematics Seminar to a coffee table book. It is appropriate for either It could also be used as a starting point for undergraduate research topics or a place to find a short undergraduate seminar talk. This is a wonderful book that is not only fun to read, but gives the reader new ideas to think about. Elementary Differential Equations with Boundary Value Problems Written in a clear and accurate language that students can understand, this book minimizes the number of explicitly stated theorems and definitions. It deals with concepts in a conversational style that engages students. It includes more than 250 illustrated, worked examples for easy reading and comprehension. Applied Discrete Structures Applied Discrete Structures is a two semester undergraduate text in discrete mathematics, focusing on the structural properties of mathematical objects. These include matrices, functions, graphs, trees, lattices and algebraic structures. The algebraic structures that are discussed are monoids, groups, rings, fields and vector spaces. Discrete Mathematics for Computer Science This book gives an introduction to discrete mathematics for beginning undergraduates. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. Many examples of formal and informal proofs are given. Exploring Discrete Mathematics With Maple This is the first supplement in discrete mathematics to concentrate on the computational aspects of the computer algebra system Maple. Detailed instructions for the use of Maple are included in an introductory chapter and in each subsequent chapter. Book of Proof, 2nd Edtion This book is an introduction to the language and standard proof methods of mathematics. | 677.169 | 1 |
-Wesley Longman
Supplemental materials are not guaranteed for used textbooks or rentals (access codes, DVDs, workbooks)....
Show More mathematical principles in this way can be both exciting and fun. Topics covered include basic calculations, statistical analysis, plotting, graphs, functions, simple algebra, trigonometry and modelling. A wealth of examples and practice exercises reinforces the student's understanding of key concepts, while challenging puzzles and brain-stretchers encourage further exploration. Numerous screendump illustrations and key press instructions provide guidance and reassurance and enable the student to exploit the powerful capabilities of the TI-83. Calculator not | 677.169 | 1 |
Mathematics
In the field of Computer Science the ability to quickly identify and use mathematical principles is a very important and necessary skill. All algorithms have their roots in Mathematics and particularly Calculus, limits and derivatives are a major part of Artificial Intelligence, Optimization and Natural Language algorithms. We as computer scientists must strive to keep up a close relationship with maths, however sometimes that relationship is strained because of other duties or just the effect of time on our brains. We often get to a point where the solution to a mathematical problem is more important than the elegance of the solution itself.
Whether it be for personal gratification, curiosity or homework related 🙂 sometimes we would like to have a step by step explanation of the problem along with its solution. This post demonstrates how to solve a univariate limit using the Maple 16 Limit Methods Tutor.
The first thing we will do is to open the Maple software.
Next enter Student[Calculus1][LimitTutor](); in the Maple editor and hit the Enter/Return key.
We will use the Limit tutor to solve the limit as taken from the book CALCULUS Ninth Edition: Dale Varberg, Edwin J. Purcell, Steven E. Rigdon: Page 64, example 3. -> (2*x^2-3*x-2)/(x-2)
Enter the equation as shown into the Function box, set "at" to 2 then hit the Start button.
Clicking on the Next Step button will proceed to the next step of the solution while providing helpful hints on the way.
The All Steps button will display all the steps and the final solution. Have fun with math 🙂 | 677.169 | 1 |
Browse related Subjects ...
Read More in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 11th edition includes new problems, updated figures and examples to help motivate students. The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two or three semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.
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Customer Reviews
Text Book
It's a text book. It meets the requirements of the course.
Starwalk
Apr 28, 2007
First/Second Year Differential Equations
I bought this book because I found I was hogging a copy from the university library. If you are doing differential equations courses in either a mathematics or possibly a physics degree, then I would recommend this as it has in pretty much of anything I would imagine you would need, at least at first and second year level (I'm second year, so cannot give an opinion about use for third/fourth years) There are sections covering both numerical and analytical solutions. Yes, the text can be a bit dry in places, but it never treats you like an idiot, and I don't think many people read books like this for fun.
Looking for extra problems to work through? Buy this book. Lecturer is an idiot and/or lecture notes are poor? Buy | 677.169 | 1 |
1236 p. Col. Illustrations, col. maps. Intended for college/higher education audience. This text is designed for a three-semester or four-quarter calculus course (math, engineering, and science majors). 'Thomas' Calculus' helps you reach today's students by developing conceptual understanding while offering the right mix of relevant applications, conceptual exercises, and skills practice. | 677.169 | 1 |
Archive Pages Design$type=blogging
Udemy Coupon 100% Off - MATLAB 2015
Udemy Coupon : Learn MATLAB programming and how to use basic math, vectors, matrices, calculus, graphics and much more. In one course!
MATLAB is a programming language developed by MathWorks and is one of the most important computation languages. The course content covers all the required topics for starting MATLAB. In this course, you will learn how to program with MATLAB and after you finish this course you can:
Do basic calculation with MATLAB
Create variables, vectors and matrices
Do advanced operations with matrices and vectors
Create simple and advanced programs to solve your problems
Plot your data set in 2D and 3D graphs
solve algebraic equations and systems
Solve calculus problems like limits
we add bonus lecture every month
finally you will get all the source codes you need
This is your complete guide. I guarantee you the full-money back,If you are not satisfied with the course! No Questions Asked! | 677.169 | 1 |
...
Show More of the relationship between mathematics and physical problems. It also emphasizes problem solving and provides a thorough approach to solutions. The third edition of , Elementary Applied Partial Differential Equations; With Fourier Series and Boundary Value Problems has been revised to include a new chapter covering dispersive waves. It also includes new sections covering fluid flow past a circular cylinder; reflection and refraction of light and sound waves; the finite element method; partial differential equations with spherical geometry; eigenvalue problems with a continuous and discrete spectrum; and first-order nonlinear partial differential equations. An essential reference for any technical or mathematics professional | 677.169 | 1 |
Algebra I: Functions as Graphs
Be sure that you have an application to open
this file type before downloading and/or purchasing.
2 MB|14 pages
Product Description
This zipped file contains three documents:
1. A smartboard notebook file that can be integrated with the Larson Algebra lesson on Functions as Graphs
2. A Do Now
3. A graph/table recording sheet
Through this lesson students begin to understand that there are 5 different represents for a function: a set of input/output tables, a graph, an equation, a verbal description, and a set of coordinates.
Pull tabs along the side of each page give you instructions for how to work with each page.
To make the smartboard file completely interactive you must have the math tools installed. | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
5 MB|18 pages
Product Description
Reason about and solve one-variable equations and inequalities be analyzing real life situations and direct analysis.
In this bundle you get.
Lesson 1/Classwork
Homework
Lesson 2/Classwork - this has an increase in rigor
Homework
Assessment (check-in)
The emphasis here is on the process.
Standard: 6.EE.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | 677.169 | 1 |
Math 2 and 2T
UCSC currently offers two versions of Math 2: Adaptive Learning and Collaborative Learning. If both versions fit your schedule, choose the one that best fits your learning style. If only one version fits with your other courses, that's fine; both cover the same core algebra skills.
Math 2: Adaptive Instruction
Instructional studio sessions five days per week. No conventional lectures!
Group discussions and projects focus on the social context of mathematics and mathematical communication; projects combine creative writing, interpretation of images, and traditional mathematical language. Individual study, supported by one-on-one mentoring, follows a personalized learning path in ALEKS.
Math 2: Collaborative Learning
Lectures three days per week and mandatory discussion sections twice per week.
Students actively participate in lecture, using iClickers to check their comprehension of new material. In section, study teams work collaboratively on worksheets that complement the online homework assignments.
Math 2T: Preparatory Math
Math 2T is a two unit course consisting primarily of individual study in ALEKS, supported by one-on-one mentoring.
Math 2T is designed to develop productive learning strategies and habits of mind. Satisfaction of weekly time and progress goals is essential for success in the course.
Math 2T is not recommended for students whose best ALEKS PPL score after reassessment is below 45.
Topics covered in Math 2 and 2T
Both Math 2, College Algebra, and Math 2T,Preparatory Math, use the adaptive learning system ALEKS for individual practice and learning. Projects, worksheets, etc. vary from course to course—contact the instructor for more information. (See the Mathematics Class Schedule for information about instructors, scheduling, etc. for a specific academic term—check the fine print below the page title to be sure you're looking at the relevant quarter's listings.)
Topics covered in the ALEKS College Algebra course:
Real Numbers
Fractions
Percents and Proportions
Signed Numbers
Properties of Real Numbers | 677.169 | 1 |
High School
Mathematics is a top priority course in all High School programs. Use Online Math Tutors to show you how you can succeed in Math.
To be succesful in most high school math courses one needs a good calculator. For algebra 1 and algebra 2, one does not need a graphing calculator but one that has a square root function and pi is a necessity. Graphing calculators are essential for Pre-calculus and Calculus.
Check out this resources for the best calculator to meet your needs.
It is truely the best calculator for algebra
You can store up to 15 functions at runtime and give them names.
You can do all sorts of algebriac operations
You can do integration and differentation
You can write your own programs that manipulate algebriac functions
You can save/edit your work
You can print out your work
Find a great software to calculate expressions, roots, extremes, derivatives, integrals, and much more.
This is EDITION: DCG Graphing Edition
FEATURES: The Graphing Edition of the DreamCalc virtual calculator for your PC or laptop, is ideal for students and scientific users. You'll get the intuitive feel and productivity of using a professional hand-held calculator, but one which adapts to your way of working.
With DreamCalc, you'll be able to graph functions and plot list data more simply than ever before. In fact, it is a match for many dedicated graphing packages, but far easier to use. Select from Reverse Polish Notation or two styles of algebraic input, and with the optional ability to run in your Windows system tray--DreamCalc will always be there whenever you need to reach for a calculator.
DreamCalc also offers you a full range of scientific functions, statistics, complex numbers, base-n logic, unit conversions, built-in constants and a powerful polynomial solver. Because it is software, it allows you to exchange your results, graphs and lists with your other applications.
It is a "must" for Business, Science, Engineering and Education.
It uses a different approach from a conventional calculator, which is more
natural to the way people calculate. When you calculate on a conventional
calculator, each time you enter a new number you lose sight of the previous
one. And if you make a mistake in the middle of a long expression, you have
to start all over. In CalcSharp, you can see all of what you are calculating,
and expressions can easily be edited. | 677.169 | 1 |
Mathematics
Mathematics is an international language; It is the language of the future. As we understand that we are part of a greater, global community, the U math department partners with our UHSA community to make real connections to prior learning to stir academic curiosity.
Our curriculum is a traditional college-prep sequence. The general pattern is Algebra 1, Geometry, Algebra 2, PreCalculus, and AP Calculus AB. Additionally the U offers AP Statistics and Functions Statistics and Trigonometry (FST). Knowing that our students will be expected to perform, we plan all of our courses with that goal in mind. We expect students to do well on SAT, ACT, MME and college entrance and placement exams. We are committed to student understanding and well-being. We balance the rigor of the material, with the reality of growing teen life. | 677.169 | 1 |
Proximity Learning's Pre Calculus course will begin with general overview topics in Algebra and trigonometry concepts. The course will introduce analytic geometry and calculus concepts such as limits, derivatives, and integrals.
Students will learn:
Review extensions of Integrated Algebra II
Review Probability and Statistics
Assist in further development and knowledge of Trigonometry and Pre Calculus | 677.169 | 1 |
Find yourself stuck on the tracks when two trains are traveling at different speeds? Help has arrived! Math Word Problems Demystified, Second Edition is your ticket to problem-solving success.
Based on mathematician George Polya's proven four-step process, this practical guide helps you master the basic procedures and develop a plan of action you can use to solve many different types of word problems. Tips for using systems of equations and quadratic equations are included. Detailed examples and concise explanations make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning.
It's a no-brainer! You'll learn to solve:
Decimal, fraction, and percent problems
Proportion and formula problems
Number and digit problems
Distance and mixture problems
Finance, lever, and work problems
Geometry, probability, and statistics problems
Simple enough for a beginner, but challenging enough for an advanced student, Math Word Problems Demystified, Second Edition helps you master this essential mathematics skill | 677.169 | 1 |
Help Me Find Out About the Sources with Calculus Homework Solver
Calculus is a branch of math studying change; a very brief description for a very complex subject. It is a subject that many students need help with due to the complexity of the problems. There are several sources you can use to help you with your calculus homework. Some of these sources even offer a homework solver; a simple java based program that allows you to input your problem then helps you solve the problem. They offer step-by-step instructions on how the answer was found.
Calculus can be challenging for any student, especially if they are just starting. This complex subject usually has tutors available at your local school. There are also online tutors who can help you learn how to do your calculus homework. This is the best choice since someone sits down and shows you step by step how the problem was solved. You will not have a tutor or application to help you solve calculus problems during an exam. Learning the material and how to solve the problem is your best bet for gaining credit for the class.
There are many websites that offer calculus problem solvers in the form of a web browser application. Some of them are java script while others are HTML5 technology. Either way, you are able to input your problem into the application to receive the answer. Some of these offer the student step by step instruction on how to solve the problem, other just give you the answer without and explanation of how to get the answer. It is not recommended to use these types of applications since you will not be able to use the application during an exam.
There are some applications that help you solve the problem but make you do the steps to come to the answer. This can help you learn the material much like a tutor can. When you input the problem, you are prompted to go to the first step to solving the problem. If you do not take the right step, the application will point you in the right direction. This guides the student and helps them learn how they can get the answer on their own. Calculus is a challenging subject but does not have to be impossible to learn. | 677.169 | 1 |
Ultimate guide to word problems
Everything you need to know about solving word problems!
off original price!
The coupon code you entered is expired or invalid, but the course is still available!
Welcome to Ultimate guide to maths word problems. You know worded problems have been problematic for students since the dawn of time. Everyone knows that 2 + 2 gives you 4, but you ask how many apples Jenny has, and all of a sudden you may as well be asking to solve Einstein's riddle.
So because of that big gap in the understanding of these pesky little words draining students' lives everywhere, I decided to create this course. It covers problem solving type questions across a myriad of topics and is split into four different levels. You will work your way up from level 1 (easiest level) and as you gain in confidence, you can take on some of the more challenging questions at levels 3 and 4.
Now if you're someone who has never come across worded problems before because you call home the space below an over-sized rock or because you're the luckiest person on the planet, then let me briefly explain word problems to you.
What are word problems?
We all learn basic concepts like fractions, decimals, algebra etc. at school. We know how to add fractions or simplify algebraic expressions, but what stumps us is having to apply our knowledge to "real life scenarios". These are called word problems. For example, if Lisa gets paid $17 an hour on the weekends and $13.50 per hour on weekdays, how much does she earn if she works for 5 hours during the week and 12 hours on Sunday? Show full working.
This is a typical algebra-style word problem. No idea how to do this? Then let me help :)
A lot of students find these types of practical questions quite challenging to answer as they have trouble linking their basic knowledge of algebra and numbers in this case to the worded problem will the course have?
As I mentioned earlier, the course will have four different levels, slowly working its way up in difficulty. I will go through a number ofproblem solving / word problem questions, split into the different topics listed below. I go through each question in a systematical step by step manner making it super easy to follow along and adopt in your own workings | 677.169 | 1 |
This course can help you to use math in your everyday life; from improving your accounting skills, advancing your career or carefully putting together your household budget
Receive a certificate on successful completion of course
more info
Even Einstein started somewhere. Let Of Course Learning bring you on a journey of discovery within the field of mathematics.
This course offers a unique and interactive approach to learning mathematics. Learn how to critically apply mathematics to real world situations, as well as understand many mathematic concepts and theorems in detail.
Whether you are interested in mathematics as a subject, or if you are a teacher who wishes to improve your classroom approach, this course will be beneficial to you. It doesn't take a mathematical genius to add up that this is a good deal!
The Fine Print
Enjoy by 15 July 2017
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Maximum of one voucher per transaction
All services must be used by one customer
Usual price based on currency conversion checked on 10/04/17. Price is subject to change due to exchange rate fluctuation
Of Course Learning is not a Registered Training Organisation within New Zealand
Not valid in conjunction with any other offer and not redeemable for cash | 677.169 | 1 |
Interactive Algebra!
Intro to Algebra: Algebra Interactive Notebook Pages
15 Easy to use interactive notebook pages on introduction to Algebra. Includes topics such as combining like terms, order of operations, expressions, and more. Photographic examples and instructions included for each page! Print, Cut, Glue, Use! #Algebra
Midpoint and Distance Formula Foldable
Distance and Midpoint Formula Foldable - a great addition to geometry or algebra interactive notebooks - 3 versions for differentiation | 677.169 | 1 |
Building
on their work with linear, quadratic, and exponential functions in Algebra I,
students extend their repertoire of functions to include polynomial,
rational, and radical functions. In this Algebra II course, rational
functions are limited to those whose numerators are of degree at most 1 and
denominators of degree at most 2.Radical functions are limited to square roots or cube roots of at most
quadratic polynomials.Certain
standards in this course require students to revisit the topics of linear,
quadratic and/or exponential functions to build conceptual understanding.
Unit One develops the structural similarities between the system of polynomials
and the system of integers. Students draw on analogies between polynomial
arithmetic and base-ten computation, focusing on properties of operations,
particularly the distributive property. Students connect multiplication of
polynomials with multiplication of multi-digit integers, and division of
polynomials with long division of integers. Students identify zeros of
polynomials, including complex zeros of quadratic polynomials, and make
connections between zeros of polynomials and solutions of polynomial
equations. The unit culminates with the fundamental theorem of algebra.
Rational numbers extend the arithmetic of integers by allowing division by
all numbers except 0. Similarly, rational expressions extend the arithmetic
of polynomials by allowing division by all polynomials except the zero
polynomial. A central theme of this unit is that the arithmetic of rational
expressions is governed by the same rules as the arithmetic of rational
numbers.Students revisit the topic of
systems of linear and quadratic equations to continue to build conceptual
understanding. (Mathematics Appendix A, p.36-38, with adjustments)
·Reason quantitatively
and use units to solve problems.
·Perform arithmetic
operations with complex numbers.
·Use complex numbers
in polynomial identities and equations.
·Interpret the
structure of expressions.
·Perform arithmetic
operations on polynomials.
·Understand the
relationship between zeros and factors of polynomials.
·Use polynomial
identities to solve problems.
·Rewrite rational
expressions.
·Understand solving
equations as a process of reasoning and explain the reasoning.
·Solve equations and
inequalities in one variable.
·Solve systems of
equations.
·Analyze functions
using different representations.
·Translate between the
geometric description and the equation for a conic section.
State
Standards:
N-Q.A.2Define appropriate quantities for the
purpose of descriptive modeling.
N-CN.A.1Know there is a complex number i such
that i2 = −1, and every complex number has the form a
+ bi with a and b real.
N-CN.A.2Use the relation i2 = –1
and the commutative, associative, and distributive properties to add,
subtract, and multiply complex numbers.
N-CN.C.7Solve quadratic equations with real
coefficients that have complex solutions.
N-CN.C.9(+)Know the Fundamental Theorem of Algebra;
show that it is true for quadratic polynomials.
A-SSE.A.2Use the structure of an expression to
identify ways to rewrite it. For example, see x4 – y4
as (x2)2 – (y2)2, thus
recognizing it as a difference of squares that can be factored as (x2
– y2)(x2 + y2).
A-APR.B.2Know and apply the Remainder Theorem: For
a polynomial p(x) and a number a, the remainder on
division by x– a is p(a), so p(a) =
0 if and only if (x – a) is a factor of p(x).
A-APR.B.3Identify zeros of polynomials when
suitable factorizations are available, and use the zeros to construct a rough
graph of the function defined by the polynomial.
A-APR.C.4Prove polynomial identities and use them
to describe numerical relationships. For example, the polynomial identity
(x2 + y2)2 = (x2 – y2)2
+ (2xy)2 can be used to generate Pythagorean triples.
A-APR.D.6Rewrite simple rational expressions in
different forms; write a(x)/b(x) in the form q(x)
+ r(x)/b(x), where a(x), b(x),
q(x), and r(x) are polynomials with the degree of
r(x) less than the degree of b(x), using
inspection, long division, or, for the more complicated examples, a computer
algebra system.
A-APR.D.7(+)Understand that rational expressions form
a system analogous to the rational numbers, closed under addition,
subtraction, multiplication, and division by a nonzero rational expression;
add, subtract, multiply, and divide rational expressions.
A-REI.A.1Explain each step in solving a simple
equation as following from the equality of numbers asserted at the previous
step, starting from the assumption that the original equation has a solution.
Construct a viable argument to justify a solution method.
A-REI.A.2Solve simple rational and radical
equations in one variable, and give examples showing how extraneous solutions
may arise.
A-REI.B.4bSolve quadratic equations in one variable.
b.
Solve quadratic equations by inspection (e.g., for x2 =
49), taking square roots, completing the square, the quadratic formula and
factoring, as appropriate to the initial form of the equation. Recognize when
the quadratic formula gives complex solutions and write them as a ± bi
for real numbers a and b.
A-REI.C.6Solve systems of linear equations exactly
and approximately (e.g., with graphs), focusing on pairs of linear equations
in two variables.
A-REI.C.7Solve a simple system consisting of a
linear equation and a quadratic equation in two variables algebraically and
graphically. For example, find the points of intersection between the line
y = –3x and the circle x2 + y2 = 3.
F-IF.C.7c Graph functions expressed symbolically and
show key features of the graph, by hand in simple cases and using technology
for more complicated cases.★ | 677.169 | 1 |
Algebra, Past and Present
What exactly is Algebra? Algebra is among the several branches of math. This branch of math employs numerical equations to make sense to the relationships linking any 2 elements that can change over a period of time. Each time a mathmatical equation has to represent something that fluctuates and doesn't remain exactly the same, letters and also symbols tend to be used to characterize the varying amount. This letter or symbol is known as a variable because it changes.
The start of contemporary algebra started out with the ancient Babylonians. These folks were the primary individuals who utilized an progressed mathmatical system for their time. This particular math made use of an algebraic system to accomplish calculations, also it enabled the Babylonians to make use of formulas and estimate the solution by using unknown values for the problem category that would be solved while in the modern-day by means of quadratic equations, linear equations, and indeterminate linear equations. During this same stage in the earliest millennium BC, many of the mathematicians in India, China, Greece, and Egypt were by means of geometric processes to resolve these kinds of equations. The idea of algebra is produced by the Arabic phrase al-jabar, which means reunion of the cracked pieces. A Persian Islamic mathematician called Muhammad ibn Masa al-khwarizmi composed a book in 820 whose title is actually translated to mean The book of Outline Concerning Computing by Transposition and Elimination. There does exist discussion in math communities upon if the mathematician Diophantes, who happens to be referred to as the father of Algebra, or Al-Khwarizmi ought to actually be referred to as the father of Algebra. There is support for Al-Khwarizmi rather than Diophantes since lots of his work with reduction is still currently available these days. He also explained how to solve a quadratic very extensively.
Within the 14 century the phrase algeber referred to bone setting, which is very similar to the initial significance of the actual term. The mathematical form of the phrase which is spelled algebra had made its advent by the sixteenth century. A man by the name of Robert Recorde has been the very first person to utilize the term algebra in its mathematical interpretation. He was additionally the designer of the actual equality symbol(=). The first interpretation of the expression Algebra would pertain today to the sub class of primary algebra. You'll find so many different areas of algebra within the contemporary classification of the term.
A number of of these additional mathematical fields include elementary algebra, s-algebra, abstract algebra, geometric algebra, boolean algebra, linear algebra, and more. The mastery of basic algebra has to be attained before calculus or several additional advanced math can be mastered. Several careers today require that at the very least fundamental algebra has been mastered. The symbolism that is first presented in elementary algebra expands all through all of the fields of mathematics. This is because the symbolism is definitely the letters in the language of math. Modern fields of algebra usually are enormously much more advanced versus the primary equations and expressiona that were by the Babylonians. Several jobs and careers in the modern-day work place demand that algebra is learned as a legitimate requirement for recruitment.
Please contact me if you want more information concerning how to do algebra or algebra test online or also 8th grade homework. | 677.169 | 1 |
New stuff in the Filebox
I just finished the latest update in the Filebox–topics updated include algebraic expressions, the distributive property, combinations, proportions, functions, basic trigonometry, equations for circles, systems of equations, exponential functions, fraction operations, geometric transformations, and the intermediate value theorem. PLUS some interactive applets that are useful for building a concrete sense of everything from basic operations up to advanced algebra, and a list of potential differentiation strategies. There's so much great stuff out there on the Internet!
Go check it all out, and shoot me an email if you're trying to track down a particular resource or topic | 677.169 | 1 |
Math Reference Guide
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this file type before downloading and/or purchasing.
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Product Description
A collection of Math Reference Materials for your students. Students can keep these sheets in a folder or binder to create their own "Math Reference Guide" to used as a quick reference for Math concepts. Use these sheets to go along with your Math lessons.
Many of these sheets include places for students to write definitions and notes. | 677.169 | 1 |
Print Edition of "Math, Better Explained" Now Available
With the magic of print-on-demand, you can order the book with overnight shipping (Amazon Prime!), and be reading full-color insights tomorrow. Yowza.
I've often been asked if a print version can be made, and I'm beaming to say it's now a reality:
12 chapters (~100 pages) of full-color explanations
Professional-quality typesetting & layout
Gorgeous, high-resolution text and diagrams
Compact, easy-to-carry size with comfortable margins (7″ x 10″)
The best part? There's no garish marketing fluff needed by traditional books that compete for shelf space (testimonials, callouts, those can go in the Amazon description!). The book is my take on a simple, friendly presentation of the math essentials. It's what I wish I had in high school (and college, and afterwards), and a tremendous value for the time and frustration it will save you.
Unlike a textbook you're afraid to open, this book is meant to be accessible. Years later, flip back to that diagram that helped imaginary numbers click. Show that curious young student how the Pythagorean theorem goes way beyond triangles. Math is meant to be seen and felt, not just thought about.
The full-color format does increase the printing costs, but I wanted to share the highest-quality version I could (hey, I'm a reader too!). The introductory price (under $20) is heavily discounted and will change soon, so grab your copy today!
As always, happy math.
PS: Reviews are sincerely appreciated, and if you're a math reviewer (or willing to be one!), contact me and I'll get a copy your way. Thanks for your support! are a parent, a student, or a formal teacher you must buy this book. Really…you must. The textbooks and websites will come and go but if introduced to your high school student early it will open up a whole new world that only those with the most insightful of teachers are ever exposed support Mark — I'm happy to say the book got up to #660 overall on Amazon, and #7 overall in math! got mine! Actually, I just ordered mine. Looking forward to reading it an excellent book. Actually for a deep understanding of the topics presented it is quite exceptional. I purchased this for my 12 year old son (math enthusiast) as I wished I had access to this when I was at school. A great feature of the book and website is discussion on how concepts (e.g. negative numbers, imaginary numbers etc.) emerged in history. In many cases this is a powerful approach to begin to reveal the intuition and one which I incorporate in my (medical and science) teaching. Well done and thank you Kalid.
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4 years 3 daysMS: Thanks!
@Matthew: Really appreciate the thoughtful comments. My goal was to share the book I wish I could have read when entering high school / college, really happy that resonated with you! Exactly, we often have to look at the history of an idea to figure out why it's so useful. Teaching it in a vacuum, without any context, is not a great way to get a deep understanding. Appreciate the note!
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4 years 3 days ago
R there anyway I could obtain the table of contents of the book interested in reading your book "Math Better Explained" and would be willing to supply you with feedback and support through sales to my private students. Perhaps you might consider forwarding me a paperback copy Gordon, I've just emailed you.
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3 years 11 months ago
dip ranjan chatterjeehi!
i teach a few physics student and am a fan of your writing.. i am willing to provide feedback if you could mail me a soft or hard copy of the book. thank youdip ranjan chatterjee: Thanks, I'll reach out to you!
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3 years 10 months ago
Arc Hope to have the iBooks version out by the holidays :).
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3 years 7 months ago
Rat I could obtain the "Introduction To Calculus" chapter of the book in PDF version in free, it would be much pleasure comment or block the author. And please don't worry, your report will be anonymous.
Forward me a paperback copy if you may. I really really need that.
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2 years 11 months ago
us^comment made no sense…
The book is not available in middle east.
How can I get this book Usama, you can buy from Amazon (US or Europe) and have it delivered. Also, you can try buying it directly from the CreateSpace eStore, here: | 677.169 | 1 |
In the Math Forum's Online Algebraic Reasoning Workshop we investigate some mathematics topics common to middle school curricula within the theme of algebraic reasoning. In this context participants explore the Math Tools digital library and several software tools that contribute in some way to mathematical understanding, problem solving, reflection and discussion.
Since January 7, 2007, we have completed thirteen sections (six workshops) of the workshop. Four additional workshops are scheduled starting January 14, 2008 with applications open on December 24, 2007.
Orienting Questions
What are the various forms of algebraic reasoning?
What helps students to develop these as much as they can?
What are some of the key hurdles or difficult transitions for students to make? | 677.169 | 1 |
Geometric Algebra Primer - Jaap Suter - 2003
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Description: Adopted with great enthusiasm in physics, geometric algebra slowly emerges in computational science. Its elegance and ease of use is unparalleled. By introducing two simple concepts, the multivecto...
AdGeometric Algebra Primer
Jaap Suter March 12, 2003
Abstract AdCopyright c 2003 Jaap Suter. Permission to make digital or hard copies of part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permission through email at clifford@jaapsuter.com. See for more information.
Information about geometric algebra is widely available in the field of physics. Knowledge applicable to computer science, graphics in particular, is lacking. As Leo Dorst [1] puts it: ". . . A computer scientist first pointed to geometric algebra as a promising way to 'do geometry' is likely to find a rather confusing collection of material, of which very little is experienced as immediately relevant to the kind of geometrical problems occurring in practice. . . . . . After perusing some of these, the computer scientist may well wonder what all the fuss is about, and decide to stick with the old way of doing things . . . " And indeed, disappointed by the mathematical obscurity many people discard geometric algebra as something for academics only. Unfortunately they miss out on the elegance and power that geometric algebra has to offer. Not only does geometric algebra provide us with new ways to reason about computational geometry, it also embeds and explains all existing theories including complex numbers, quaternions, matrix-algebra, and Pl¨ckerspace. Geou metric algebra gives us the necessary and unifying tools to express geometry and its relations without the need for tricks or special cases. Ultimately, it makes communicating ideas easier.
1.2
Overview
The layout of the paper is as follows; I start out by talking a bit about subspaces, what they are, what we can do with them and how traditional vectors or one-dimensional subspaces fit in the picture. After that I will define what a 4
geometric algebra is, and what the fundamental concepts are. This chapter is the most important as all other theory builds upon it. The following chapter will introduce some common and handy concepts which I call tools. They are not fundamental, but useful in many applications. Once we have mastered the fundamentals, and armed with our tools, we can tackle some applications of geometric algebra. It is this chapter that tries to demonstrate the elegance of geometric algebra, and how and where it replaces traditional methods. Finally, I wrap things up, and provide a few references and a roadmap on how to continue a study of geometric algebra..
1.3
Acknowledgements
I would like to thank David Hestenes for his books [7] [8] and papers [10] and Leo Dorst for the papers on his website [6]. Anything you learn from this introduction, you indirectly learned from them. My gratitude to Per Vognsen for explaining many of the mathematical obscurities that I encountered, and providing me with some of the proofs in this paper. Thanks to Kurt Miller, Conor Stokes, Patrick Harty, Matt Newport, Willem De Boer, Frank A. Krueger and Robert Valkenburg for comments. Finally, I am greatly indebted to Dirk Gerrits. His excellent skills as an editor and his thorough proofreading allowed me to correct many errors.
1.4
Disclaimer
Of course, any mistakes in this text are entirely mine. I only hope to provide an easy-to-read introduction. Proofs will be omitted if the required mathematics are beyond the scope of this paper. Many times only an example or an intuitive outline will be given. I am certain that some of my reasoning won't hold in a thorough mathematical review, but at least you should get an impression. The enthusiastic reader should pick up some of the references to extend his knowledge, learn about some of the subtleties and find the actual proofs.
5
Chapter 2
Subspaces
It is often neglected that vectors represent 1-dimensional subspaces. This is mainly due to the fact that it seems the only concept at hand. Hence we abuse vectors to form higher-dimensional subspaces. We use them to represent planes by defining normals. We combine them in strange ways to create oriented subspaces. Some papers even mention quaternions as vectors on a 4-dimensional unit hypersphere. For no apparent reason we have been denying the existence of 2-, 3- and higher-dimensional subspaces as simple concepts, similar to the vector. Geometric algebra introduces these and even defines the operators to perform arithmetic with them. Using geometric algebra we can finally represent planes as true 2-dimensional subspaces, define oriented subspaces, and reveal the true identity of quaternions. We can add and subtract subspaces of different dimensions, and even multiply and divide them, resulting in powerful expressions that can express any geometric relation or concept. This chapter will demonstrate how vectors represent 1-dimensional subspaces and uses this knowledge to express subspaces of arbitrary dimensions. However, before we get to that, let us consider the very basics by using a familiar example.
a
b
Figure 2.1: The dot product What if we project a 1-dimensional subspace onto another? The answer is well known; For vectors a and b, the dot product a · b projects a onto b resulting in the scalar magnitude of the projection relative to b's magnitude. This is 6
depicted in figure 2.1 for the case where b is a unit vector. Scalars can be treated as 0-dimensional subspaces. Thus, the projection of a 1-dimensional subspace onto another results in a 0-dimensional subspace.
2.1
Bivectors
Geometric algebra introduces an operator that is in some ways the opposite of the dot product. It is called the outer product and instead of projecting a vector onto another, it extends a vector along another. The ∧ (wedge) symbol is used to denote this operator. Given two vectors a and b, the outer product a ∧ b is depicted in figure 2.2.
a b
Figure 2.2: Vector a extended along vector b The resulting entity is a 2-dimensional subspace, and we call it a bivector. It has an area equal to the size of the parallelogram spanned by a and b and an orientation depicted by the clockwise arc. Note that a bivector has no shape. Using a parallelogram to visualize the area provides an intuitive way of understanding, but a bivector is just an oriented area, in the same way a vector is just an oriented length.
a b
Figure 2.3: Vector b extended along vector a If b were extended along a the result would be a bivector with the same area but an opposite (ie. counter-clockwise) orientation, as shown in figure 2.3. In mathematical terms; the outer product is anticommutative, which means that: a ∧ b = −b ∧ a (2.1)
Figure 2.4: A two dimensional basis Now, recall equation (2.2) which says that the outer product of a vector with itself equals zero. Thus we are left with: a ∧ b =(α1 β2 e1 ∧ e2 )+ (α2 β1 e2 ∧ e1 ) Now take another look at figure 2.4. There, I represents the outer product e1 ∧ e2 . This will be our choice for the basis bivector. Because of (2.1) this means that e2 ∧ e1 = −I. Using this information in the previous equation, we obtain: a ∧ b = (α1 β2 − α2 β1 )I (2.6) Which is how to calculate the outer product of two vectors a = (α1 , α2 ) and b = (β1 , β2 ). Thus, in two dimensions, we express bivectors in terms of a basis bivector called I. In the Euclidian plane we use I to represent e12 = e1 ∧ e2 .
It is worth noticing that the choice between using either eij or eji as a basis bivector is completely arbitrary. Some people prefer to use {e12 , e23 , e31 } because it is cyclic, but this argument breaks down in four dimensions or higher; e.g. try making {e12 , e13 , e14 , e23 , e24 , e34 } cyclic. I use {e12 , e13 , e23 } because it solves some issues [12] in computational geometric algebra implementations. The outer product of two vectors will result in a linear combination of the three basis bivectors. I will demonstrate this by using two vectors a and b: a = α1 e1 + α2 e2 + α3 e3 b = β1 e1 + β2 e2 + α3 e3 The outer product a ∧ b becomes: a ∧ b = (α1 e1 + α2 e2 + α3 e3 ) ∧ (β1 e1 + β2 e2 + β3 e3 )
which is the outer product of two vectors in 3-dimensional Euclidian space. For some, this looks remarkably like the definition of the cross product. But they are not the same. The outer product works in all dimensions, whereas the cross product is only defined in three dimensions.1 Furthermore, the cross product calculates a perpendicular subspace instead of a parallel one. Later we will see why this causes problems in certain situations2 and how the outer product solves these.
cross product definitions are valid in all spaces with uneven dimension. you ever tried transforming a plane, you will remember that you had to use an inverse of a transposed matrix to transform the normal of the plane.
2 If
1 Some
11
2.2
Trivectors
Until now, we have been using the outer product as an operator of two vectors. The outer product extended a 1-dimensional subspace along another to create a 2-dimensional subspace. What if we extend a 2-dimensional subspace along a 1-dimensional one? If a, b and c are vectors, then what is the result of (a∧b)∧c? Intuition tells us this should result in a 3-dimensional subspace, which is correct and illustrated in figure 2.6.
b c
a
Figure 2.6: A Trivector A bivector extended by a third vector results in a directed volume element. We call this a trivector. Note that, like bivectors, a trivector has no shape; only volume and sign. Even though a box helps to understand the nature of trivectors intuitively, it could have been any shape. In 3-dimensional Euclidian space R3 , there is one basis trivector equal to e1 ∧ e2 ∧ e3 = e123 . Sometimes, in Euclidian space, this trivector is called I. We already saw this symbol being used for e12 in the Euclidian plane, and we'll return to it when we discuss pseudo-scalars. The result of the outer product of three arbitrary vectors results in a scalar multiple of this basis trivector. In 4-dimensional space R4 , there are four basis trivectors e123 , e124 , e134 , and e234 , and consequently an arbitrary trivector will be a linear combination of these four basis trivectors. But what about the Euclidian Plane? Obviously, there can be no 3-dimensional subspaces in a 2dimensional space R2 . The following informal proof demonstrates why trivectors do not exist in two dimensions. We need to show that for arbitrary vectors a, b, and c ∈ R2 the following holds: (a ∧ b) ∧ c = 0
So far we have seen scalars, vectors, bivectors and trivectors representing 0-, 1-, 2- and 3-dimensional subspaces respectively. Nothing stops us from generalizing all of the above to allow subspaces with arbitrary dimension. Therefore, we introduce the term k-blades, where k refers to the dimension of the subspace the blade spans. The number k is called the grade of a blade. Scalars are 0-blades, vectors are 1-blades, bivectors are 2-blades, and trivectors are 3-blades. In other words, the grade of a vector is one, and the grade of a trivector is three. In higher dimensional spaces there can be 4-blades, 5-blades, or even higher. As we have shown for n = 2 in the previous section, in an n-dimensional space the n-blade is the blade with the highest grade. Recall how we expressed vectors as a linear combination of basis vectors and bivectors as a linear combination of basis bivectors. It turns out that every k-blade can be decomposed onto a set of basis k-blades. The following tables contain all the basis blades for subspaces of dimensions 2, 3 and 4. k 0-blades (scalars) 1-blades (vectors) 2-blades (bivectors) basis k-blades {1} {e1 , e2 } {e12 } total 1 2 1
Figure 2.9: Basis blades in 4 dimensions Generalizing this; how many basis k-blades are needed in an n-dimensional space to represent arbitrary k-blades? It turns out that the answer lies in the 14
binomial coefficient:
n k
=
n! (n − k)!k!
This is because a basis k-blade is uniquely determined by the k basis vectors from which it is constructed. There are n different basis vectors in total. n is k the number of ways to choose k elements from a set of n elements and thus it is easily seen that the number of basis k-blades is equal to n . k Here are a few examples which you can compare to the tables above. The number of basis bivectors or 2-blades in 3-dimensional space is: 3 2 = 3! =3 (3 − 2)!2!
Geometric Algebra
All spaces Rn generate a set of basis blades that make up a Geometric Algebra of subspaces, denoted by C n .1 For example, a possible basis for C 2 is: { 1 , e1 , e 2 , I }
basis scalar basis vectors basis bivector
Here, 1 is used to denote the basis 0-blade or scalar-basis. Every element of the geometric algebra C 2 can be expressed as a linear combination of these basis blades. Another example is a basis of C 3 which could be: { 1 , e1 , e 2 , e 3 , e12 , e13 , e23 e123 }
basis scalar basis vectors
basis bivectors basis trivector
The total number of basis blades for an algebra can be calculated by adding the numbers required for all basis k-blades:
n k=0
n k
= 2n
(3.1)
The proof relies on some combinatorial mathematics and can be found in many places. You can use the following table to check the formula for a few simple geometric algebras. C C C C C C
n 0 1 2 3 4
1 The reason we use C n is because geometric algebra is based on the theory of Clifford Algebras, a topic within mathematics beyond the scope of this paper
16
3.1
The Geometric Product
Until now we have only used the outer product. If we combine the outer product with the familiar dot product we obtain the geometric product. For arbitrary vectors a, b the geometric product can be calculated as follows: ab = a · b + a ∧ b (3.2)
Wait, how is that possible? The dot product results in a scalar, and the outer product in a bivector. How does one add a scalar to a bivector? Like complex numbers, we keep the two entities separated. The complex number (3 + 4i) consists of a real and imaginary part. Likewise, ab = a · b + a ∧ b consists of a scalar and a bivector part. Such combinations of blades are called multivectors.
3.2
Multivectors
A multivector is a linear combination of different k-blades. In R2 it will contain a scalar part, a vector part and a bivector part: α1
scalar part
+ α2 e1 + α3 e2 +
vector part
α4 I
bivector part
Where αi are real numbers, e.g. the components of the multivector. Note that αi can be zero, which means that blades are multivectors as well. For example, if a1 and a4 are zero, we have a vector or 1-blade. In R2 we need 22 = 4 real numbers to denote a full multivector. A multivector in R3 can be defined with 23 = 8 real numbers and will look like this: α1
scalar part
In the same way, a multivector in R4 will have 24 = 16 components. Unfortunately, multivectors can't be visualized easily. Vectors, bivectors and trivectors have intuitive visualizations in 2- and 3-dimensional space. Multivectors lack this way of thinking, because we have no way to visualize a scalar added to an area. However, we get something much more powerful than easy visualization. A multivector, as a linear combination of subspaces, turns out to be extremely expressive, and can be used to convey many different concepts in geometry.
And in the same way as we did when we wrote out the outer product, we may reorder the scalar multiplications (3.4) to obtain: AB = α1 β1 + α1 β2 e1 + α1 β3 e2 + α1 β4 I + α2 β1 e1 + α2 β2 e1 e1 + α2 β3 e1 e2 + α2 β4 e1 I + α3 β1 e2 + α3 β2 e2 e1 + α3 β3 e2 e2 + α3 β4 e2 I + α4 β1 I + α4 β2 Ie1 + α4 β3 Ie2 + α4 β4 II This looks like a monster of a calculation at first. But if you study it for a while, you will notice that it is fairly structured. The resulting equation demonstrates that we can express the geometric product of arbitrary multivectors as a linear combination of geometric products of basis blades. So what we need, is to understand how to calculate geometric products of basis blades. Let's look at a few different combinations. For example, using equation (3.2) we can write: e1 e1 = e1 · e1 + e1 ∧ e1 But remember from equation (2.2) that a ∧ a = 0 because it has no area. Also, the dot product of a vector with itself is equal to its squared magnitude. If we choose the magnitude of the basis vectors e1 , e2 , etc. to be 1, we may simplify the above to: e1 e1 = e1 · e1 +e1 ∧ e1 = e1 · e1 +0 =1 +0 =1 Another example is, again in C 2 : e1 e2 = e1 · e2 + e1 ∧ e2 Now remember that e1 is perpendicular to e2 so the dot product e1 · e2 = 0. This leaves us with: e1 e2 = e1 · e2 +e1 ∧ e2 =0 +e1 ∧ e2 =0 +I =I A more complicated example involves the geometric product of e1 and I. The previous example showed us that I = e12 is equal to e1 e2 . We can use this (3.6)
19
and equation (3.3) to write: e1 I = e1 e12 = e1 (e1 e2 ) = (e1 e1 )e2 = 1e2 = e2 You might begin to see a pattern. Because the basis blades are perpendicular, the dot and outer product have trivial results. We use this to simplify the result of a geometric product with a few rules. 1. Basis blades with grades higher than one (bivectors, trivectors, 4-blades, etc.) can be written as an outer product of perpendicular vectors. Because of this, their dot product equals zero, and consequently, we can write them as a geometric product of vectors. For example, in some high dimensional space, we could write: e12849 = e1 ∧ e2 ∧ e8 ∧ e4 ∧ e9 = e1 e2 e8 e4 e9 2. Equation (2.1) allows us to swap the order of two non-equal basis vectors if we negate the result. This means that we can write: e1 e2 e3 = −e2 e1 e3 = e2 e3 e1 = −e3 e2 e1 3. Whenever a basis vector appears next to itself, it annihilates itself, because the geometric product of a basis vector with itself equals one. ei ei = 1 Example: e112334 = e24 Using these three rules we are able to simplify any geometric product of basis blades. Take the following example: e1 e23 e31 e2 = e1 e2 e3 e3 e1 e2 = e1 e2 e1 e2 = −e1 e1 e2 e2 = −1 using rule one using rule three using rule two using rule three twice (3.8) We can now create a so-called multiplication table which lists all the combinations of geometric products of basis blades. For C 2 it would look like figure 3.1. 20 (3.7)
1 e1 e2 I
1 1 e1 e2 I
e1 e1 1 −I −e2
e2 e2 I 1 e1
I I e2 −e1 −1
Figure 3.1: Multiplication Table for basis blades in C
2
According to this table, the multiplication of I and I should equal −1, which can be calculated as follows: I 2 = e12 e12 = e1 e2 e1 e2 = −e2 e1 e1 e2 = −e2 e2 = −1 by definition using rule one using rule two using rule three using rule three
We defined the geometric product for vectors as a combination of the dot and outer product: ab = a · b + a ∧ b We can rewrite these equations to express the dot product and outer product in terms of the geometric product: a∧b= 1 (ab − ba) 2 1 a · b = (ab + ba) 2 (3.10) (3.11)
(α2 β2 + α3 β3 ) − (β2 α2 + β3 α3 ) + (α2 β3 − α3 β2 )I − (β2 α3 − β3 α2 )I 2 Notice the scalar part results in zero, which leaves us with: (α2 β3 − α3 β2 )I − (β2 α3 − β3 α2 )I 2 Subtracting the two bivectors we get: (α2 β3 − α3 β2 − β2 α3 + β3 α2 )I 2 This may be rewritten as: (2α2 β3 − 2α3 β2 )I 2 And now dividing by 2 we obtain: A ∧ B = (α2 β3 − α3 β2 )I for multivectors A and B with zero scalar and bivector part. Compare this with equation (2.6) that defines the outer product for two vectors a and b. If you remember that the vector part of a multivector ∈ C 2 is in the second and third component, you will realize that these equations are the same. Note that (3.10) and (3.11) only hold for vectors. The inner and outer product of higher order blades is more complicated, not to mention the inner and outer product for multivectors. Yet, let us try to see what they could mean.
3.5
The Inner Product
I informally demonstrated what the outer product of a vector and a bivector looks like when I introduced trivectors. What about the dot product? What could the dot product of a vector and a bivector look like? Figure 3.3 depicts the result. Notice how the inner product is the vector perpendicular to the actual projection. In more general terms, it is the complement (within the subspace of B) of the orthogonal projection of a onto B. [2] We will no longer call this generalization a dot product. The generic notion of projections and perpendicularity is captured by an operator called the inner product. 23
c
b c (a ∧ b)
a
Figure 3.3: The dot product of a bivector and a vector Unfortunately, there is not just one definition of the inner product. There are several versions floating around, their usefulness depending on the problem area. They are not fundamentally different however, and all of them can be expressed in terms of the others. In fact, one could say that the flexibility of the different inner products is one of the strengths of geometric algebra. Unfortunately, this does not really help those trying to learn geometric algebra, as it can be overwhelming and confusing. The default and best known inner product [8] is very useful in Euclidian mechanics, whereas the contraction inner product [2], also known as the Lounesto inner product, is more useful in computer science. Other inner products include the semi-symmetric or semi-commutative inner product, also known as the Hestenes inner product, the modified Hestenes or (fat)dot product and the forced Euclidean contractive inner product. [13] [5] Obviously, because of our interest for computer science, we are most interested in the contraction inner product. We will use the symbol to denote a contraction. It may seem a bit weird at first, but it will turn out to be very useful. Luckily, for two vectors it works exactly as the traditional inner product or dot product. For different blades, it is defined as follows [2]: scalars vector and scalar scalar and vector vectors vector, multivector distribution α a α a a β = αβ β=0 b = αb b = a · b (the usual dot product) (b ∧ C) = (a b) ∧ C − b ∧ (a C) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19)
(A ∧ B) C = A (B C)
Try to understand how the above provides a recursive definition of the contraction operator. There are the basic rules for vectors and scalars, and there is (3.18) for the contraction between a vector and the outer product of a vector and 24
a multivector. Because linearity holds over the contraction, we can decompose contractions with multivectors into contractions with blades. Now, remember that any blade D with grade n can be written as the outer product of a vector b and a blade C with grade n − 1. This means that the contraction a D can be written as a (b ∧ C) and consequently as (a b) ∧ C − b ∧ (a C) according to (3.18). We know how to calculate a b by definition, and we can recursively solve a C until the grade of C is equal to 1, which reduces it to a contraction of two vectors. Obviously, this is not a very efficient way of calculating the inner product. Fortunately, the inner product can be expressed in terms of the geometric product (and vice versa as we've done before), which allows for fast calculations. [12] I will return to the inner product when I talk about grades some more in the tools chapter. In the chapter on applications we will see where and how the contraction product is useful. From now on, whenever I refer to the inner product I mean any of the generalized inner products. If I need the contraction, I will mention it explicitly. I will allow myself to be sloppy, and continue to use the · and symbol interchangeably.
3.6
Inner, Outer and Geometric
We saw in equation (3.2) that the geometric product for vectors could be defined in terms of the dot (inner) and outer product. What if we use (3.10) and (3.11) combined: 1 1 (ab + ba) + (ab − ba) (ab + ba) + (ab − ba) = 2 2 2 (ab + ba + ab − ba) = 2 2ab = 2 = ab This demonstrates the two possible approaches to introduce geometric algebra. Some books [7] give an abstract definition of the geometric product, by means of a few axioms, and derive the inner and outer product from it. Other material [8] starts with the inner and outer product and demonstrates how the geometric product follows from them. You may prefer one over the other, but ultimately it is the way the geometric product, the inner product and the outer product work together that gives geometric algebra its strength. For two vectors a and b we have: ab = a · b + a ∧ b as a result, they are orthogonal if ab = -ba because the inner product of two perpendicular vectors is zero. And they are collinear if ab = ba because the 25
wedge of two collinear vectors is zero. If the two vectors are neither collinear nor orthogonal the geometric product is able to express their relationship as 'something in between'.
26
Chapter 4
Tools
Strictly speaking, all we need is an algebra of multivectors with the geometric product as its operator. Nevertheless, this chapter introduces some more definitions and operators that will be of great use in many applications. If you are tired of all this theory, I suggest you skip over this section and start with some of the applications. If you encounter unfamiliar concepts, you can refer to this chapter.
4.1
Grades
I briefly talked about grades in the chapter on subspaces. The grade of a blade is the dimension of the subspace it represents. Thus multivectors have combinations of grades, as they are linear combinations of blades. We denote the blade-part with grade s of a multivector A using A s . For multivector A = (4, 8, 5, 6, 2, 4, 9, 3) ∈ C 3 we have: A A A A
0 1 2 3
=4 = (8, 5, 6) = (2, 4, 9) =3
n
scalar part vector part bivector part trivector part
Any multivector A in C did informally:
n
can be denoted as a sum of blades, like we already
A
k=0
k
= A
0
+ A
1
+ ... + A
n
Using this notation I can demonstrate what the inner and outer product mean for grades. For two vectors a and b the inner product a · b results in a scalar c. The vectors are 1-blades, the scalar is a 0-blade. This leads to: a
1
· b
1
= ab
0
27
In figure 3.3 we saw that a vector a projected onto a bivector B resulted in a vector. Here, we'll be using the contraction product. So, in other words the contraction product of a 2-blade and a 1-blade results in a 1-blade. Using a multivector notation: a 1 B 2 = aB 2−1 Generalizing this for blades A and B with grade s and t respectively: A
s
B
t
= AB
u
where u =
s > t, 0 s ≤ t, t - s
We might say that the contraction inner product is a 'grade-lowering' operation. And, of course, the outer product is its opposite as a grade-increasing operation. Recall that for two 1-blades or vectors the outer product resulted in a 2-blade or bivector: a 1 ∧ b 1 = ab 2 The outer product between a 2-blade and a 1-blade results in a 2 + 1 = 3-blade or trivector. Generalizing we get for two blades A and B with grade s and t: A
s
∧ B
t
= AB
s+t
Note that A and B have to be blades. These equations do not hold when they are arbitrary multivectors.
4.2
The Inverse
Most multivectors have a left inverse satisfying A−1L A = 1 and a right inverse satisfying AA−1R = 1. We can use these inverses to divide a multivector by another. Recall that the geometric product is not commutative therefore the A left and right inverse may or may not be equal. This means that the B notation −1L −1R is ambiguous since it can mean both B A and AB . Unfortunately calculating the inverse of a geometric product is not trivial, much like calculating inverses of matrices is complicated for all but a few special cases. Luckily there is an important set of multivectors for which calculating the inverse is very straightforward. These are called the versors and they have the property that they are a geometric product of vectors. A multivector A is a versor if it can be written as: A = v1 v2 v3 ...vk where v1 ...vk are vectors, i.e. 1-blades. As a fortunate consequence, all blades are versors too.1 For a versor A we define its reverse, using the † symbol, as: A† = vk vk−1 ...v2 v1 (4.1)
1 Remember that we use vectors to create subspaces of higher dimension, using the outer product.
28
This means that, because of equation (2.1), the reverse of a blade is only a possible sign change. Remember that each swap of indices in the product produces a sign change, thus if k is uneven the reverste of A is equal to itself, and if it's uneven the reverse of A is the −A. Note that this does not apply to versors in general. The left and right inverse of a versor are the same and can be calculated as follows: A† (4.2) A† A To understand this, we have to start by realizing that the denominator is always a scalar because: A−1 = A† A = v1 v2 ...vk−1 vk vk vk−1 ...v2 v1 = |v1 |2 |v2 |2 ...|vk−1 |2 |vk |2 And since a scalar divided by itself equals one, this means that: A† A A† A= † =1 A† A A A Furthermore it also proves that the left and right inverse are the same. Division by a scalar α is multiplication by 1/α, which is, according to equation (3.3) commutative, proving that the left and right inverses of a versor are indeed equal. This means that for a versor A, we have A−1L = A−1R = A−1 and therefore the following: A−1L A = AA−1R = A−1 A = AA−1 = 1 A−1 A = It is important to notice that in the case of vectors, the scalar represents the squared magnitude of the vector. As a consequence, the inverse of a unit vector is equal to itself. Not many people are comfortable with the idea of division by vectors, bivectors, or multivectors. They are only accustomed to division by scalars. But if we have a geometric product and a definition for the inverse, nothing stops us from division. Later we will see that this is extremely useful.
4.3
Pseudoscalars
In equation (2.3) we saw how to calculate the number of basis blades for a given grade. From this it follows that every geometric algebra has only one basis 0-blade or basis-scalar, independent of the dimension of the algebra: n 0 = n! n! = =1 (n − 0)!0! n!
More interesting is the basis blade with the highest dimension. For a geometric algebra C n the number of blades with dimension n is: n n = n! n! = =1 (n − n)!n! n! 29
In C 2 this was e1 e2 = I as shown in figure 2.4. In C 3 this is the trivector e1 e2 e3 = e123 . In general every geometric algebra has a single basis blade of highest dimension. This is called the pseudoscalar.
4.4
The Dual
Traditional linear algebra uses normal vectors to represent planes. Geometric algebra introduces bivectors which can be used for the same purpose. By using the pseudoscalar we can get an understanding of the relationship between the two representations. The dual A∗ of a multivector A is defined as follows: A∗ = AI −1 (4.3)
where I represents the pseudoscalar of the geometric algebra that is being used. The pseudoscalar is a blade (the blade with highest grade) and therefore its left and right inverse are the same, and hence the above formula is not ambiguous. Let us consider a simple example in C 3 , calculating the dual of the basis bivector e12 . The pseudoscalar is e123 . Pseudoscalars are blades and thus versors. You can check yourself that its inverse is e3 e2 e1 . We'll use this to calculate the dual of e12 : e∗ = e12 e3 e2 e1 12 = e1 e2 e3 e2 e1 = −e1 e3 e2 e2 e1 = −e1 e3 e1 = e1 e1 e3 = e3 (4.4) Thus, the dual is basis vector e3 , which is exactly the normal vector of basis bivector e12 . In fact, this is true for all bivectors. If we have two arbitrary vectors a and b ∈ C 3 : a = α1 e1 + α2 e2 + α3 e3 b = β1 e1 + β2 e2 + β3 e3 According to equation (2.7) their outer product is: a ∧ b = (α1 β2 − α2 β1 )e12 + (α1 β3 − α3 β1 )e13 + (α2 β3 − α3 β2 )e23
Which is exactly the traditional cross product. We conclude that in three dimensions, the dual of a bivector is its normal. The dual can be used to convert between bivector and normal representations. But the dual is even more, because it is defined for any multivector.
4.5
Projection and Rejection
If we have a vector a and bivector B we can decompose a in two parts. One part a||B that is collinear with B. We call this the projection of a onto B. The other part is a⊥B , and orthogonal to B. We call this the rejection 2 of a from B. Mathematically: a = a||B + a⊥B (4.6) This is depicted in figure 4.1. Such a decomposition turns out to be very useful and I will demonstrate how to calculate it. First, equation (3.2) says that the geometric product of two vectors is equal to the sum of the inner and outer product. There is a generalization of this saying that for arbitrary vector a and k-blade B the geometric product is: aB = a · B + a ∧ B (4.7)
2 This term has been introduced by David Hestenes in his New Foundations For Classical Mechanics [8]. To quote: "The new term 'rejection' has been introduced here in the absence of a satisfactory standard name for this important concept."
31
a B a⊥B
a||B
Figure 4.1: Projection and rejection of vector a in bivector B Note that a has to be a vector, and B a blade of any grade. That is, this doesn't hold for multivectors in general. Proofs can be found in the references. [14] [8] Using (4.7) we can calculate the decomposition (4.6) of any vector a onto a bivector B. By definition, the inner and outer product of respectively orthogonal and collinear blades are zero. In other words, the inner product of a vector orthogonal to a bivector is zero: a⊥B · B = 0 Likewise the outer product of a vector collinear with a bivector is zero: a||B ∧ B = 0 (4.9) (4.8)
Thus, the perpendicular part of vector a times bivector B is equal to the outer product of a and B. Now all we need to do is divide both sides of the equation by B to obtain the perpendicular part of a: a⊥B B = a ∧ B a⊥B BB −1 = (a ∧ B)B −1 a⊥B = (a ∧ B)B −1
equation (4.11) and (4.10) equation (3.3) equation (4.7) by definition of the inverse
33
4.6
Reflections
Armed with a way of decomposing blades in orthogonal and collinear parts we can take a look at reflections. We will get ahead of ourselves and take a specific look at the geometric algebra of the Euclidian space R3 denoted with C 3 . Suppose we have a bivector U . Its dual U ∗ will be the normal vector u. What if we multiply a vector a with vector u, projecting and rejecting a onto U at the same time: ua = u(a||U + a⊥U ) = ua||U + ua⊥U Using (3.2) we write it in full: ua = (u · a||U + u ∧ a||U ) + (u · a⊥U + u ∧ a⊥U ) Note that (because u is the normal of U ) the vectors a||U and u are perpendicular. This means that the inner product a||U · u equals zero. Likewise, the vectors a⊥U and u are collinear. This means that the outer product a||U ∧ u equals zero. Removing these two 0 terms: ua = u ∧ a||U + u · a⊥U = u · a⊥U + u ∧ a||U Recall that the inner product between two vectors is commutative, and the outer product is anticommutative, so we can write: ua = a⊥U · u − a||U ∧ u We can now insert those 0-terms back in (putting in the form of equation (3.2)): ua = (a⊥U · u + a⊥U ∧ u) − (a||U · u + a||U ∧ u) Writing it as a geometric product now: ua = a||U u − a⊥U u = (a||U − a⊥U )u Meaning that: −ua = −(a||U − a⊥U )u = (a⊥U − a||U )u Notice how we changed the addition of the perpendicular part into a subtraction by multiplying with −u. Now, if we add a multiplication with the inverse we obtain the following, depicted in figure 4.2: −uau−1 = −u(a||U + a⊥U )u−1 = (a||U − a⊥U )uu−1 = a||U − a⊥U 34
u
a U a⊥U
a||U
a||U − a⊥U = −uau−1
Figure 4.2: Reflection In general, if we sandwich a vector a in between another vector −u and its inverse u−1 , we obtain a reflection in the dual u∗ . Note that in many practical cases u will be a unit vector, which means its inverse is u itself. Thus the reflection of a vector a in a plane with unit-normal u is simply −uau. Later, we will see that reflections are not only useful by themselves, but combined they allow us to do rotations.
35
4.7
The Meet
The meet operator is an operator between two arbitrary blades A and B and defined as follows: A ∩ B = A∗ · B In other words, the meet A ∩ B is the inner product of the dual of A with B. It is no coincidence that the ∩ symbol is used to denote the meet. The result of a meet represents the smallest common subspaces of blades A and B. Let's see, in an informal way, what the meet operator does for two bivectors. Looking at figure 4.3 we see two bivectors A and B.
A∗ a A∩B A
B
Figure 4.3: The Meet In this figure, the dual of bivector A will be the normal vector A∗ . Then, as we've already seen, the inner product of this vector with bivector B will create the vector perpendicular to the projected vector a . This is exactly the vector that lies on the intersection of the two bivectors. A more formal proof of the above, or even the full proof that the meet operator represents the smallest common subspace of any two blades, is far from trivial and beyond this paper. Here, I just want to demonstrate that there is a meet operator, and that it is easily defined using the dual and the inner product. We will be using the meet operator later when we talk about intersections between primitives.
36
Chapter 5
Applications
Up until now I haven't focused on the practical value of geometric algebra. With an understanding of the fundamentals, we can start applying the theory to real world domains. This is where geometric algebra reveals its power, but also its difficulty. Geometric algebra supplies us with an arithmetic of subspaces, but it is up to us to interpret each subspace and operation and relate it to some real-life concept. This chapter will demonstrate how different geometric algebras combined with different interpretations can be used to explain traditional geometric relations.
5.1
The Euclidian Plane
For an easy start, we'll consider the two dimensional Euclidian Plane to learn about some of the things its geometric algebra C 2 has to offer.
5.1.1
Complex Numbers
If you recall the geometric algebra of the Euclidian Plane, you might remember we used I to denote the basis bivector. Then, in table 3.1 we saw that I 2 = −1. Thus, we might say: I= √ −1
Suppose we interpret a multivector with a scalar and bivector blade as a complex number. The scalar corresponds to the real part, and the bivector to the imaginary part. Thus, we can interpret a multivector (α1 , α2 , α3 , α4 ) from C 2 as a complex number α1 + iα4 as long as α2 and α3 are zero. Not surprisingly, multivector addition and subtraction corresponds directly with complex number addition and subtraction. But even more so, the geometric product is exactly the multiplication of complex numbers, as the following will prove. 37
I will discuss rotations in two dimensions very briefly. When we return to rotations in three dimensions I will introduce the more general dimension-free theory and give several longer and more formal proofs. If we want to rotate a vector a = α2 e1 + α3 e2 over an angle θ into vector a = α2 e1 + α3 e2 , we can employ the following well known formulas: α2 = cos(θ)α2 − sin(θ)α3 α3 = sin(θ)α2 + cos(θ)α3 (5.1) (5.2)
Thus we see that the geometric product of a complex number and a vector results in a vector with components: α2 = β1 α2 + β4 α3 α3 = β4 α2 − β1 α3 (5.3) Compare this with equations (5.1) and (5.2). If we take β1 = cos(θ) and a β4 = sin(θ) we can use complex numbers to do rotations, because then: α2 = cos(θ)α2 − sin(θ)α3 = β1 α2 − β4 α3 = α2 α3 = sin(θ)α2 + cos(θ)α3 = β4 α2 + β1 α3 = α3 At this point we no longer talk about complex numbers but we call B a spinor. In general, spinors are n-dimensional rotators, and in C 2 they are represented by a linear combination of a scalar plus a bivector. Equation (3.2) says that the geometric product of two vectors is a scalar plus a bivector. So let's find a p and q that generate B: B = β1 + β4 I = p · q + p ∧ q = pq Traditional vector algebra tells us that the angle between two unit vectors can be expressed using the dot product, i.e. p · q = cos(θ). It also tells us that the magnitude of the cross product of two unit vectors is equal to the sine of the same angle, |p × q| = sin(θ). I already demonstrated that the cross product is related to the outer product through the dual. In fact, it turns out that the outer product between two unit vectors is exactly the sine of their angle times the basis bivector I. In other words: p · q = cos(θ) p ∧ q = sin(θ)I A thorough proof can be found in Hestenes's New Foundations For Classical Mechanics [8]. The consequence is that, because of equation (3.2), for unit vectors p and q: pq = cos(θ) + sin(θ)I with θ being the angle between the two vectors. Concluding, a spinor in C 2 is a scalar plus a bivector. Its components correspond directly to the sine and cosine of the angle. The geometric product of two unit vectors generates a spinor. We can derive this in a similar way by creating the following identity: a p = q a
39
Which is not ambiguous, and can also be written as: pq −1 = a a−1 Basically, this says that "what p is to q, is a to a." We can rewrite this to: a = pq −1 a But the inverse of a unit vector is equal to itself, and thus: a = pq a
spinor
If the spinor pq represents a clockwise rotation, then qp represents a counterclockwise rotation. This is thanks to the fact that the geometric product is not commutative or anticommutative. As a result it can convey more information removing much of the ambiguities of traditional methods where certain representations can only identify the rotation over the smallest angle. It's interesting to look at the rotation over 180 degrees. We can do it by constructing a spinor out of the basis vector e1 and its negation −e1 . Obviously there is a 180 degree angle between them. If we multiply them (see table 3.1 for a refresher), we get: e1 (−e1 ) = −(e1 e1 ) = −1 which makes sense because a rotation over 180 degrees reverses the signs. But this becomes more interesting if we do it through two successive rotations over 90 degrees. A rotation by 90 degrees is a multiplication with the basis bivector I. As an example, consider the following geometric products between the basis blades e1 , e2 and I, taken directly from the multiplication table for C 2 as in figure 3.1. Ie1 = −e2 Ie2 = e1 I(−e1 ) = e2 I(−e2 ) = −e1 Now doing two successive rotations: I 2 e1 = I(Ie1 ) = I(−e2 ) = −e1 I 2 e2 = I(Ie2 ) = I(e1 ) = −e2 This provides yet another demonstration that I 2 equals −1. But it also demonstrates how we can combine rotations. It turns out that, as our intuition dictates, that the geometric product of two rotations form a new rotation. This demonstrates that we can use the geometric product to concatenate rotations, and that spinors form a subgroup of C 2 . 40
And we conclude, as expected, that the multiplication of two spinors results in a spinor in C 2 .
5.1.3
Lines
Equation (2.2) told us that the outer product of a vector with itself equals zero. We can use this to construct a line through the origin at (0, 0). For a given direction vector u all points x on the line satisfy: x∧u=0 The proof is easy if you realize that every x can be written as a scalar multiple of u. For lines trough an arbitrary point a we can write the following: (x − a) ∧ u = 0 We can rewrite this equation in several different forms: (x − a) ∧ u = 0 (x ∧ u) − (a ∧ u) = 0 (x ∧ u) = (a ∧ u) (x ∧ u) = U Where u is the direction vector of the line and U is the bivector a ∧ u. This is depicted in figure 5.1. Again, note that the bivector has no specific shape. It is only the area that indirectly defines the distance from origin to the line. If we want to calculate the distance directly, we need the vector d as illustrated in figure 5.1. We can calculate this vector easily by doing: d= 41 U u
u
a u U
U d u
Figure 5.1: Lines in the Euclidian plane The magnitude of this vector will be the distance from the line to the origin. This is trivial to prove. From the above we see that du = U From equation (3.2) we can write: du = d · u + d ∧ u = U But remember that U is a bivector equal to d ∧ u and that consequently d · u must be zero. Well, if the inner product of two vectors equals zero, they are perpendicular, and hence d is perpendicular to the line, and thus the shortest vector from the origin to the line. Things are even better if u is a unit vector. In applications this is often the case, and it allows us to write d = U u because the inverse of a unit vector is the unit vector itself, hence d = U = U u−1 = U u. Even more so, in the case that u u is a unit vector, then |U | = |d|, exactly the distance from the origin to the line.
42
5.2
Euclidian Space
For those with an interest in computer graphics, the world of three dimensions is the most interesting of course. I will look at some common 3D operations and concepts, and show how they relate to traditional methods. However, we will learn about homogeneous space later, which deals with a four dimensional model of three dimensional space. This model provides significant advantages over C 3 . Fortunately, most of the theory presented in this section will be presented in a dimension free manner, and can readily be applied there as well.
5.2.1
Rotations
When I discussed rotations in the Euclidian plane, I showed you that a spinor cos θ + sin θI rotates a vector over an angle θ. I then showed that the geometric product of two unit vectors s and t generates a spinor because s · t = cos θ and s ∧ t = sin θI. I will now prove that such a rotation in a plane works for any dimension. After that I will extend it so we can do arbitrary rotations. Finally, we will see how this relates to a traditional method for rotations. You might be surprised. Rotations in a plane We will use a unit-bivector B to define our plane of rotation. Given a vector v lying in the same plane as the bivector, we want to rotate it over an angle θ into a vector v . Using a spinor R of the form cos θ + sin θB, the proposition is: v = Rv (5.4)
We need to prove that v lies in the same plane as B and v, by demonstrating that v ∧B =0 (5.5)
And we need to prove that the angle between v and v equals θ by showing that the well known dot product equality holds: v · v = |v ||v| cos θ (5.6)
Note that nothing is said about the dimension of the space these vectors and bivector reside in. This makes sure our proof holds for any dimensions Let's start out by writing the spinor in full: v = (cos θ + sin θB)v = cos θv + sin θBv Because v is a vector and B is a bivector, we can write: v = cos θv + sin θ(B · v + B ∧ v) 43
Because v lies in the same plane as B, we know that B ∧ v = 0, resulting in: v = cos θv + sin θB · v If you refer back to image 3.3 you will see that the inner product between a vector and a bivector returns the complement of the projection. In this case v is already in the bivector plane, and thus its own projection. The resulting complement B · v is a vector in the same plane, but perpendicular to v. We will denote this vector with v⊥ . This leads to the final result, which will allow us to prove (5.5) and (5.6): v = cos θv + sin θv⊥ The proof of (5.5) is easy. If you look at the last equation, you will see that v is an addition of two vectors v and v⊥ . Both these vectors are in the B plane, and hence any addition of these vectors will also be in the same plane. Hence v ∧ B = 0. Understanding the proof of (5.6) will be easier if you to take a look at figure 5.2. There, I've depicted the three relevant vectors in the B plane.
The dot product is distributive over vector addition: cos θv · v + sin θv⊥ · v = |v|2 cos θ Because v⊥ and v are perpendicular, their dot product equals zero: cos θv · v = |v|2 cos θ And finally, the dot product of a vector with itself is equal to its squared magnitude: cos θ|v|2 = |v|2 cos θ This concludes our proof that a spinor cos θ + sin θB rotates any vector in the B-plane over an angle θ. Since we made no assumptions on the dimension anywhere, this shows that spinors work in any dimension. In the Euclidian Plane we didn't have much choice regarding the B-plane, because there is only one plane, but in three or more dimensions we can use any arbitrary bivector. Yet, the above only works when the vector we wish to rotate lies in the actual plane. Obviously we also want to rotate arbitrary vectors. I will now demonstrate how we can use reflections to achieve this in a dimension free way. Arbitrary Rotations Let's see what happens if we perform two reflections on any vector v. One using unit vector s and then another using unit vector t: v = −t(−svs)t = tsvst We will denote st using R, and because it is a versor we can write ts = (st)† = R† : v = R† vR Note that R and R† have the form: R = st = s · t + s ∧ t R† = ts = s · t − s ∧ t Hence, they are spinors -scalar plus bivector- and can be used to rotate in the s ∧ t plane. It is the combination with the reverse that allows us to do arbitrary rotations, removing the restriction that v has to lie in the rotation plane. Let's denote the plane of rotation that is specified by the bivector s ∧ t, with A. Now, decompose v into a part perpendicular to A and a part collinear with A: v = v||A + v⊥A
45
Now the following relations hold:
=0
v||A A = v||A · A + v||A ∧ A = v||A · A = −A · v||A = −A · v||A − A ∧ v||A = −Av||A It may be worth pointing out why v||A · A = −A · v||A . The inner product for vectors is commutative whereas the inner product between a vector and a bivector is anticommutative. It turns out that the sign of the commutativity depends on the grade of the blade. For vector p and blade Q with grade r we have: p · Q = (−1)r+1 Q · p I will not prove this here. It can be found in Hestenes New Foundations for Classical Mechanics [8] as well as other papers. For the perpendicular part of v, we have:
=0
v⊥A A = v⊥A · A +v⊥A ∧ A = v⊥ A ∧ A = A ∧ v⊥A = A · v⊥A + A ∧ v⊥A = Av⊥A If you are confused by the fact that the outer product between a vector and a bivector turns out to be commutative, then remember that the outer product between a vector and a vector was anticommutative. Hence: v⊥A ∧ A = v⊥A ∧ s ∧ t = −s ∧ v⊥A ∧ t = s ∧ t ∧ v⊥ A = A ∧ v⊥A In fact, the commutativity of the outer product between a vector p and a blade Q with grade r is related much like the inner product: p ∧ Q = (−1)r Q ∧ p For this, the proof is easy. The sign simply depends on the amount of vectorswaps necessary, and hence on the number of vectors in the blade. This is directly related to the grade of the blade. 46
Remember that s and t are unit vectors and hence: R† R = tsst = 1 This allows us to write: v = v||A RR + v⊥A Now, notice that we are multiplying the collinear part v||A by R twice, and that the perpendicular part v⊥A is untouched. Recall that R was a scalar and a bivector and thus a spinor that rotates vectors in the bivector plane. Well, v||A lies in that plane. This means that if we make sure that R rotates over an angle 1 θ, a double rotation will rotate 2 over angle θ. Thus the operation R† vR does exactly what we want. This is illustrated in figure 5.3.
A∗
v
v
v⊥A
θ
1 θ 2
v⊥A
v||A
v||A
A
Figure 5.3: An arbitrary rotation It is worth mentioning that the dual of the A plane, denoted by A∗ , is obviously the normal to the plane, and hence the rotation axis. Conclusion If, by now, you think that rotations are incredibly complicated in geometric algebra you are in for a surprise. All of the above was just to prove that for any angle θ and bivector A, the spinor: 1 1 R = cos θ + sin θA 2 2 can perform a rotation of an arbitrary vector relative to the A plane, by doing: v = R† vR 48
In other words, the spinor performs a rotation around the normal of the A plane, denoted by the dual A∗ . Best of all, spinors don't just work for vectors, but for any multivector. This means that we can also use spinors to rotate bivectors, trivectors or complete multivectors. For example, if we have a bivector B, we can rotate it by θ degrees around an axis a or relative to a bivector a∗ = A by : B = R† BR 1 1 1 1 B = (cos θ − sin A)B(cos θ + sin A) 2 2 2 2 Proofs of this generalization of spinor theory can be found in most of the references. Now that we've seen the dimension free definition of spinors, and a proof on why they perform rotations, we can return to Euclidian space and see what it means in three dimensions. Recall the bivector representation in C 3 . We used three basis bivectors e12 , e13 and e23 and denoted any bivector B using a triplet (β1 , β2 , β3 ). If B is a unit-bivector and introduce the γ symbol for the scalar, then we can represent a spinor R ∈ C 3 over an angle θ as follows: 1 1 R = cos θ + sin θB 2 2
(γ, β1 ,β2 ,β3 )
R = (w, xi, yj, zk) where 1 w = cos θ 2 1 x = sin θβ1 2 1 y = sin θβ2 2 1 z = sin θβ3 2 Hopefully this is starting to look familiar by now. Compare my definition of a spinor, including the way i, j and k multiply, with any textbook on quaternions and you will easily see the resemblance. In fact, in three dimensions, spinors are exactly quaternions. A quaternion rotation is exactly a vector sandwiched in between the quaternion and its inverse. But because we are using unitquaternions (or unit-spinors) the inverse is the same as the reverse. Don't assume that all of this is a coincidence, or merely a striking resemblance. Quaternions and spinors are exactly the same: A quaternion is a scalar plus a bivector Apart from the fact that quaternions have four components, there is nothing four-dimensional or imaginary about a quaternion. The first component is a scalar, and the other three components form the bivector-plane relative to which the rotation is performed. If you would look at the conversion from an axis-angle representation to a quaternion, you would see that the angle is divided by two. This makes sense now, because the spinor rotates the collinear part of a vector twice. Also, the conversion takes the dual of the axis and multiplies it by sin 1 θ to create 2 exactly the bivector we need. If you would write out the geometric product of two spinors, you would notice that it is exactly the same as a quaternion multiplication. Finally, the inverse of a spinor (w, x, y, z) is (w, -x, -y, -z) which follows from the fact that the inverse rotates over the opposite angle combined with the fact that cos(−θ) = cos θ yet sin(−θ) = − sin θ. Spinors may be quaternions in 3D, but geometric algebra has given us much more: • There is nothing four dimensional or imaginary to a quaternion. It is simply a scalar plus a real bivector. This gives us a way to actually visualize quaternions. Compare this to the many failed attempts to draw four dimensional unit hyperspheres in textbook examples on quaternions. • Spinors rotate more than just vectors. They can rotate bivectors, trivectors, n-blades and any multivector. If you ever tried rotating the normal of plane, you will readily appreciate the ability to rotate the plane directly. 50
• Unlike quaternions, spinors are dimension-free and work just as well in 2D, 4D and any other dimension. Their theory extends to any space without changes. Back to Reflections At the start of this section we combined two reflections to generate a rotation. Let's look at that again, to get another way of looking at rotations. So, we perform two reflections on any vector v. One using unit vector s and then another using unit vector t: v = −t(−svs)t = tsvst We denoted st using R and ts using R† and worked with those from then. Instead, we will now look at what those two reflections do in a geometric interpretation. This is illustrated in figure 5.4 using a top down view.
v
θ v v s
1 θ 2
t
Figure 5.4: A rotation using two reflections In this figure, we reflect v in the plane perpendicular to s to obtain vector v . Then we reflect that vector in the plane perpendicular to t to obtain the final rotated vector v . We can actually easily prove this by adding a few angles. We will measure the angles using the bivector dual of s as the frame of reference. Then we denote the angle of vector v as θ(v). Thus, if we reflect v in this bivector dual we obtain vector v with an angle of θ(v ) = −θ(v). Then we mirror this vector v in the bivector dual of t to obtain an angle θ(v ) equal to: θ(v ) = θ(t) + θ(t) − θ(v ) = θ(t) + θ(t) + θ(v) = 2θ(t) + θ(v)
51
This proves that the angle between v and v is twice the angle between s and t. That this works for vectors even outside of the s ∧ t plane is left as an exercise for the reader. Also, try to convince yourself that reflecting in t followed by s is precisely the opposite rotation over the angle −θ.
52
5.2.2
Lines
We've discussed lines before when we looked at the Euclidian plane. The interesting thing is that no reference to any specific dimension was made. As a result, we can use the same equations to describe a line in 3D. That is, for a line through point a with direction-vector u, every point x on that line will satisfy all of the following: (x − a) ∧ u = 0 (x ∧ u) − (a ∧ u) = 0 (x ∧ u) = (a ∧ u) (x ∧ u) = U Where U is the bivector a ∧ u. Also, the shortest distance from the origin is again given by vector d = U . The short proof given in the previous section on u lines is still valid. Again, if u is a unit vector, then d = U u because u−1 = u.
5.2.3
Planes
Talking about planes in the Euclidian plane wasn't very useful as there is only one plane. In three dimensions, things are more interesting. Of course, the concept we use to model a plane should not come as a surprise. We've been using it implicitly in our talk about rotations already. We can model a plane by using a bivector. That is, any point x on the plane will satisfy: x∧B =0 Remember that the outer product between a bivector and a vector collinear with the bivector will equal zero, and the above will be obvious. If we wish to describe a plane through a point a, we can use the following: (x − a) ∧ B = 0 Compare this with the equations for describing a line. It's interesting to see that the outer product with a vector describes a line, and the outer product with a bivector describes a plane. In three dimensions the equation above requires six components. Three for bivector B, and three for vector a. Fortunately, we can rewrite it, so it requires four components, comparable to the traditional way of storing a plane where you use the normal and the shortest distance to the origin. It's as follows: (x − a) ∧ B = 0 x∧B−a∧B =0 x∧B =a∧B x∧B =U Notice that the part on the right side is a trivector (denoted by U ). In 3D trivectors are represented by just one scalar, allowing us to store a plane using 53
four components. This scalar represents the volume of the pyramid spanned by bivector B and the origin. In most applications, we can easily makes sure that B is a unit-bivector. The consequence is that the scalar is exactly the distance from the origin to the plane.
54
5.3
Homogeneous Space
So far, we have used a point on a plane and point on a line to characterize the actual position of the plane and line in the Euclidian Plane and Space. For example, to describe a plane in three dimensions we needed a bivector to define its orientation and a coordinate-vector to define its position. In this section, we will embed an n-dimensional space in a higher n + 1-dimensional space. This builds upon the traditional homogeneous model and it will enable us to use pure blades to describe elements from the n-space without the need for a supporting position.
5.3.1
Three dimensional homogeneous space
Let's look at the Euclidian plane and embed it in a three dimensional space. All we have to do is extend our vector basis e1 , e2 with a third unit vector e that is perpendicular to the other two: e1 · e2 = 0 e1 · e = 0 e2 · e = 0 e1 · e1 = |e1 | = 1 e2 · e2 = |e2 | = 1 e · e = |e| = 1
x y We will then describe every coordinate (x, y) in the euclidian plane as ( w , w , w ). w More commonly we make sure that w equals one so we can simply write (x, y, 1). By doing this, we have effectively taken the origin (0, 0) and placed it a dimension higher. This means that the origin is no longer a special case, allowing us to do positional geometry in easier ways, without the need for supporting coordinates.
Lines Let's construct the line through points P and Q. Instead of taking the direction vector P − Q and a supporting coordinate P or Q, we can simply take the bivector P ∧ Q and that's it. This is shown in figure 5.5. The resulting bivector uniquely defines the line through P and Q and conversely Q ∧ P defines the line with opposite direction. Recall that a bivector has no shape, only orientation and area. Hence, figure 5.5 displays three possible representations for the bivector P ∧ Q which are all the same, and all define the same line. Simply put, the bivector defines a plane, and the intersection of the plane with the Euclidian plane (depicted with the dashed plane) defines the line. The question is what this representation buys us. First, we can transform lines a lot easier now. For example, instead of having to rotate two entities (the position vector and the direction vector) we simply rotate the bivector. 55
This is shown in figure 5.6. Obviously we can already reason that the intersection of the two lines is at (2, 2, 1). The bivectors that correspondingly define
Another benefit shows up when we want to calculate the intersection point of two lines. This is where the meet operator becomes incredibly useful. Let's look at two lines in the homogeneous model, and see what their meet results in. We have a line through points P and Q defined by the bivector P ∧ Q, and a line through points N and M defined by the bivector N ∧ M . The intersection point will be exactly at:
dual of P ∧ Q using equation (4.5): (P ∧ Q)∗ = (8, 0, −4)∗ = (−4, 0, 8) = −4e1 + 0e2 + 8e Now to complete the meet operator: (P ∧ Q)∗ · (N ∧ M ) = (8, 0, −4)∗ · (−8, −4, 0) = (−4, 0, 8) · (−8, −4, 0) It is important to realize that the right hand side of the inner product is a bivector and not a vector. Thus, we have the inner product between a vector (on the left hand side) and a bivector. We will use the earlier definition of the contraction inner-product to calculate this. To be precise, we employ definition (3.18). The meet of the two lines is: (P ∧ Q)∗ (N ∧ M ) = ((P ∧ Q)∗ N ) ∧ M − N ∧ ((P ∧ Q)∗ M ) Now we notice two inner products between vectors, which we know how to calculate. They will result in scalars. Then we are left with outer products between scalars and bivectors. As it happens to be, the outer product of a scalar and a bivector is simply the intuitive scalar product. Hence, we calculate: ((−4, 0, 8) (0, 2, 1)) ∧ (4, 2, 1) − (0, 2, 1) ∧ ((−4, 0, 8) (4, 2, 1)) = 8 ∧ (4, 2, 1) − (0, 2, 1) ∧ −8 = (32, 16, 8) − (0, −16, −8) = (32, 32, 16)
x y Now remember that in homogeneous space every coordinate is defined as ( w , w , w), hence the intersection point of the two lines is:
(
32 32 16 , , ) = (2, 2, 1) 16 16 16
Which obviously, and according to figure 5.6, is correct. The complete calculation we just did is rather verbose and not something you would usually do. Instead, we just accept the meet operator and its definition using the inner product and the dual operator. Using the meet we can simply denote the intersection of two lines using the ∩ symbol.
5.3.2
Four dimensional homogeneous space
We have used a three dimensional space to model the Euclidean plane. We will now use C 4 to embed Euclidian space. Once again, we use a fourth basis vector e orthogonal to the other three basis vectors e1 , e2 and e3 . All coordinates x y z (x, y, z) will be denoted using ( w , w , w , w ). Again, if w equals one we can write w (x, y, z, 1). 58
Lines in 4D homogeneous space Constructing a line in this model is no different than we did before. For a line through points P and Q we take the outer product to construct a bivector that uniquely defines the line in three dimensions. It's rather complicated to depict this in a figure since we have no means to effectively display a four dimensional space. Fortunately, the same rules still apply. If we want to rotate a line we use a spinor. Of course such a spinor would now be a spinor in C 4 . Also, if we want the intersection of two lines we simply take the meet of the two bivectors. Let us take a brief moment to look at the bivector representation in the homogeneous model. As discussed before, there are six basis bivectors in C 4 . Instead of the basis vector e4 we have the special basis vector e. This creates the following set of basis bivectors: e1 ∧ e2 = e12 e1 ∧ e3 = e13 e2 ∧ e3 = e23 e1 ∧ e e2 ∧ e e3 ∧ e Thus, in a four dimensional geometric algebra, a bivector forms a linear combination of the six basis bivectors. Hence we need six components to define a three dimensional line. As it turns out, these six scalars are what other theories commonly refer to as Pl¨ckerspace coordinates. However, the only thing that is u six-dimensional about Pl¨ckerspace is the fact that we use six scalars. Everyu thing becomes much more intuitive once we realize that in the four dimensional homogeneous model bivectors define lines and arbitrary bivectors are linear combinations of the six basis bivectors. In fact, the well known Pl¨ckerspace u operations to define relations among lines can be expressed in terms of the meet and the related join operator which defines the union of two subspaces. Unfortunately, a more detailed discussion of the join and meet operators is beyond this paper. Planes Lines are defined by two coordinates, and we can express them by taking the outer product of the two coordinates. We can easily extend this. Planes are defined by three coordinates, and the homogeneous representation of a plane is simply the outer product of these three points. For example, the plane through points P , Q and R is defined by the trivector: P ∧Q∧R For example, given a triangle defined by those three points there is no need to calculate the edge vectors to perform a cross product, after which the distance 59
to the origin still needs to be calculated. We simply take the outer product of the three vertices to obtain the bivector which defines the plane. As with lines before, transforming the plane can simply be done by transforming the associated trivector. And not surprising, and very elegantly, the intersection of two planes or the intersection of a plane and a line can simply be done using the meet operator. The actual proof is beyond this paper, but trying out some numerical examples yourself should easily convince you that the meet operator does not discriminate among subspaces.
5.3.3
Concluding Homogeneous Space
Traditional Euclidian vector algebra always uses a supporting position vector to define the location of elements in space. This complicates all calculations since they have to act on two separate concepts (position and orientation) instead of just one. Furthermore, because the origin is treated as a special kind of position vector, we often run into special cases and exceptions in our algorithms. Homogeneous space allows us to use k-blades in an n-dimensional space to define positional (k − 1)-blades in an (n − 1)-dimensional space. This greatly simplifies our algorithms and it gives us the ability to reason about subspaces without the need for special cases or exceptions. At first sight, this elegance and ease of use comes with a considerable overhead; namely the extra storage required for the added dimension. However, at a second glance we notice that it's not entirely true. A vector takes three components in C 3 and four in C 4 . However, most often the last scalar is equal to one and could be optimized away. There does not seem to be much need for this though. The advantages of homogeneous vectors have already been acknowledged in many traditional applications. An obvious example is matrix-algebra where four-by-four matrices are used to store rotations and translations combined. In C 4 a bivector takes six scalars, and a trivector takes four scalars. Refer back to figure 2.3 for a reminder. We've used a bivector to represent a line, and a trivector to represent a plane. The amount of components required is therefore exactly the same as it would be in Euclidian space. There, we use two three-component vectors to represent a line, and four scalars to represent a plane. Admittedly, the homogeneous model is not optimal in all cases. There are certain operations and algorithms where a traditional approach is more suitable and other contexts where the homogeneous model is advantageous. Geometric algebra allows us to use both models alongside each other and use the one that best fits the situation.
60
Chapter 6
Conclusion
Geometric Algebra introduces two new concepts. First, it acknowledges that a vector is a one-dimensional subspace and hence that there should be higherdimensional subspaces as well. It then defines bivectors, trivectors and k-blades as the generalization of subspaces. Multivectors are linear combinations of several blades of different grades. Secondly, it defines the geometric product for multivectors. Because the geometric product embodies both the inner and outer product it combines the notion of orthogonality and collinearity into one operation. Combined with the fact that the geometric product gives most multivectors an inverse it becomes an extremely powerful operator capable of expressing many different geometric relations and ideas. Using blades and multivectors and the inner, outer and geometric product we can build up a set of tools consisting of the dual, inverses, projections and rejections, reflections, the meet and potentially much more. The fundamentals and the tools give us new ways of defining common elements like lines and planes. Both in traditional models and in the homogeneous model. It also gives us new ways to reason about the relations between elements. Furthermore, geometric algebra provides the full background for quaternion theory, finally demonstrating that there is nothing four-dimensional about them. Instead, they are just a linear combination of a scalar and a bivector. It also explains Pl¨ckerspace in terms of the homogeneous model and the join and meet u operators. Best of all, the entire theory is independent of any dimension. This allows us to provide independent proofs and algorithms that can be applied to any situation. In fact, it is often intuitive to sketch a proof in a lower dimension and then extend it to the general case. Geometric algebra greatly simplifies this process. Still, it is not just the mere elegance and power of geometric algebra that makes it interesting, but it is the simple fact that we don't lose any of our traditional methods. Geometric algebra explains and embraces them, and then enriches and unifies those existing theories into one. 61
6.1
The future of geometric algebra
Currently, there are two major obstacles that prohibit the mainstream acceptance of geometric algebra as the language of choice. First of all, the learning curve for geometric algebra is too steep. Existing introductory material assumes the reader has a certain mathematical knowledge that many doing geometry in practical environments lack. Furthermore, old habits die hard. So instead of presenting geometric algebra as a replacement of traditional methods, it should simply be a matter of language. Geometric algebra is a language to express old and new concepts in easier ways. Hopefully this paper succeeded in providing a relatively easy introduction and demonstrating that it can explain and expand on existing theories. Secondly, there is still a lot of work to be done when it concerns implementations and applications. Even though geometric algebra looks promising for computational geometry at first sight, it turns out that a mapping from theory to practice is not as straightforward as one would hope. Obviously the old methods and theories have been used in practice for years and have undergone severe tweaking and tuning. Geometric algebra still has a long way to go before we will see implementations that allow us to benefit from the theoretical expressiveness and elegance without sacrificing performance. At the moment it is best used in theory, after which actual implementations give up on the generality and simply provide specialized algorithms for specific problem domains. Fortunately, there are several initiatives to work on this second obstacle. The first is a library by Daniel Fontijne, Leo Dorst and Tim Bouma called Gaigen [11]. This library comes in the form of a code generator that generates optimized code given a certain algebra signature. Another initiative is the Clifford library [12] which uses meta programming techniques to transform generic function calls into specialized and optimized implementations.
6.2
Further Reading
This paper merely tries to provide a simple introduction to geometric algebra and some of its applications. Hopefully it gives the reader a small foundation on which to pursue more detailed and thorough material. I can strongly recommend reading chapters one, two and five of Hestenes' New Foundations for Classical Mechanics [8], which is still considered the best and most intuitive introduction to geometric algebra by many. All of the material on Leo Dorst's website [6] is very informative and useful, especially the two papers published in the IEEE Computer Graphics and Applications [2] [3]. Finally, the material on Hestenes' website [10] provides some rather advanced, but useful, material. Especially the Old Wine in New Bottles paper which introduces a new and very elegant model called the conformal split. If you wonder why planes are spheres through infinity and how this can be useful, this [9] is the paper to read. | 677.169 | 1 |
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