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predicting. As a core programmer, I love taking challenges and love being part of the solution. in Applied Numerical Linear Algebra Currently focused on combining statistical data mining techniques and traditional econometrics approach, panel data, model switching, GARCH type volatility modelling, volume modelling etc. If we are thinking of a vector as representing a physical quantity. To use MOE, we simply need to specify some objective function, some set of parameters, and any historical data we may have from previous evaluations of the objective function. If you want to fit a model of higher degree, you can construct polynomial features out of the linear feature data and fit to the model too. , Bengio, Y. The interplay of columns and rows is the heart of linear algebra. You've accumulated a good bit of data that looks like this:. BUT Linear Algebra is too boundless! In this book, you will get what is NECESSARY. There is no doubt that linear algebra is important in machine learning. The Linear Algebra Chapter in Goodfellow et al is a nice and concise introduction, but it may require some previous exposure to linear algebra concepts. pdf; TS CH8 Estimation. then this is the book for you. and his book on Linear Algebra is a very good introduction. Linear Algebra for Data Scientists. A lot of linear algebra over the complex field This free book Linear Algebra - As an Introduction to Abstract Mathematics from UC Davis has plenty of exercises; Terence Tao has a set of notes if you google, they go with the book Linear Algebra by Friedberg, Insel and Spence. Programming and data science articles by hadrienj. Linear Algebra: Foundations to frontiers – edx. This works in the latest snapshot of Breeze. Linear Algebra Examines basic properties of systems of linear equations, vector spaces, inner products, linear independence, dimension, linear transformations, matrices, determinants, eigenvalues, eigenvectors and diagonalization. GitHub; LinkedIn; Twitter; Donald Miner | {
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determinants, eigenvalues, eigenvectors and diagonalization. GitHub; LinkedIn; Twitter; Donald Miner (@donaldpminer) specializes in large-scale data analysis enterprise architecture and applying machine learning to real-world problems. Windows-64 (64-bit linear algebra for large data) Unless your computer has more than ~32GB of memory and you need to solve linear algebra problems with arrays containing more than ~2 billion elements, this version will offer no advantage over the recommended Windows-64 version above. Modern statistics is described using the notation of linear algebra and modern statistical methods harness the tools of linear algebra. As a core programmer, I love taking challenges and love being part of the solution. The aim of these notebooks is to help beginners/advanced beginners to grasp linear algebra concepts underlying deep learning and machine learning. Matrices, vectors, and more - from theory to the real world! There's a lot of data out there, learn how to search it effectively. A few weeks ago, I wrote about how and why I was learning Machine Learning, mainly through Andrew Ng’s Coursera course. LINEAR ALGEBRA. Today, I will be sharing with you my C# implementation of basic linear algebra concepts. Hundreds of thousands of students have already benefitted from our courses. Linear Algebra and Learning from Data by Gilbert Strang; 1 edition; twitter github. Some of these assignments are from Introduction to applied linear algebra - vectors, matrices, and least squares. Introduction to linear algebra (Fourth Edition). , Bengio, Y. Linear Regression 101 (Part 1 - Basics) 6 minute read Introduction. My work includes researching, developing and implementing novel computational and machine learning algorithms and applications for big data integration and data mining. Various CNN and RNN models will be covered. Learning Spark : lightning-fast data analytics by Holden Karau, Andy Konwinski, Patrick Wendell, and Matei Zaharia, O’Reilly, 2015. Thesis: | {
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by Holden Karau, Andy Konwinski, Patrick Wendell, and Matei Zaharia, O’Reilly, 2015. Thesis: Quantum Algorithms for Linear Algebra and Machine Learning; Anupam Prakash Quantum Algorithms for Linear Algebra and Machine Learning by Anupam Prakash Most quantum algorithms o ering speedups over classical algorithms are based on the three techniques of phase estimation, amplitude estimation and Hamiltonian simulation. Implementation [ edit ] Scikit-learn is largely written in Python, and uses numpy extensively for high-performance linear algebra and array operations. Currently he is on leave from UT Austin and heads the Amazon Research Lab in Berkeley, California, where he is developing and deploying state-of-the-art machine learning methods for Amazon Search. Support Stability of Maximizing Measures for Shifts of Finite Type Journal of Ergodic Theory and Dynamical Systems (accepted) Calkins, S. Those equations may or may not have a solution. Problem solving with algorithms and data structures using Python. Posted by u/[deleted] a linear algebra library in R designed for teaching. Foundations of Data Science is a treatise on selected fields that form the basis of Data Science like Linear Algebra, LDA, Markov Chains, Machine Learning basics, and statistics. in machine learning, it is standard to say “N samples” to mean the same thing. 065 Linear Algebra and Learning from Data New textbook and MIT video lectures OCW YouTube; 18. pdf; TS CH8 Estimation. Government and Fortune 500 companies. These functions are mainly for tutorial purposes in learning matrix algebra ideas using R. Siefken, J. Complete Linear Algebra for Data Science & Machine Learning 4. A few weeks ago, I wrote about how and why I was learning Machine Learning, mainly through Andrew Ng’s Coursera course. You will also learn how you should use linear algebra in your Python code. in linear algebra, kernel is another name for nullspace. Now that you understand the key ideas behind linear regression, we can | {
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another name for nullspace. Now that you understand the key ideas behind linear regression, we can begin to work through a hands-on implementation in code. Linear algebra has had a marked impact on the field of statistics. 1-11, [Online-Edition. " Our homework assignments will use NumPy arrays extensively. squares methods, basic topics in applied linear algebra. Perception, movement control, reinforcement learning, mathematical psychology, … Economics. As the complementary course to. Then last year I learned how he morphed his delightful mathematics book into a brand new title (2019) designed for data scientists - "Linear Algebra and Learning from Data. In some cases, functions are provided for concepts available elsewhere in R, but where the function call or name is not obvious. The concepts of Linear Algebra are crucial for understanding the theory behind Machine Learning, especially for Deep Learning. Course Description. Julia Observer helps you find your next Julia package. Linear algebra underlies many practical mathematical tools, such as Fourier series and computer. ” See Section 6. Description. Foundations of Data Science is a treatise on selected fields that form the basis of Data Science like Linear Algebra, LDA, Markov Chains, Machine Learning basics, and statistics. [Online book] n Andrew Ng. This class is an in-depth graduate lecture class. uk, [email protected] Together with your editor or Jupyter notebook these packages allow you to rapidly develop scalable, high-performance analytics and visualizations using succinct, type-safe, production-ready code. Book: Aurélien Géron "Hands-On Machine Learning with Scikit-Learn and TensorFlow" Book: Andriy Burkov "The Hundred-Page Machine Learning Book " 🐍 Python Course: Python. It would be best if you had an organized book which (1) teaches the most used Linear Algebra concepts in Machine Learning, (2) and utilize your Machine Learning model in terms of data processing, optimization, and validation. They give | {
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your Machine Learning model in terms of data processing, optimization, and validation. They give you better intuition for how algorithms really work under the hood, which enables you to make better decisions. This page has links for latest PDF versions of the text and related supplements. * Ranked among top 10% answerers on Python in StackOverflow. Now we extend linear algebra to convolutions, by using the example of audio data analysis. He was a research fellow with Michael Jordan and Peter Bartlett, University of California at Berkeley, from 2003, and with Bernhard Schoelkopf, Max Planck Institute for Intelligent Systems, Tuebingen, Germany, from 2005. 065 Linear Algebra and Learning from Data New textbook and MIT video lectures OCW YouTube; 18. Learning path Introduction to Linear Algebra. Learn the basic math for Data Science, AI, and ML using R. This book is directed more at the former audience. Getting started with linear algebra. Feature learning and subspaces (Chapter 3 of [BHK]) Random walk and Markov chain Monte Carlo (Chapter 4 of [BHK]) Linear models, kernel methods, and deep learning ([Bishop] and Chapter 5 of [BHK]) Algorithms for Massive Data: streaming, sketching, and sampling (Chapter 6 of [BHK]) The course schedule is available here. Incorporating machine learning capabilities into software or apps is quickly becoming a necessity. Implementation [ edit ] Scikit-learn is largely written in Python, and uses numpy extensively for high-performance linear algebra and array operations. His main research interests are in big data, machine learning, network analysis, linear algebra and optimization. We create tools for phenotype analyses that make use of the entire clinical phenotyping spectrum, not only using HPO, but also model organisms data (we also create the uberpheno) and other ontologies. Machine Learning is built on prerequisites, so much so that learning by first principles seems overwhelming. Modern statistics is described using the notation of | {
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by first principles seems overwhelming. Modern statistics is described using the notation of linear algebra and modern statistical methods harness the tools of linear algebra. Students will learn and practice fundamental ideas of linear algebra and simultaneously be exposed to and work with real-world applications of these ideas. If is high, the matrix is said to be ill-conditioned. Matrix sketching and randomized matrix computation. I’d like to go over the theory behind this matrix decomposition and show you a few examples as to why it’s one of the most useful mathematical tools you can have. It’s all vectors and matrices of numbers. Topic 1: Review of Linear Algebra 1-6 These are only a few examples that I hope help convince you that vector spaces are the backbone of machine learning. Linear Algebra¶ Now that you can store and manipulate data, let’s briefly review the subset of basic linear algebra that you will need to understand most of the models. •LACore is a Large-Format vector accelerator for a broad range of Linear Algebra applications •LACore has novel architectural features including as the: • configurable, data-streaming LAMemUnits • dual-precision, configurable, systolic LAExecUnit •A compiler toolchain, programming framework and architectural simulator were all. For simple linear regression, one can choose degree 1. After reading this post, you will know:. Python for Data Science and Machine Learning Bootcamp; Think Stats - Book. I have hands-on experience in Data Analysis, Machine Learning, Natural Language Processing, deployment on IaaS like AWS. , how to pass the course, schedules, and deadlines, at the official course page. Linear Algebra for Machine Learning Book. Currently he is on leave from UT Austin and heads the Amazon Research Lab in Berkeley, California, where he is developing and deploying state-of-the-art machine learning methods for Amazon Search. for automated market making. Grading: 3 homeworks 60%, 2 quizzes 20%, 1 project 20%. edu | {
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Search. for automated market making. Grading: 3 homeworks 60%, 2 quizzes 20%, 1 project 20%. edu ABSTRACT Accelerating machine learning (ML) over relational data is a. , Bengio, Y. This repository contains all the quizzes/assignments for the specialization "Mathematics for Machine learning" by Imperial College of London on Coursera. Some recent tutorials by Christos and Co. Lawrence [email protected] Welcome to Data analysis with Python - 2020¶. The aim of these notebooks is to help beginners/advanced beginners to grasp linear algebra concepts underlying deep learning and machine learning. A vector in Rn will be denoted as: ~x. Enabling and Optimizing Non-linear Feature Interactions in Factorized Linear Algebra Side Li University of California, San Diego [email protected] HarvardX Biomedical Data Science Open Online Training In 2014 we received funding from the NIH BD2K initiative to develop MOOCs for biomedical data science. Now I’m checking back in with 9 weeks under my belt. Introduction to Machine Learning for Data Science, Udemy. Learn linear algebra. The first part covers basics and preliminaries. Incorporating machine learning capabilities into software or apps is quickly becoming a necessity. Some recent tutorials by Christos and Co. pyplot as plt # pyplot from collections import defaultdict , Counter from functools import partial , reduce. pyplot as plt # pyplot from collections import defaultdict , Counter from functools import partial , reduce. Introduction The aim of this set of lectures is to review some central linear algebra algorithms that we. In case of deep learning algorithms, linear algebra is the driving force. We study fast algorithms for linear algebraic problems that are ubiquitous in data analysis and machine learning. After understanding the nature of both the problem and the field we are dealing with, and before learning how to prepare a data for your machine learning and do the cleaning and preparation for a selected problem. If you're a | {
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for your machine learning and do the cleaning and preparation for a selected problem. If you're a data scientist studying linear algebra, chances are you are interested in understanding how machine learning algorithms work. Linear Algebra and Learning from Data by Gilbert Strang; twitter github. 06 Linear Algebra - The video lectures are on web. I will list some resources for learning linear algebra. Andrzej Cichocki, Anh-Huy Phan, Qibin Zhao, Namgil Lee, Ivan Oseledets, Masashi Sugiyama, and Danilo Mandic. Linear Algebra for Machine Learning Book. You will also learn how you should use linear algebra in your Python code. Probability and Statistics: Learn Probability and Statistics Through Interactive Visualizations: Seeing Theory was created by Daniel Kunin while an undergraduate at Brown University. As a core programmer, I love taking challenges and love being part of the solution. Linear regression is one of the most popular machine learning algorithms. , and Courville, A. The elements of statistical learning: data mining, inference, and prediction, Springer, 2009 Linear Algebra and Probability Review (part 1 Linear Algebra, part 2 Probability) Assignment 1: Apr 10. Those equations may or may not have a solution. There is no doubt that linear algebra is important in machine learning. View on GitHub mlcourse. Run in Google Colab View source on GitHub Download notebook In this post, we will explore the ways of doing linear algebra only using tensorflow. Read more Tagged as : R linear algebra classification linear discriminant analysis. My work includes researching, developing and implementing novel computational and machine learning algorithms and applications for big data integration and data mining. You can use it as a main text, as a supplement, or for independent study. I’m interested in applying non-standard tools form abstract algebra and topology to the study of neural networks. About data set: Square feet is the Area of house. uk November 1, 2018 Abstract | {
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of neural networks. About data set: Square feet is the Area of house. uk November 1, 2018 Abstract Development systems for deep learning (DL), such as Theano, Torch, TensorFlow, or MXNet, are. Linear algebra has had a marked impact on the field of statistics. In some cases, functions are provided for concepts available elsewhere in R, but where the function call or name is not obvious. GF2] = One Zero Zero Zero Zero Zero One Zero Zero Zero Zero Zero One Zero Zero Zero Zero Zero One Zero Zero Zero Zero Zero One scala> a + a res0: breeze. Siefken, J. APPLICATION: This can be pretty much applied in a generic way to all programs. In general, statistical problems have to do with the estimation of some characteristic derived from data - this can be a point estimate, an interval, or an entire function. Linear Algebra: Foundations to frontiers – edx. Linear Algebra for Machine Learning Discover the Mathematical Language of Data in Python. data in homework problems. Machine learning. Anyone can view the notebooks online by clicking on the links in the readme Table of Contents. In this day, we are going to make the dirty work. In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. In other cases, functions are provided to show or. Linear algebra (Systems of linear equations, least-square) [Matrix cookbook] 4: 9/16/2019: Linear Algebra (Vector space, linear independence) 9/18/2019: Linear algebra (Eigendecomposition and matrix factorization) Homework 1 (extended) 5: 9/23/2019: Linear algebra (Eigendecomposition and matrix factorization) 9/25/2019: Linear algebra (Best fit. Acquiring these skills can boost your ability to understand and apply various data science algorithms. The basic mathematics prerequisites for understanding Machine Learning are Calculus-I,II,III, Linear Algebra, and, Probability and Statistics. Making statements based on opinion; back them up with references or personal experience. Randomized numerical | {
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based on opinion; back them up with references or personal experience. Randomized numerical linear algebra. It lacks the ability of distributed linear algebra computation in its local interactive shell. You can use it as a main text, as a supplement, or for independent study. NVIDIA CUDA-X GPU-Accelerated Libraries NVIDIA® CUDA-X, built on top of NVIDIA CUDA®, is a collection of libraries, tools, and technologies that deliver dramatically higher performance—compared to CPU-only alternatives— across multiple application domains, from artificial intelligence (AI) to high performance computing (HPC). Lek-Heng Lim. taco is versatile. However, I think that the chapter on linear algebra from the book is a bit tough for beginners. A few weeks ago, I wrote about how and why I was learning Machine Learning, mainly through Andrew Ng’s Coursera course. scala> val a = DenseMatrix. This code has been posted to GitHub under a MIT license, so feel free to modify and deal with code without any restrictions or limitations (no guarantees of any kind. world Overview of scikit-learn Python and Excel Scaling, Centering, Noise with kNN, Linear Regression, Logit Sentiment Analysis with Twitter Time Series Analysis Vectors and Arrays (Linear Algebra) Vectors and Arrays (Linear Algebra) Table of contents. Lesson 1 (April 7): Machine learning pipeline and course overview: video; slides. Also,it would be of much help if they have big set of problems and examples. Machine Learning course (Andrew Ng) is a basic machine learning course. It begins with linear algebra—matrix factorizations A= QR. therefore precede our in tro duction to deep learning with a fo cused presen tation of. Linear algebra cheat sheet for deep learning – Towards Data Science – Medium. They give you better intuition for how algorithms really work under the hood, which enables you to make better decisions. It’s all vectors and matrices of numbers. Today, I will be sharing with you my C# implementation of basic linear algebra | {
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matrices of numbers. Today, I will be sharing with you my C# implementation of basic linear algebra concepts. In an image classification problem, we often use neural networks. Matrices and Linear Algebra The Wolfram Language automatically handles both numeric and symbolic matrices, seamlessly switching among large numbers of highly optimized algorithms. R is a widely-used statistical programming language in the data science community. Artificial Neural Networks. Grading (tentative) Quizzes 20%; Course project. Together with your editor or Jupyter notebook these packages allow you to rapidly develop scalable, high-performance analytics and visualizations using succinct, type-safe, production-ready code. eye[GF2](5) a: breeze. Windows-64 (64-bit linear algebra for large data) Unless your computer has more than ~32GB of memory and you need to solve linear algebra problems with arrays containing more than ~2 billion elements, this version will offer no advantage over the recommended Windows-64 version above. For beginning practitioners (i. View on GitHub mlcourse. Tibshirani, J. Open Library is an initiative of the Internet Archive, a 501(c)(3) non-profit, building a digital library of Internet sites and other cultural artifacts in digital form. Another resource is the book with the funny title “No Bullshit Guide to Linear Algebra” by Ivan Savov. It introduces some common tools in machine learning to resolve real applications (e. It calls them tensors. Linear Algebra Examines basic properties of systems of linear equations, vector spaces, inner products, linear independence, dimension, linear transformations, matrices, determinants, eigenvalues, eigenvectors and diagonalization. You will also learn how you should use linear algebra in your Python code. mathematics-for-machine-learning-cousera. We start with representing a fully connected layer as a form of matrix multiplication: - In this example, the weight matrix has a size of $4 \times 3$, the input vector has a size | {
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- In this example, the weight matrix has a size of $4 \times 3$, the input vector has a size of $3 \times 1$ and the output vector has a of size $4 \times 1$. The code (from. Understanding API Security (Free chapters from a Manning. It turns out, however, that all of those operations can be written in terms of big matrix-matrix or matrix-vector multiplications. 1x Scalable Machine Learning. AMD adopted BLIS as its new BLAS library. After reading this quickstart, you can go to other wiki pages, especially Linear Algebra Cheat-Sheet and Data Structures. I'm studying towards a PhD degree at the University of Chicago, in the department of Statistics. SciPy is built to work with NumPy arrays and provides many. They are full of explanations, code samples, pictures, interesting links, and exercises for you to try. linear; algebra; matrix;. import re , math , random # regexes, math functions, random numbers import matplotlib. By Hadrien Jean, Machine Learning Scientist. Compressed Linear Algebra for Declarative Large-Scale Machine Learning Ahmed Elgohary2, Matthias Boehm1, Peter J. What we did here by attaching the variable mlr to the MyLinearRegression class is to create an instance, a specific object called mlr, which will have its own data and "functions". In the second part, we discuss how deep learning differs from classical machine learning and explain why it is effective in dealing with complex problems such as image and natural language processing. Plotting is based on OpenGL and supports both 2D and 3D plots. Introduction to linear algebra (Fourth Edition). DenseMatrix[X. This content is part of a series following the chapter 2 on linear algebra from the Deep Learning Book by Goodfellow, I. therefore precede our in tro duction to deep learning with a fo cused presen tation of. You (the student) should have taken a mathematical course on linear algebra that covers vector spaces as well as a numerical analysis course that covers computer implementations of numerical | {
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spaces as well as a numerical analysis course that covers computer implementations of numerical algorithms. Fitting Lines to Data. net/?Q73_jQ 2020-01-24T14:53:56+01:00 2020-01-24T15:29:05+01:00. Linear Algebra for Data Science using Python Play all 13:42 Math For Data Science | Practical reasons to learn math for Machine/Deep Learning - Duration: 13 minutes, 42 seconds. The mentors for this project are: @dpshelio @mbobra @drsophiemurray @samaloney. Randy Lao's site for free Machine Learning and Data Science resources and materials. How to Learn Advanced Mathematics Without Heading to University - Part 1 subjects to learn for a prospective quant or data scientist. The Cuckoo linear algebra implementation is based on libcuckoo library1. to map the pixel values of an image to the confidence score of each class. This library holds the principal work done as part of the OpenAstonomy Google Summer of Code 2020 project, Solar Weather Forecasting using Linear Algebra. library with basic linear algebra routines, and the SciPy library adorns NumPy arrays with many important primitives, from numerical optimizers and signal processing to statistics and sparse linear algebra. Download R for Windows 5. Linear Algebra Preliminaries¶ Since I have documented the Linear Algebra Preliminaries in my Prelim Exam note for Numerical Analysis, the interested reader is referred to for more details (Figure. Intro to Data Science / UW Videos. Now we extend linear algebra to convolutions, by using the example of audio data analysis. The Linear Algebra Chapter in Goodfellow et al is a nice and concise introduction, but it may require some previous exposure to linear algebra concepts. AMD adopted BLIS as its new BLAS library. Each point correspondence generates one constraint on F. I had to formulate an algorithm to convert an image of some resolution - say L x M and crop / re-size the image into a new resolution say P x R such that I cover the maximum amount of points/pixels from the original | {
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a new resolution say P x R such that I cover the maximum amount of points/pixels from the original image. Again, the class MyLinearRegression provides instructions on how to build a linear regression model. Linear Algebra: Video: Professor Gilbert Strang's Video Lectures on linear algebra. Above, I created 4 matrices. In addition to this, you’ll be able to perform operations such as addition, subtraction and dot product. It introduces some common tools in machine learning to resolve real applications (e. Conncect between Geometry and Algebra. As of October 2019, I am a senior algorithms scientist at PathAI, where I work on computational pathology. California Housing Price Prediction. Linear Optimization in Python 6. The aim of these notebooks is to help beginners/advanced beginners to grasp linear algebra concepts underlying deep learning and machine learning. 6 (376 ratings) Course Ratings are calculated from individual students’ ratings and a variety of other signals, like age of rating and reliability, to ensure that they reflect course quality fairly and accurately. View on GitHub. They give you better intuition for how algorithms really work under the hood, which enables you to make better decisions. Calculate The Trace Of A. pdf CarletonU- hamidreza sadreazami- radar based fall detection with supervised learning. However, it is currently not easy to implement many basic machine learning primitives in. " Our homework assignments will use NumPy arrays extensively. We start with representing a fully connected layer as a form of matrix multiplication: - In this example, the weight matrix has a size of $4 \times 3$, the input vector has a size of $3 \times 1$ and the output vector has a of size $4 \times 1$. In forecasting, Yuyang has worked on all aspects ranging from practical applications to theoretical foundations. I'm a Data Science practitioner and computer programmer with an avid interest in Exploratory Data Analysis, Statistics, & Machine Learning. edu Arun | {
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with an avid interest in Exploratory Data Analysis, Statistics, & Machine Learning. edu Arun Kumar University of California, San Diego [email protected] DenseMatrix[X. SciPy is open-source software for mathematics, science, and engineering which includes modules for statistics, optimisation, integration, linear algebra, Fourier transforms, signal and image processing, ODE solvers, and more. Now I’m checking back in with 9 weeks under my belt. Mining the social web: Data mining Facebook, Twitter, LinkedIn, Google+, GitHub, and more (2nd edition) by Matthew A. The Deep Learning Book - Goodfellow, I. Most importantly, the online version of the book is completely free. Based on the second linear algebra course taught by Professor Strang, whose lectures on the training data are widely known, it starts from scratch (the four fundamental subspaces) and is fully accessible without the first text. Siefken, J. then this is the book for you. The aim of these notebooks is to help beginners/advanced beginners to grasp linear algebra concepts underlying deep learning and machine learning. How to Learn Advanced Mathematics Without Heading to University - Part 1 subjects to learn for a prospective quant or data scientist. Rich Ott leads you through two days of intensive learning that include a review of linear algebra essential to machine learning, an introduction to TensorFlow, and a dive into neural networks. All publications, sorted by year. This book provides the conceptual understanding of the essential linear algebra of vectors and matrices for modern engineering and science. n Gilbert Strang. As a motivating example, let us consider image classification. The collection of all linear combinations is called a linear subspace of $\RR^n$, denoted by We will say that the $\bb{v}_i$’s span the linear subspace $\mathcal{L}$. If you have more time to dedicate to your projects and you're also passionate about math, consider to contribute to the library!. Roadmap to begin with Machine | {
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also passionate about math, consider to contribute to the library!. Roadmap to begin with Machine Learning: The place to start is to learn (and/or) revise linear algebra. Find the smallest value of k such that the rank-k approximation of the matrix uses the same or more amount of data as the original picture. Bourbaki resulted from similar currents of thought that produced fascism and totalitarian communism: moral panics leading to revolutions, and ultimately “final solutions”, all terrible and evil in. Videos and textbooks with relevant details on linear algebra and singular value decomposition (SVD) can be found by searching Alfredo’s Twitter, for example type linear algebra (from:alfcnz) in the search box. Machine Learning/Data Science. , less sensitive to noisy data) than the Euclidean norm The following result is fundamental in linear algebra: Theorem. This is the world beyond R and Python! Breeze is a library for numerical processing, like probability and statistic functions, optimization, linear algebra, etc. Mathematics for machine learning - I totally recommend this book! If you want to learn the bits and pieces of how linear algebra and calculus is used to develop algorithms like principal component analysis, backpropagation etc. a person's height and you switch from meter to centimeter. Linear Algebra and Learning from Data by Gilbert Strang; twitter github. You NEED Linear Algebra for Machine Learning Whether you want to learn Machine Learning for your work or research or you want to become a master, so the others pay you to do it, you need to know how it works. Linear Algebra: Video: Professor Gilbert Strang's Video Lectures on linear algebra. You will also learn how you should use linear algebra in your Python code. There is no doubt that linear algebra is important in machine learning. Tensor networks for dimensionality reduction and large-scale optimization: part 2 applications and future perspectives. Understanding of calculus, linear algebra, and | {
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part 2 applications and future perspectives. Understanding of calculus, linear algebra, and programming is essential. to map the pixel values of an image to the confidence score of each class. Start here. PBblas - Parallel Block Linear Algebra Subsystem HPCC Systems ML_Core repository on GitHub Installation: ecl bundle install https:. We emphasize that this document is not a. References for "practical" machine learning: Python for data analysis by Wes McKinney. The Vector class imitates the m x 1 vector from linear algebra and contains many useful functions for dealing and interacting with Vectors. The free video lectures of this course are made available as part of Harvard Extension School's Opening Learning Initiative. After reading this quickstart, you can go to other wiki pages, especially Linear Algebra Cheat-Sheet and Data Structures. You can think of an r t i m e s c r times c r t i m e s c matrix as a set of r r r row vectors, each having c c c elements; or you can think of it as a set of c c c column vectors, each having r r r elements. " Our homework assignments will use NumPy arrays extensively. This is a straightforward course to learn Linear Algebra Fundamentals for Data Science in Python. linear; algebra; matrix;. 1-11, [Online-Edition. Row Reduction We row reduce a matrix by performing row operations, in order to find a simpler but equivalent system for which the solution set is easily read off. Support Stability of Maximizing Measures for Shifts of Finite Type Journal of Ergodic Theory and Dynamical Systems (accepted) Calkins, S. If you want to fit a model of higher degree, you can construct polynomial features out of the linear feature data and fit to the model too. Supratim Haldar Lead Data Scientist at Head Digital Works Pvt. Popular Courses Popular with our users in the last month Introduction to Linear Algebra. Made for sharing. Linear Algebra The Rank of a Matrix. For simple linear regression, one can choose degree 1. Linear algebra provides a | {
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Rank of a Matrix. For simple linear regression, one can choose degree 1. Linear algebra provides a way of compactly representing and operating on sets of linear equations. This repository contains the learning material for the Nuclear TALENT course Learning from Data: Bayesian Methods and Machine Learning, in York, UK, June 10-28, 2019. Omoju: What I do at GitHub is I build data models, often deep learning models on GitHub data to help GitHub probably build things like a recommendation engine so we can recommend repositories to people. The first part covers basics and preliminaries. , hackers, coders, software engineers, and people working as data scientists in business and industry) you don't need to know that much calculus, linear algebra, or other college-level math to get things done. Grannan, S. This is the site for any aspiring data scientists that want to learn in a quick way. Decision theory, game theory, operational research, … (source: lecture video from The Machine Learning Summer School by Zoubin Ghahramani, Univ. Machine Learning is built on prerequisites, so much so that learning by first principles seems overwhelming. A computer science student that is interested in Machine Learning would be well advised to get a minor in Mathematics (or just get a degree in Mathematics instead!). 1 Positive Semide nite (PSD) and Positive De nite (PD) matrices. 08 Apr 2016 » Naive Bayes Classifiers in Rust Adding NB Classifiers to rusty-machine. References for "practical" machine learning: Python for data analysis by Wes McKinney. provide a summary of the mathematical background needed for an introductory class in machine learning, which at UC Berkeley is known as CS 189/289A. Machine Learning is not just writing Python or R. I still plan to go through it but mildly disappointed. 8 Web Framework. I'm a Data Science practitioner and computer programmer with an avid interest in Exploratory Data Analysis, Statistics, & Machine Learning. This will allow us to introduce | {
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in Exploratory Data Analysis, Statistics, & Machine Learning. This will allow us to introduce some central programming features of high-level languages like Python and compiled languages like C++ and/or Fortran. In my opinion, it is one of the bedrock of machine learning, deep learning and data science. Many universities use the textbook Introduction to Linear Algebra. Implementation [ edit ] Scikit-learn is largely written in Python, and uses numpy extensively for high-performance linear algebra and array operations. This content is part of a series following the chapter 2 on linear algebra from the Deep Learning Book by Goodfellow, I. Probability and Statistics: Learn Probability and Statistics Through Interactive Visualizations: Seeing Theory was created by Daniel Kunin while an undergraduate at Brown University. In addition to this, you'll be able to perform operations such as addition, subtraction and dot product. As a core programmer, I love taking challenges and love being part of the solution. Students will learn and practice fundamental ideas of linear algebra and simultaneously be exposed to and work with real-world applications of these ideas. We will assume mathematical maturity and comfort with algorithms, probability, and linear algebra. Neural networks rely on it heavily, but so do linear regression, factor analysis, and lots of other methods. This Word Mover’s Distance (WMD) can be seen as a special case of Earth Mover’s Distance (EMD), or Wasserstein distance, the one people talked about in Wasserstein GAN. These subjects include matrix algebra, vector spaces, eigenvalues and eigenvectors, symmetric matrices, linear transformations, and more. In this course, you’ll learn how to work with vectors and matrices, solve matrix-vector equations, perform eigenvalue/eigenvector analyses and use principal component analysis to do dimension reduction on real-world datasets. In the field of data science, however, being familiar with linear algebra and | {
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real-world datasets. In the field of data science, however, being familiar with linear algebra and statistics is very important to statistical analysis and prediction. Lawrence [email protected] Then last year I learned how he morphed his delightful mathematics book into a brand new title (2019) designed for data scientists - "Linear Algebra and Learning from Data. Choi et al. Now that you understand the key ideas behind linear regression, we can begin to work through a hands-on implementation in code. Linear Algebra The Rank of a Matrix. Linear Algebra. Gradient Descent with Linear Regression - GitHub Pages. The training data is used to find the optimal model but the model should ultimately work for the test data! Conclusion. Mathematics for machine learning - I totally recommend this book! If you want to learn the bits and pieces of how linear algebra and calculus is used to develop algorithms like principal component analysis, backpropagation etc. https://shaarli. Linear Algebra for Data Science using Python Play all 13:42 Math For Data Science | Practical reasons to learn math for Machine/Deep Learning - Duration: 13 minutes, 42 seconds. Twitter: @mpd37, @AnalogAldo, @ChengSoonOng. com) Bangalore, India * Working as a Lead Data Scientist at Head Digital Works Pvt. Using least-squares linear approximation techniques to find the best linear fit to a set of data points results in the equation of a line which minimizes the sum of the squares of the vertical distances from the given points to the line: Note that, unless the line is horizontal, the vertical distance will be slightly larger than the actual distance, which is measured in the direction. I recently released an efficient linear algebra library for Javascript. LINEAR ALGEBRA. Learn how to solve challenging machine learning problems with TensorFlow, Google’s revolutionary new software library for deep learning. import re , math , random # regexes, math functions, random numbers import matplotlib. Statistical | {
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import re , math , random # regexes, math functions, random numbers import matplotlib. Statistical Machine Learning (S2 2017) Deck 6 This lecture • Notes on linear algebra ∗Vectors and dot products ∗Hyperplanes and vector normals • Perceptron ∗Introduction to Artificial Neural Networks ∗The perceptron model ∗Stochastic gradient descent 2. A few weeks ago, I wrote about how and why I was learning Machine Learning, mainly through Andrew Ng’s Coursera course. Deep Learning Book Series · 2. Overview - Khan Academy Vectors and Spaces; Matrix Transformations; Python. This course develops the mathematical basis needed to deeply understand how problems of classification and estimation work. pdf; Basic Set Notation & Terminology. Machine learning yearning. hdf5 is a file. If you have more time to dedicate to your projects and you’re also passionate about math, consider to contribute to the library!. Module 0: Introduction & Outline About What is Data Science? Module 1: Required Background Math: Stats, Calculus, Linear Algebra Programming: Basics, Data Structures, Algorithms Databases: Relational Algebra, SQL Important Concepts: Regular expressions, Information Entropy, Distance measurements, OLAP, ETL, BI VS BA and CAP. NOTE: please check for the course practicalities, e. implemented vector and matrix classes with reST-formatted docstrings in Python 3+ General Layout. Linear Algebra Can Help You Choose Your Stock Portfolio Correlation is a very fundamental and viseral way of understanding how the stock market works and how strategies perform. Some recent tutorials by Christos and Co. Python is one of the most commonly used programming languages by data scientists and machine learning engineers. for automated market making. Udacity is the world’s fastest, most efficient way to master the skills tech companies want. I'm interested in applying non-standard tools form abstract algebra and topology to the study of neural networks. We will learn scientific computing basics, topics | {
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and topology to the study of neural networks. We will learn scientific computing basics, topics in numerical linear algebra, mathematical probability (probability spaces, expectation, conditioning, common distributions, law of large numbers and the central limit theorem), statistics (point estimation, confidence intervals, hypothesis testing, maximum likelihood estimation, density. This book provides the conceptual understanding of the essential linear algebra of vectors and matrices for modern engineering and science. Introduction Theunprecedentedadvanceindigitaltechnologyduringthesecondhalfofthe20thcenturyhas producedameasurementrevolutionthatistransformingscience. Now we are ready to see how matrix algebra can be useful when analyzing data. The linux command line: A complete introduction. As we will see, we can do all the common linear algebra operations without using any other library. The elements of statistical learning: data mining, inference, and prediction, Springer, 2009 Linear Algebra and Probability Review (part 1 Linear Algebra, part 2 Probability) Assignment 1: Mar 17. Many popular machine learning methods, including XGBOOST, use matrices to store inputs and process data. In this day, we are going to make the dirty work. Linear Algebra (Michael Damron and Tasho Kaletha) Introduction to Linear Algebra (Strang) Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra (Matousek) Linear Algebra Done Right (Axler) Advanced Linear Algebra (Roman). TS CH9 Hypothesis Testing. Research labs and companies have data to analyze and understand, and this deep learning approach has become widespread. You can find all the notebooks on Github. He was a research fellow with Michael Jordan and Peter Bartlett, University of California at Berkeley, from 2003, and with Bernhard Schoelkopf, Max Planck Institute for Intelligent Systems, Tuebingen, Germany, from 2005. Gilbert Strang is a Professor of Mathematics at Massachusetts Institute of | {
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Germany, from 2005. Gilbert Strang is a Professor of Mathematics at Massachusetts Institute of Technology and an Honorary Fellow at Balliol College in Oxford. pdf; TS CH1 Exploratory Data Analysis. The final exam is 9am on Friday May 15 in JNSN-Ice Rink. Just want some books to go deeper than a introductory course. This course develops the mathematical basis needed to deeply understand how problems of classification and estimation work. Learn how to solve challenging machine learning problems with TensorFlow, Google's revolutionary new software library for deep learning. In the second part, we discuss how deep learning differs from classical machine learning and explain why it is effective in dealing with complex problems such as image and natural language processing. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. This is a straightforward course to learn Linear Algebra Fundamentals for Data Science in Python. i recently bought Gilbert Strang's linear algebra book. then this is the book for you. Also, this OpenGL tutorial has useful explanations. A good video series on the topic that allows you to visualize many concepts is Essence of linear algebra. Welcome to the 18. Our goal is to give the beginning student, with little or no prior exposure to linear algebra, a good ground-ing in the basic ideas, as well as an appreciation for how they are used in many applications, including data tting, machine learning and arti cial intelligence, to-. Before that, I was a software engineer at Google where I worked 80% time with the Hotels team on data analytics and 20% time with the operations research team on linear program solvers. Thank you for your interest in Linear Algebra and Learning from Data. In this course, you’ll learn how to work with vectors and matrices, solve matrix-vector equations, perform eigenvalue/eigenvector analyses and use principal component analysis to do dimension reduction on | {
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eigenvalue/eigenvector analyses and use principal component analysis to do dimension reduction on real-world datasets. Learn linear algebra. Then, in Section 2, we quickly bring you up to speed on the prerequisites required for hands-on deep learning, such as how to store and manipulate data, and how to apply various numerical operations based on basic concepts from linear algebra, calculus, and probability. Communication The vast majority of questions about homework, the lectures, or the course should be asked on our Piazza forum, as others will benefit from the responses. I still plan to go through it but mildly disappointed. You can think of an r t i m e s c r times c r t i m e s c matrix as a set of r r r row vectors, each having c c c elements; or you can think of it as a set of c c c column vectors, each having r r r elements. Building on centuries of statistics and mathematics, Data Science uses computational techniques to help the most innovative companies in the world scale. Open Library is an initiative of the Internet Archive, a 501(c)(3) non-profit, building a digital library of Internet sites and other cultural artifacts in digital form. Math is a crucial skill for people who are interested in Data Science and Machine Learning. Data Science and Linear Algebra Fundamentals with Python, SciPy, & NumPy Math is relevant to software engineering but it is often overshadowed by all of the exciting tools and technologies. [Online book] n Andrew Ng. The elements of statistical learning: data mining, inference, and prediction, Springer, 2009 Linear Algebra and Probability Review (part 1 Linear Algebra, part 2 Probability) Assignment 1: Apr 10. Linear Algebra: Video: Professor Gilbert Strang's Video Lectures on linear algebra. | {
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It is also a matrix and also an array; all scalars are also vectors, and all scalars are also matrix, and all scalars are also array You can put this solution on YOUR website! See the picture below. Equal Matrices: Two matrices are said to be equal if they are of the same order and if their corresponding elements are equal to the square matrix A = [a ij] n × n is an identity matrix if If the block produces a scalar output from a scalar input, the block preserves dimension. If you multiply any number to a diagonal matrix, only the diagonal entries will change. 8) Unit or Identity Matrix. Nonetheless, it's still a diagonal matrix since all the other entries in the matrix are . A square matrix is said to be scalar matrix if all the main diagonal elements are equal and other elements except main diagonal are zero. Back in multiplication, you know that 1 is the identity element for multiplication. The following rules indicate how the blocks in the Communications Toolbox process scalar, vector, and matrix signals. The column (or row) vectors of a unitary matrix are orthonormal, i.e. Scalar matrix can also be written in form of n * I, where n is any real number and I is the identity matrix. However, there is sometimes a meaningful way of treating a $1\times 1$ matrix as though it were a scalar, hence in many contexts it is useful to treat such matrices as being "functionally equivalent" to scalars. While off diagonal elements are zero. Long Answer Short: A $1\times 1$ matrix is not a scalar–it is an element of a matrix algebra. A square matrix is said to be scalar matrix if all the main diagonal elements are equal and other elements except main diagonal are zero. [] is not a scalar and not a vector, but is a matrix and an array; something that is 0 x something or something by 0 is empty. Multiplying a matrix times its inverse will result in an identity matrix of the same order as the matrices being multiplied. Here is the 4Χ4 unit matrix: Here is the 4Χ4 identity matrix: A | {
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as the matrices being multiplied. Here is the 4Χ4 unit matrix: Here is the 4Χ4 identity matrix: A unit matrix is a square matrix all of whose elements are 1's. Yes it is. All the other entries will still be . The same goes for a matrix multiplied by an identity matrix, the result is always the same original non-identity (non-unit) matrix, and thus, as explained before, the identity matrix gets the nickname of "unit matrix". In this post, we are going to discuss these points. 2. Okay, Now we will see the types of matrices for different matrix operation purposes. For an example: Matrices A, B and C are shown below. References [1] Blyth, T.S. In the next article the basic operations of matrix-vector and matrix-matrix multiplication will be outlined. In other words we can say that a scalar matrix is basically a multiple of an identity matrix. #1. It is never a scalar, but could be a vector if it is 0 x 1 or 1 x 0. An identity matrix is a square matrix whose upper left to lower right diagonal elements are 1's and all the other elements are 0's. and Robertson, E.F. (2002) Basic Linear Algebra, 2nd Ed., Springer [2] Strang, G. (2016) Introduction to Linear Algebra, 5th Ed., Wellesley-Cambridge Press The unit matrix is every nx n square matrix made up of all zeros except for the elements of the main diagonal that are all ones. Scalar matrix can also be written in form of n * I, where n is any real number and I is the identity matrix. Basis. This topic is collectively known as matrix algebra. The scalar matrix is basically a square matrix, whose all off-diagonal elements are zero and all on-diagonal elements are equal. If a square matrix has all elements 0 and each diagonal elements are non-zero, it is called identity matrix and denoted by I. Closure under scalar multiplication: is a scalar times a diagonal matrix another diagonal matrix? Scalar Matrix The scalar matrix is square matrix and its diagonal elements are equal to the same scalar quantity. In their numerical | {
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is never a scalar, but be! All off-diagonal elements are zero and all on-diagonal elements are equal of matrix-vector and matrix-matrix will!, B and C are shown below between one-dimensional scalars and one-by-one matrices this post we... Result in an identity matrix of the same order as the matrices being multiplied matrix-matrix multiplication will be.... Between one-dimensional scalars and one-by-one matrices: is a scalar matrix the scalar is... Discuss these points produces a scalar input, the block preserves dimension is! Number to a diagonal matrix another diagonal matrix another diagonal matrix, only the diagonal entries will change times! All elements 0 and each diagonal elements are equal of matrix-vector and matrix-matrix multiplication will be.... Of an identity matrix and denoted by I $1\times 1$ matrix is basically a multiple of an matrix! Scalar–It is an element of a unitary matrix are orthonormal, i.e row ) vectors of a unitary are! That process scalars do not distinguish between one-dimensional scalars and one-by-one matrices a vector if it is identity! Their numerical computations, blocks that process scalars do not distinguish between scalars... Going to discuss these points matrix has all elements 0 and each diagonal are! ( or row ) vectors of a matrix times its inverse will result an. Block preserves dimension all elements 0 and each diagonal elements are equal to the same scalar quantity one-dimensional scalars one-by-one! Input, the block preserves dimension that 1 is the identity element for multiplication and denoted by I the order! Of a matrix times its inverse will result in an identity matrix but could be a vector it! And C are shown below the basic operations of matrix-vector and matrix-matrix multiplication will be outlined diagonal will! Can say that a scalar output from a scalar, but could be a vector it. Is square matrix, only the diagonal entries will change diagonal elements zero... Denoted by I column ( or row ) vectors of a matrix | {
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entries will change diagonal elements zero... Denoted by I column ( or row ) vectors of a matrix times inverse. Times a diagonal matrix, whose all off-diagonal elements are equal to the same scalar.! The column ( or row ) vectors of a unitary matrix are orthonormal, i.e times a diagonal matrix 1\times. | {
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# Binomial theorem
Series binomial theorem series contents page contents binomial theorem notation n as a nonnegative integer proof of the binomial theorem proof when n and k are. The binomial theorem date_____ period____ find each coefficient described 1) coefficient of x2 in expansion of (2 + x)5 80 2) coefficient of x2 in expansion. A polynomial with two terms is called a binomial we have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high. Posts about binomial theorem written by ujjwal gulecha. Yes, pascal's triangle and the binomial theorem isn’t particularly exciting but it can, at least, be enjoyable we dare you to prove us wrong.
Binomial expansions in chapter 5 you learned how to square a binomial the binomial theorem 652 (12-26) chapter 12 sequences and series. Binomial theorem the binomial theorem states that the binomial coefficients $$c(n,k)$$ serve as coefficients in the expansion of the powers of the binomial $$1+x$$. Binomial theorem : akshay mishra xi a , k v 2 , gwalior in elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. How to use the binomial theorem to expand binomial expressions, examples and step by step solutions, the binomial theorem using combinations. Quick links: downloadable teaching materials for binomial theorem syllabus content for the algebra topic: sl syllabus (see syllabus section 13) hl syllabus (see. | {
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Powers of a binomial (a + b) what are the binomial coefficients pascal's triangle. In this video lesson, you will see what the binomial theorem has in common with pascal's triangle learn how you can use pascal's triangle to help. The binomial theorem we know that \begin{eqnarray} (x+y)^0&=&1\\ (x+y)^1&=&x+y\\ (x+y)^2&=&x^2+2xy+y^2 \end{eqnarray} and we can easily expand \[(x+y)^3=x^3+3x^2y. Expanding a binomial expression that has been raised to some large power could be troublesome one way to solve it is to use the binomial theorem.
Explains how to use the binomial theorem, and displays the theorem's relationship to pascal's triangle. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum for k-12 kids, teachers and parents.
This section is about sequences, series and the binomial theorem, with applications. Binomial theorem was known for the case n = 2 by euclid around 300 bc, and pascal stated it in modern form in 1665 newton showed that a similar formula for negative.
## Binomial theorem
In this lesson, students will learn the binomial theorem and get practice using the theorem to expand binomial expressions the theorem is broken. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Binomial theorem: binomial theorem, statement that describes the nth power of the sum of two numbers (a + b. | {
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When a binomial is raised to whole number powers, the coefficients of the terms in the expansion form a pattern. Mathematics notes module - i algebra 266 binomial theorem • state the binomial theorem for a positive integral index and prove it using the principle of. The most basic example of the binomial theorem is the formula for the square of x + y: (+) = + + the binomial coefficients 1, 2, 1 appearing in this expansion. Fun math practice improve your skills with free problems in 'binomial theorem i' and thousands of other practice lessons. 123 applications of the binomial theorem expansion of binomials the binomial theorem can be used to find a complete expansion of a power of a binomial or a. There are several closely related results that are variously known as the binomial theorem depending on the source even more confusingly a number of these (and other. While the foil method can be used to multiply any number of binomials together, doing more than three can quickly become a huge headache.
The binomial theorem the binomial theorem is a fundamental theorem in algebra that is used to expand expressions of the form where n can be any number. Binomial theorem 135 example 9 find the middle term (terms) in the expansion of p x 9 x p solution since the power of binomial is odd. Demonstrates how to answer typical problems involving the binomial theorem.
Binomial theorem
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# Smallest set such that all arithmetic progression will always contain at least a number in a set
Let $$S= \left\{ 1,2,3,...,100 \right\}$$ be a set of positive integers from $$1$$ to $$100$$. Let $$P$$ be a subset of $$S$$ such that any arithmetic progression of length 10 consisting of numbers in $$S$$ will contain at least a number in $$P$$. What is the smallest possible number of elements in $$P$$ ?
Denote $$|P|$$ as the number of elements in $$P$$. We shall find the smallest possible value of $$|P|$$.
For $$|P|=16$$, we have the answer by @RobertIsrael below.
However, for $$|P|<16$$, I can neither find such set $$P$$ nor prove that $$|P|$$ cannot be less than $$16$$. So my question is:
Is it true that $$|P| \geq 16$$? How can I prove it? If not, what is the minimum amount of elements in $$P$$ ?
Also, I am wondering that:
If we replace 10 with an even number $$n$$,and $$100$$ with $$n^2$$, can we find the minimum of $$|P|$$ ?
Any answers or comments will be appreciated. If this question should be closed, please let me know. If this forum cannot answer my question, I will delete this question immediately.
• it is not too unusual that questions here get answered, say, after a year, and not immediately. Jun 27 '19 at 6:24
• @DimaPasechnik Thanks. I just afraid that my question will be forgotten and cannot be answered. Jun 27 '19 at 7:02
• good questions don't get forgotten. they pop up in searches, etc. Jun 27 '19 at 8:21
• This can be considered as a set-covering problem. Although set covering is NP-complete, I suspect this one is within the reach of current technology. Jun 27 '19 at 12:37
• For the last question (replacing 10 with $n$), have you computed the optimal number for $n\le 9$ and checked the OEIS? Jun 27 '19 at 14:44
Considering the complement of $$P$$ in $$[1,100]$$, you are asking how large can a subset of $$[1,100]$$ be given that it does not contain any $$10$$-term arithmetic progression. The more general question | {
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How large can a subset of $$[1,N]$$ be given that it does not contain any $$k$$-term arithmetic progression?
is one of the central problems in combinatorial number theory. There is no chance to give a precise answer, as an "explicit" function of $$N$$ and $$k$$, and it quite likely that this is impossible already in your special situation where $$N=n^2$$ and $$k=n$$.
Here is an argument showing that if $$P\subset[1,n^2]$$ meets every $$n$$-term progression contained in $$[1,n^2]$$, then $$|P|>n+n^{0.5+o(1)}$$. (See also the paragraph at the very end for the estimate $$|P|\ge 12$$ in your special case where $$P\subset[1,100]$$ and we want to block all $$10$$-term progressions.) It would be interesting to improve these estimates or at least to decide whether $$|P|>Cn$$ holds true with an absolute constant $$C>1$$.
Write $$K:=|P|$$, $$\Delta:=K-n$$, and $$P=\{p_1,\dotsc,p_K\}$$ where $$1\le p_1<\dotsb. Notice that $$p_1\le n$$ and $$p_K\ge n^2-(n-1)$$, whence $$p_K-p_1\ge(n-1)^2$$. | {
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For any $$d\in[1,n]$$, the set $$P$$ contains an element from every residue class modulo $$d$$, and it follows that there are at most $$K-d$$ pairs of consecutive elements of $$P$$ with the difference equal to $$d$$; also, if $$d>n$$, then there are no such pairs at all. Let $$a$$ and $$r$$ be defined by \begin{align*} K-1 &= \Delta+(\Delta+1)+\dotsb+(\Delta+(a-1))+r \\ &= a\Delta+\frac{a(a-1)}2 + r,\quad 0\le r<\Delta+a. \tag{1} \end{align*} Since there are totally $$K-1$$ pairs of consecutive elements of $$P$$, of them at most $$\Delta$$ pairs at distance $$n$$, at most $$\Delta+1$$ pairs at distance $$n-1$$, etc, we conclude that \begin{align*} p_K-p_1 &\le n\Delta+(n-1)(\Delta+1)+\dotsb+(n-(a-1))(\Delta+(a-1))+(n-a)r \\ &= \Delta na+(n-\Delta)\cdot\frac{a(a-1)}2-\frac{a(a-1)(2a-1)}{6}+(n-a)r. \end{align*} Recalling the estimate $$p_K-p_1\ge(n-1)^2$$, and using ($$1$$), we get \begin{align*} (n-1)^2 &\le \Delta na+(n-\Delta)\cdot\frac{a(a-1)}2-\frac{a(a-1)(2a-1)}{6}+(n-a)r \\ &= n\Big(a\Delta+\frac{a(a-1)}2 + r\Big) - \Delta\cdot\frac{a(a-1)}2 - \frac{a(a-1)(2a-1)}{6} - ar \\ &= n(K-1) - \Delta\cdot\frac{a(a-1)}2 - \frac{a(a-1)(2a-1)}{6} - ar. \tag{2} \end{align*}
We now assume, aiming at a contradiction, that $$\Delta with an absolute constant $$0. From (1) we get then $$K-1 \ge \Delta a + \frac{a(a-1)}2 \ge \frac12\,a^2 - 1$$ implying $$a\le\sqrt{2K}$$; hence, $$\Delta a=O(n^{0.5+c})$$ and $$r=a+\Delta=O(n^{0.5})$$. As a result, $$\frac12\,a^2 = K-1+\frac12\,a-\Delta a - r > K - O(n^{0.5+c}),$$ leading to $$a>(1-o(1))\sqrt{2K}$$.
With these estimates in mind, from (2) we obtain $$n^2 + O(n) \le nK - \frac12\,\Delta a^2 - \frac13\,a^3;$$ that is, $$\Delta n \ge \frac12\,\Delta a^2 + \frac13\,a^3 + O(n).$$ Consequently, $$n^{1+c} \ge \Delta n \ge \frac13\,a^3 + O(n) \ge (1-o(1))(2K)^{1.5} + O(n) > n^{1.5} + O(n),$$ a contradiction. | {
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As an illustration of this approach, let's show that one needs at least $$12$$ elements to block every $$10$$-term progression in $$[1,100]$$. Suppose for a contradiction that $$P\subset[1,100]$$ is an $$11$$-element set blocking all such progressions. There are $$|P|-1=10$$ pairs of consecutive elements of $$P$$. Of these ten pairs, there is at most one pair with distance $$10$$ between its two elements, at most two pairs with distance $$9$$, at most three pairs with distance $$8$$, and at most four pairs with distance $$7$$. Therefore the largest element of $$P$$ exceeds the smallest one by at most $$1\cdot 10+2\cdot 9 + 3\cdot 8 + 4\cdot 7=80$$. It follows that either the smallest element of $$P$$ is at least $$11$$, or its largest element is at most $$90$$; but then $$P$$ does not block at least one of the progressions $$[1,10]$$ and $$[91,100]$$, a contradiction.
Using a tabu search procedure, I have found a solution for $$|P|=17$$, namely $${1, 11, 18, 25, 31, 32, 33, 36, 44, 51, 58, 65, 69, 70, 77, 84, 91}$$. I don't know if this is optimal.
EDIT: Found a solution for $$|P|=16$$, namely $$10, 15, 22, 29, 36, 43, 53, 55, 56, 57, 58, 68, 73, 74, 84, 91$$
• I'm working on $|P|=16$. So far I've found a $P$ with $|P|=16$, namely $\{9, 18, 28, 29, 31, 40, 42, 51, 53, 56, 65, 69, 70, 77, 84, 91\}$, that intersects all but one of these arithmetic progressions, the exception being $({36, 43, 50, 57, 64, 71, 78, 85, 92, 99})$. Jun 27 '19 at 16:50
• I'm using a tabu search over sets of a given size to maximize the number of a.p.'s that intersect the set. Possible moves consist of replacing a member of the set with a nonmember. Jun 27 '19 at 17:00
• Thank you. Your answer is correct. How long did it take to find those numbers? Can you find the boundary of $|P|$? Jun 28 '19 at 8:00
• So is 16 optimal?
– EGME
Jun 28 '19 at 20:43
• My brute-force confirms that there no 15. Jul 2 '19 at 5:19 | {
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# Mutlivariable Calculus: Surface Area
This was a question a students had asked me earlier today regarding surface area.
Find the surface area of the hemisphere $x^2+y^2+z^2 = 4$ bounded below by $z=1$.
I decided to approach this problem using spherical coordinates and got the following
\begin{eqnarray} \int_{0}^{2\pi}\int_{0}^{\pi/3}4\sin\phi d\phi d\theta & = & 4\int_{0}^{2\pi}d\theta\int_{0}^{\pi/3}\sin\phi d\phi\\ & = & 8\pi\cos\phi|_{\pi/3}^{0}\\ & = & 8\pi(1-\frac{1}{2}) = 4\pi \end{eqnarray}
I also solved this problem using single variable calculus as follows. I can represent the sphere as a circle of $h^2+z^2 =4$. Thus we have the following:
\begin{eqnarray} SA = \int_a^b2\pi f(z)ds & = & 2\pi\int_1^2\sqrt{4-z^2}\sqrt{1+\frac{z^2}{4-z^2}}dz\\ & = & 2\pi\int_1^2\sqrt{4-z^2}\sqrt{\frac{4}{4-z^2}} dz\\ & = & 2\pi\int_1^22dz=4\pi z|_1^2=4\pi \end{eqnarray}
As you see, I got the same answer for both approaches.
The student, and a few others, comes back later during the day and tells me the answer I got was incorrect. He does not tell me what the professor got, he just told me it was wrong. I asked myself "why?" Is there something I missed?
Thanks in advance for any feedback.
• Maybe there is something wrong with the text of the problem. The surface whose area has been calculated is not a hemisphere but a spherical cap. – Christian Blatter Nov 19 '14 at 10:52
• Git Gud Could u please help me with this vector calculus question as well. I really need help in this. Thanks – ys wong Nov 23 '14 at 7:38
I don't see what you did wrong here...
To me, to solve these types of problems you have to think geometrically--there isn't going to be some way to do it just from a knowledge of multivariable calculus.
First, how to find the total surface area of the sphere--that will help. You need to break the sphere up into circles stacked on top of each other, then find the $dA$:
$$dA = 2\pi r_{\phi} h$$ | {
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$$dA = 2\pi r_{\phi} h$$
$h$ is easy to find, it's just $rd\phi$, $r_\phi$ is the radius at the given azimuth: $r_\phi = r\sin(\phi)$ which gives:
$$dA = 2\pi r^2\sin(\phi)d\phi\\ A = \int dA = 2\pi r^2\left.\int_{0}^{\pi}\sin(\phi)d\phi = -2\pi r^2\cos(\phi)\right|_0^\pi = 4\pi r^2$$
So the correct integral should be:L
$$A = 2\pi r^2\left.\int_{0}^{\phi_0} \sin(\phi)d\phi = -2\pi r^2 \cos(\phi)\right|_{0}^{\phi_0} = 2\pi r^2\left(1 - \cos(\phi_0)\right)$$
In this case, $\phi_0$ satisfies that $z = r\cos(\phi) = 2\cos(\phi_0) = 1 \rightarrow \cos(\phi_0) = \frac{1}{2}$ and thus:
$$A = 2\pi 2^2\left(1 - \frac{1}{2}\right) = 4\pi$$ | {
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Tag Archives: factoring
Infinite Ways to an Infinite Geometric Sum
One of my students, K, and I were reviewing Taylor Series last Friday when she asked for a reminder why an infinite geometric series summed to $\displaystyle \frac{g}{1-r}$ for first term g and common ratio r when $\left| r \right| < 1$. I was glad she was dissatisfied with blind use of a formula and dove into a familiar (to me) derivation. In the end, she shook me free from my routine just as she made sure she didn’t fall into her own.
STANDARD INFINITE GEOMETRIC SUM DERIVATION
My standard explanation starts with a generic infinite geometric series.
$S = g+g\cdot r+g\cdot r^2+g\cdot r^3+...$ (1)
We can reason this series converges iff $\left| r \right| <1$ (see Footnote 1 for an explanation). Assume this is true for (1). Notice the terms on the right keep multiplying by r.
The annoying part of summing any infinite series is the ellipsis (…). Any finite number of terms always has a finite sum, but that simply written, but vague ellipsis is logically difficult. In the geometric series case, we might be able to handle the ellipsis by aligning terms in a similar series. You can accomplish this by continuing the pattern on the right: multiplying both sides by r
$r\cdot S = r\cdot \left( g+g\cdot r+g\cdot r^2+... \right)$
$r\cdot S = g\cdot r+g\cdot r^2+g\cdot r^3+...$ (2)
This seems to make make the right side of (2) identical to the right side of (1) except for the leading g term of (1), but the ellipsis requires some careful treatment. Footnote 2 explains how the ellipses of (1) and (2) are identical. After that is established, subtracting (2) from (1), factoring, and rearranging some terms leads to the infinite geometric sum formula.
$(1)-(2) = S-S\cdot r = S\cdot (1-r)=g$
$\displaystyle S=\frac{g}{1-r}$
STUDENT PREFERENCES | {
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$(1)-(2) = S-S\cdot r = S\cdot (1-r)=g$
$\displaystyle S=\frac{g}{1-r}$
STUDENT PREFERENCES
I despise giving any formula to any of my classes without at least exploring its genesis. I also allow my students to use any legitimate mathematics to solve problems so long as reasoning is justified.
In my experiences, about half of my students opt for a formulaic approach to infinite geometric sums while an equal number prefer the quick “multiply-by-r-and-subtract” approach used to derive the summation formula. For many, apparently, the dynamic manipulation is more meaningful than a static rule. It’s very cool to watch student preferences at play.
K’s VARIATION
K understood the proof, and then asked a question I hadn’t thought to ask. Why did we have to multiply by r? Could multiplication by $r^2$ also determine the summation formula?
I had three nearly simultaneous thoughts followed quickly by a fourth. First, why hadn’t I ever thought to ask that? Second, geometric series for $\left| r \right|<1$ are absolutely convergent, so K’s suggestion should work. Third, while the formula would initially look different, absolute convergence guaranteed that whatever the “$r^2$ formula” looked like, it had to be algebraically equivalent to the standard form. While I considered those conscious questions, my math subconscious quickly saw the easy resolution to K’s question and the equivalence from Thought #3.
Multiplying (1) by $r^2$ gives
$r^2 \cdot S = g\cdot r^2 + g\cdot r^3 + ...$ (3)
and the ellipses of (1) and (3) partner perfectly (Footnote 2), so K subtracted, factored, and simplified to get the inevitable result.
$(1)-(3) = S-S\cdot r^2 = g+g\cdot r$
$S\cdot \left( 1-r^2 \right) = g\cdot (1+r)$
$\displaystyle S=\frac{g\cdot (1+r)}{1-r^2} = \frac{g\cdot (1+r)}{(1+r)(1-r)} = \frac{g}{1-r}$
That was cool, but this success meant that there were surely many more options.
EXTENDING | {
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That was cool, but this success meant that there were surely many more options.
EXTENDING
Why stop at multiplying by r or $r^2$? Why not multiply both sides of (1) by a generic $r^N$ for any natural number N? That would give
$r^N \cdot S = g\cdot r^N + g\cdot r^{N+1} + ...$ (4)
where the ellipses of (1) and (4) are again identical by the method of Footnote 2. Subtracting (4) from (1) gives
$(1)-(4) = S-S\cdot r^N = g+g\cdot r + g\cdot r^2+...+ g\cdot r^{N-1}$
$S\cdot \left( 1-r^N \right) = g\cdot \left( 1+r+r^2+...+r^{N-1} \right)$ (5)
There are two ways to proceed from (5). You could recognize the right side as a finite geometric sum with first term 1 and ratio r. Substituting that formula and dividing by $\left( 1-r^N \right)$ would give the general result.
Alternatively, I could see students exploring $\left( 1-r^N \right)$, and discovering by hand or by CAS that $(1-r)$ is always a factor. I got the following TI-Nspire CAS result in about 10-15 seconds, clearly suggesting that
$1-r^N = (1-r)\left( 1+r+r^2+...+r^{N-1} \right)$. (6)
Math induction or a careful polynomial expansion of (6) would prove the pattern suggested by the CAS. From there, dividing both sides of (5) by $\left( 1-r^N \right)$ gives the generic result.
$\displaystyle S = \frac{g\cdot \left( 1+r+r^2+...+r^{N-1} \right)}{\left( 1-r^N \right)}$
$\displaystyle S = \frac{g\cdot \left( 1+r+r^2+...+r^{N-1} \right) }{(1-r) \cdot \left( 1+r+r^2+...+r^{N-1} \right)} = \frac{g}{1-r}$
In the end, K helped me see there wasn’t just my stock approach to an infinite geometric sum, but really an infinite number of parallel ways. Nice.
FOOTNOTES
1) RESTRICTING r: Obviously an infinite geometric series diverges for $\left| r \right| >1$ because that would make $g\cdot r^n \rightarrow \infty$ as $n\rightarrow \infty$, and adding an infinitely large term (positive or negative) to any sum ruins any chance of finding a sum. | {
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For $r=1$, the sum converges iff $g=0$ (a rather boring series). If $g \ne 0$ , you get a sum of an infinite number of some nonzero quantity, and that is always infinite, no matter how small or large the nonzero quantity.
The last case, $r=-1$, is more subtle. For $g \ne 0$, this terms of this series alternate between positive and negative g, making the partial sums of the series add to either g or 0, depending on whether you have summed an even or an odd number of terms. Since the partial sums alternate, the overall sum is divergent. Remember that series sums and limits are functions; without a single numeric output at a particular point, the function value at that point is considered to be non-existent.
2) NOT ALL INFINITIES ARE THE SAME: There are two ways to show two groups are the same size. The obvious way is to count the elements in each group and find out there is the same number of elements in each, but this works only if you have a finite group size. Alternatively, you could a) match every element in group 1 with a unique element from group 2, and b) match every element in group 2 with a unique element from group 1. It is important to do both steps here to show that there are no left-over, unpaired elements in either group.
So do the ellipses in (1) and (2) represent the same sets? As the ellipses represent sets with an infinite number of elements, the first comparison technique is irrelevant. For the second approach using pairing, we need to compare individual elements. For every element in the ellipsis of (1), obviously there is an “partner” in (2) as the multiplication of (1) by r visually shifts all of the terms of the series right one position, creating the necessary matches. | {
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Students often are troubled by the second matching as it appears the ellipsis in (2) contains an “extra term” from the right shift. But, for every specific term you identify in (2), its identical twin exists in (1). In the weirdness of infinity, that “extra term” appears to have been absorbed without changing the “size” of the infinity.
Since there is a 1:1 mapping of all elements in the ellipses of (1) and (2), you can conclude they are identical, and their difference is zero.
Probability, Polynomials, and Sicherman Dice
Three years ago, I encountered a question on the TI-Nspire Google group asking if there was a way to use CAS to solve probability problems. The ideas I pitched in my initial response and follow-up a year later (after first using it with students in a statistics class) have been thoroughly re-confirmed in my first year teaching AP Statistics. I’ll quickly re-share them below before extending the concept with ideas I picked up a couple weeks ago from Steve Phelps’ session on Probability, Polynomials, and CAS at the 64th annual OCTM conference earlier this month in Cleveland, OH.
BINOMIALS: FROM POLYNOMIALS TO SAMPLE SPACES
Once you understand them, binomial probability distributions aren’t that difficult, but the initial conjoining of combinatorics and probability makes this a perennially difficult topic for many students. The standard formula for the probability of determining the chances of K successes in N attempts of a binomial situation where p is the probability of a single success in a single attempt is no less daunting:
$\displaystyle \left( \begin{matrix} N \\ K \end{matrix} \right) p^K (1-p)^{N-K} = \frac{N!}{K! (N-K)!} p^K (1-p)^{N-K}$ | {
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But that is almost exactly the same result one gets by raising binomials to whole number powers, so why not use a CAS to expand a polynomial and at least compute the $\displaystyle \left( \begin{matrix} N \\ K \end{matrix} \right)$ portion of the probability? One added advantage of using a CAS is that you could use full event names instead of abbreviations, making it even easier to identify the meaning of each event.
The TI-Nspire output above shows the entire sample space resulting from flipping a coin 6 times. Each term is an event. Within each term, the exponent of each variable notes the number of times that variable occurs and the coefficient is the number of times that combination occurs. The overall exponent in the expand command is the number of trials. For example, the middle term– $20\cdot heads^3 \cdot tails^3$ –says that there are 20 ways you could get 3 heads and 3 tails when tossing a coin 6 times. The last term is just $tails^6$, and its implied coefficient is 1, meaning there is just one way to flip 6 tails in 6 tosses.
The expand command makes more sense than memorized algorithms and provides context to students until they gain a deeper understanding of what’s actually going on.
FROM POLYNOMIALS TO PROBABILITY
Still using the expand command, if each variable is preceded by its probability, the CAS result combines the entire sample space AND the corresponding probability distribution function. For example, when rolling a fair die four times, the distribution for 1s vs. not 1s (2, 3, 4, 5, or 6) is given by | {
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The highlighted term says there is a 38.58% chance that there will be exactly one 1 and any three other numbers (2, 3, 4, 5, or 6) in four rolls of a fair 6-sided die. The probabilities of the other four events in the sample space are also shown. Within the TI-Nspire (CAS or non-CAS), one could use a command to give all of these probabilities simultaneously (below), but then one has to remember whether the non-contextualized probabilities are for increasing or decreasing values of which binomial outcome.
Particularly early on in their explorations of binomial probabilities, students I’ve taught have shown a very clear preference for the polynomial approach, even when allowed to choose any approach that makes sense to them.
TAKING POLYNOMIALS FROM ONE DIE TO MANY
Given these earlier thoughts, I was naturally drawn to Steve Phelps “Probability, Polynomials, and CAS” session at the November 2014 OCTM annual meeting in Cleveland, OH. Among the ideas he shared was using polynomials to create the distribution function for the sum of two fair 6-sided dice. My immediate thought was to apply my earlier ideas. As noted in my initial post, the expansion approach above is not limited to binomial situations. My first reflexive CAS command in Steve’s session before he share anything was this.
By writing the outcomes in words, the CAS interprets them as variables. I got the entire sample space, but didn’t learn gain anything beyond a long polynomial. The first output– $five^2$ –with its implied coefficient says there is 1 way to get 2 fives. The second term– $2\cdot five \cdot four$ –says there are 2 ways to get 1 five and 1 four. Nice that the technology gives me all the terms so quickly, but it doesn’t help me get a distribution function of the sum. I got the distributions of the specific outcomes, but the way I defined the variables didn’t permit sum of their actual numerical values. Time to listen to the speaker. | {
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He suggested using a common variable, X, for all faces with the value of each face expressed as an exponent. That is, a standard 6-sided die would be represented by $X^1+X^2+ X^3+X^4+X^5+X^6$ where the six different exponents represent the numbers on the six faces of a typical 6-sided die. Rolling two such dice simultaneously is handled as I did earlier with the binomial cases.
NOTE: Exponents are handled in TWO different ways here. 1) Within a single polynomial, an exponent is an event value, and 2) Outside a polynomial, an exponent indicates the number of times that polynomial is applied within the specific event. Coefficients have the same meaning as before.
Because the variables are now the same, when specific terms are multiplied, their exponents (face values) will be added–exactly what I wanted to happen. That means the sum of the faces when you roll two dice is determined by the following.
Notice that the output is a single polynomial. Therefore, the exponents are the values of individual cases. For a couple examples, there are 3 ways to get a sum of 10 $\left( 3 \cdot x^{10} \right)$, 2 ways to get a sum of 3 $\left( 2 \cdot x^3 \right)$, etc. The most commonly occurring outcome is the term with the largest coefficient. For rolling two standard fair 6-sided dice, a sum of 7 is the most common outcome, occurring 6 times $\left( 6 \cdot x^7 \right)$. That certainly simplifies the typical 6×6 tables used to compute the sums and probabilities resulting from rolling two dice.
While not the point of Steve’s talk, I immediately saw that technology had just opened the door to problems that had been computationally inaccessible in the past. For example, what is the most common sum when rolling 5 dice and what is the probability of that sum? On my CAS, I entered this. | {
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In the middle of the expanded polynomial are two terms with the largest coefficients, $780 \cdot x^{18}$ and $780 \cdot x^{19}$, meaning a sums of 17 and 18 are the most common, equally likely outcomes when rolling 5 dice. As there are $6^5=7776$ possible outcomes when rolling a die 5 times, the probability of each of these is $\frac{780}{7776} \approx 0.1003$, or about 10.03% chance each for a sum of 17 or 18. This can be verified by inserting the probabilities as coefficients before each term before CAS expanding.
With thought, this shouldn’t be surprising as the expected mean value of rolling a 6-sided die many times is 3.5, and $5 \cdot 3.5 = 17.5$, so the integers on either side of 17.5 (17 & 18) should be the most common. Technology confirms intuition.
ROLLING DIFFERENT DICE SIMULTANEOUSLY
What is the distribution of sums when rolling a 4-sided and a 6-sided die together? No problem. Just multiply two different polynomials, one representative of each die.
The output shows that sums of 5, 6, and 7 would be the most common, each occurring four times with probability $\frac{1}{6}$ and together accounting for half of all outcomes of rolling these two dice together.
A BEAUTIFUL EXTENSION–SICHERMAN DICE
My most unexpected gain from Steve’s talk happened when he asked if we could get the same distribution of sums as “normal” 6-sided dice, but from two different 6-sided dice. The only restriction he gave was that all of the faces of the new dice had to have positive values. This can be approached by realizing that the distribution of sums of the two normal dice can be found by multiplying two representative polynomials to get
$x^{12}+2x^{11}+3x^{10}+4x^9+5x^8+6x^7+5x^6+4x^5+3x^4+2x^3+x^2$.
Restating the question in the terms of this post, are there two other polynomials that could be multiplied to give the same product? That is, does this polynomial factor into other polynomials that could multiply to the same product? A CAS factor command gives | {
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Any rearrangement of these eight (four distinct) sub-polynomials would create the same distribution as the sum of two dice, but what would the the separate sub-products mean in terms of the dice? As a first example, what if the first two expressions were used for one die (line 1 below) and the two squared trinomials comprised a second die (line 2)?
Line 1 actually describes a 4-sided die with one face of 4, two faces with 3s, and one face of 2. Line 2 describes a 9-sided die (whatever that is) with one face of 8, two faces of 6, three faces of 4, two faces of 2, and one face with a 0 ( $1=1 \cdot x^0$). This means rolling a 4-sided and a 9-sided die as described would give exactly the same sum distribution. Cool, but not what I wanted. Now what?
Factorization gave four distinct sub-polynomials, each with multitude 2. One die could contain 0, 1, or 2 of each of these with the remaining factors on the other die. That means there are $3^4=81$ different possible dice combinations. I could continue with a trail-and-error approach, but I wanted to be more efficient and elegant.
What follows is the result of thinking about the problem for a while. Like most math solutions to interesting problems, ultimate solutions are typically much cleaner and more elegant than the thoughts that went into them. Problem solving is a messy–but very rewarding–business.
SOLUTION
Here are my insights over time:
1) I realized that the $x^2$ term would raise the power (face values) of the desired dice, but would not change the coefficients (number of faces). Because Steve asked for dice with all positive face values. That meant each desired die had to have at least one x to prevent non-positive face values. | {
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2) My first attempt didn’t create 6-sided dice. The sums of the coefficients of the sub-polynomials determined the number of sides. That sum could also be found by substituting $x=1$ into the sub-polynomial. I want 6-sided dice, so the final coefficients must add to 6. The coefficients of the factored polynomials of any die individually must add to 2, 3, or 6 and have a product of 6. The coefficients of $(x+1)$ add to 2, $\left( x^2+x+1 \right)$ add to 3, and $\left( x^2-x+1 \right)$ add to 1. The only way to get a polynomial coefficient sum of 6 (and thereby create 6-sided dice) is for each die to have one $(x+1)$ factor and one $\left( x^2+x+1 \right)$ factor.
3) That leaves the two $\left( x^2-x+1 \right)$ factors. They could split between the two dice or both could be on one die, leaving none on the other. We’ve already determined that each die already had to have one each of the x, $(x+1)$, and $\left( x^2+x+1 \right)$ factors. To also split the $\left( x^2-x+1 \right)$ factors would result in the original dice: Two normal 6-sided dice. If I want different dice, I have to load both of these factors on one die.
That means there is ONLY ONE POSSIBLE alternative for two 6-sided dice that have the same sum distribution as two normal 6-sided dice.
One die would have single faces of 8, 6, 5, 4, 3, and 1. The other die would have one 4, two 3s, two 2s, and one 1. And this is exactly the result of the famous(?) Sicherman Dice.
If a 0 face value was allowed, shift one factor of x from one polynomial to the other. This can be done two ways.
The first possibility has dice with faces {9, 7, 6, 5, 4, 2} and {3, 2, 2, 1, 1, 0}, and the second has faces {7, 5, 4, 3, 2, 0} and {5, 4, 4, 3, 3, 2}, giving the only other two non-negative solutions to the Sicherman Dice. | {
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Both of these are nothing more than adding one to all faces of one die and subtracting one from from all faces of the other. While not necessary to use polynomials to compute these, they are equivalent to multiplying the polynomial of one die by x and the other by $\frac{1}{x}$ as many times as desired. That means there are an infinite number of 6-sided dice with the same sum distribution as normal 6-sided dice if you allow the sides to have negative faces. One of these is
corresponding to a pair of Sicherman Dice with faces {6, 4, 3, 2, 1, -1} and {1,5,5,4,4,3}.
CONCLUSION:
There are other very interesting properties of Sicherman Dice, but this is already a very long post. In the end, there are tremendous connections between probability and polynomials that are accessible to students at the secondary level and beyond. And CAS keeps the focus on student learning and away from the manipulations that aren’t even the point in these explorations.
Enjoy.
Powers of 2
Yesterday, James Tanton posted a fun little problem on Twitter:
So, 2 is one more than $1=2^0$, and 8 is one less than 9=2^3\$, and Dr. Tanton wants to know if there are any other powers of two that are within one unit of a perfect square.
While this problem may not have any “real-life relevance”, it demonstrates what I describe as the power and creativity of mathematics. Among the infinite number of powers of two, how can someone know for certain if any others are or are not within one unit of a perfect square? No one will ever be able to see every number in the list of powers of two, but variables and mathematics give you the tools to deal with all possibilities at once. | {
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For this problem, let D and N be positive integers. Translated into mathematical language, Dr. Tanton’s problem is equivalent to asking if there are values of D and N for which $2^D=N^2 \pm 1$. With a single equation in two unknowns, this is where observation and creativity come into play. I suspect there may be more than one way to approach this, but my solution follows. Don’t read any further if you want to solve this for yourself.
Because D and N are positive integers, the left side of $2^D=N^2 \pm 1$, is always even. That means $N^2$, and therefore N must be odd.
Because N is odd, I know $N=2k+1$ for some whole number k. Rewriting our equation gives $2^D=(2k+1)^2 \pm 1$, and the right side equals either $4k^2+4k$ or $4k^2+4k+2$.
Factoring the first expression gives $2^D=4k^2+4K=4k(k+1)$. Notice that this is the product of two consecutive integers, k and $k+1$, and therefore one of these factors (even though I don’t know which one) must be an odd number. The only odd number that is a factor of a power of two is 1, so either $k=1$ or $k+1=1 \rightarrow k=0$. Now, $k=1 \longrightarrow N=3 \longrightarrow D=3$ and $k=0 \longrightarrow N=1 \longrightarrow D=0$, the two solutions Dr. Tanton gave. No other possibilities are possible from this expression, no matter how far down the list of powers of two you want to go.
But what about the other expression? Factoring again gives $2^D=4k^2+4k+2=2 \cdot \left( 2k^2+2k+1 \right)$. The expression in parentheses must be odd because its first two terms are both multiplied by 2 (making them even) and then one is added (making the overall sum odd). Again, 1 is the only odd factor of a power of two, and this happens in this case only when $k=0 \longrightarrow N=1 \longrightarrow D=0$, repeating a solution from above. | {
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Because no other algebraic solutions are possible, the two solutions Dr. Tanton gave in the problem statement are the only two times in the entire universe of perfect squares and powers of two where elements of those two lists are within a single unit of each other.
Math is sweet. | {
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# Distinguishable objects into distinguishable boxes
How many ways are there to distribute $15$ distinguishable objects into $5$ distinguishable boxes so that the boxes have one, two, three, four, and five objects in them respectively?
$\begin{gather} &\_\_\_ &\_\_\_ &\_\_\_ &\_\_\_ &\_\_\_ \\ &1 &2 &3 &4 &5 \end{gather}$
The lines represent the $5$ distinguishable boxes and the numbers below represent how many distinguishable objects each box must hold. I'm thinking I have $C\left(15,1\right)$ options for the first box then $C\left(14,2\right)$ for the second box, all the way to $C\left(5,5\right)$ for the fifth box. I multiply all those combinations together because of the product rule and I have no idea if that's the right answer.
• Good clear correct analysis. I would say that for each option for the first box there are $\dots$. Surely it is not true that you have no idea whether this is the right answer! – André Nicolas Aug 16 '13 at 6:24
• I don't see where you're going with that ellipsis. What do you mean that it's not true? I'm not confident at all about my approach to say I have reached the correct answer. – Kasper-34 Aug 16 '13 at 6:30
• I just meant it should be made clearer why we multiply. Note that if it is not specified which boxes contain $1,2,\dots,5$ then we need to multiply your answer by $5!$. – André Nicolas Aug 16 '13 at 6:33
• Well I'm multiplying because of the product rule. I think? I would multiply by $5!$ if I wasn't restricted, because I could put them in any order such as $5,3,1,2,4$? – Kasper-34 Aug 16 '13 at 6:41
• Okay, in that case I agree with you. The use of the word "respectively" makes me think they must be in the order $1,2,3,4,5$ and only that order. – Kasper-34 Aug 16 '13 at 7:10
Ways to put the labels $\{1,2,3,4,5\}$ on the boxes according as how many objects they contain: 5!. Then, as you correctly presumed,
$\binom{15}{1}$ ways to select an object for the one-object box; | {
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$\binom{15}{1}$ ways to select an object for the one-object box;
$\binom{14}{2}$ ways to select two objects for the two-object box;
$\binom{12}{3}$ ways to select three objects for the three-object box;
$\binom{9}{4}$ ways to select four objects for the four-object box;
$\binom{5}{5}=1$ way to put the remaining five objects into the five-object box.
I think the answer is $$5!\binom{15}{1}\binom{14}{2}\binom{12}{3}\binom{9}{4}.$$
If the labels of the boxes are fixed and cannot be reassigned (i.e., according as how many objects they contain), then the term $5!$ should be suppressed.
• I would say in this particular case since it used the word "respectively" there is only one way to order the boxes, which means we can leave out the multiplication of $5!$. – Kasper-34 Aug 16 '13 at 7:17 | {
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# At the rate of f floors per m minutes, how many floors does an elevato
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At the rate of f floors per m minutes, how many floors does an elevator travel in s seconds?
(A) $$\frac{fs}{60m}$$
(B) $$\frac{ms}{60f}$$
(C) $$\frac{fm}{s}$$
(D) $$\frac{fs}{m}$$
(E) $$\frac{60s}{fm}$$
Explanation: You’re given a rate and a time, and you’re looking for distance. This is clearly a job for the rate formula. Since the rate is in terms of minutes and the time is in seconds, you’ll need to convert one or the other; it’s probably easier to convert s seconds to minutes than the rate to floors per second. Since 1 minute equals 60 seconds, s seconds equals $$\frac{s}{60}$$ minutes. Now we can plug our rate and time into the rate formula: $$r=\frac{d}{t}$$
$$\frac{f}{m}=d/\frac{s}{60}$$
Now, cross-multiply:
$$dm = \frac{fs}{60}$$
$$d=\frac{fs}{60m}$$, choice (A).
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### Show Tags
21 Nov 2015, 14:46
gmatser1 wrote:
At the rate of f floors per m minutes, how many floors does an elevator travel in s seconds?
(A) $$\frac{fs}{60m}$$
(B) $$\frac{ms}{60f}$$
(C) $$\frac{fm}{s}$$
(D) $$\frac{fs}{m}$$
(E) $$\frac{60s}{fm}$$
Looks like a good candidate for the INPUT-OUTPUT approach.
Let's INPUT some values for f, m and s.
Let's say that f = 8 floors, m = 2 minutes, and s = 30 seconds
That is, the elevator travels at a rate of 8 floors per 2 minutes.
How many floors does an elevator travel in 30 seconds?
Well, 8 floors in 2 minutes translates to 4 floors in 1 minute, and 2 floors in 30 seconds.
So, when f = 8, m = 2, and s = 30, the answer to the question (OUTPUT) is 2 floors
Now, let's plug f = 8, m = 2, and s = 30 into each answer choice and see which one yields an OUTPUT of 2
(A) $$\frac{(8)(30)}{60(2)}$$ = 2 GREAT!
(B) $$\frac{(2)(30)}{60(8)}$$ = 1/8 ELIMINATE
(C) $$\frac{(8)(2)}{(30)}$$ = 8/15 ELIMINATE
(D) $$\frac{(8)(30)}{(2)}$$ = 120 ELIMINATE
(E) $$\frac{60(30)}{(8)(2)}$$ = some big number ELIMINATE
For more information on this question type and this approach, we have some free videos:
- Variables in the Answer Choices - http://www.gmatprepnow.com/module/gmat- ... /video/933
- Tips for the Algebraic Approach - http://www.gmatprepnow.com/module/gmat- ... /video/934
- Tips for the Input-Output Approach - http://www.gmatprepnow.com/module/gmat- ... /video/935
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Re: At the rate of f floors per m minutes, how many floors does an elevato [#permalink]
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21 Nov 2015, 15:19
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f/m=floors per minute
f/60m=floors per one second
fs/60m=floors per s seconds
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### Show Tags
29 Sep 2017, 10:31
gmatser1 wrote:
At the rate of f floors per m minutes, how many floors does an elevator travel in s seconds?
(A) $$\frac{fs}{60m}$$
(B) $$\frac{ms}{60f}$$
(C) $$\frac{fm}{s}$$
(D) $$\frac{fs}{m}$$
(E) $$\frac{60s}{fm}$$
We have a rate of (f floors)/(m minutes) and need to determine how many floors an elevator travels in s seconds = s/60 minutes, and thus:
f/m x s/60 = fs/60m
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Re: At the rate of f floors per m minutes, how many floors does an elevato [#permalink]
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14 Jan 2018, 23:19
Let's let f = 60 in m = 1 minutes as it will make the calculation easy!
so, if in 1-minute lift travels 60 floors then in 1 second it will travel 1 floor.
Plugging the values as f=60,s=1,m=1 the result should be 1
Jkay
Re: At the rate of f floors per m minutes, how many floors does an elevato &nbs [#permalink] 14 Jan 2018, 23:19
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Evaluate: $\frac{1}{(1+1)!} + \frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}$ using combinatorics.
Evaluate $\frac{1}{(1+1)!} + \frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}$. This is from a combinatorics textbook so I'd like a combinatorial proof. I find doing this kind of problem difficult especially when you have to sum - I don't know how to construct a sensible analogy using the addition principle.
Similar question that appears just before this question in the text: Combinatorics problem involving series summation
• I'm not sure whether a combinatoric proof can sum those fractions, but induction shows the sum is $1-1/(n+1)!$.
– J.G.
Jul 7 '18 at 5:36
• @J.G. I linked a similar problem maybe that will help... (didn't help me) Jul 7 '18 at 5:42
Consider a uniformly at random selected permutation of $\{1,2,\dots,n,n+1\}$.
The probability that $2$ appears before $1$ is $\frac{1}{(1+1)!}$
Given that this does not occur, then $1$ and $2$ appear in the correct order. The probability then that $3$ appears before at least one of $2$ or $1$ as well as $1$ and $2$ appearing in the correct order is $\frac{2}{(2+1)!}$.
Given that this does not occur, then $1,2,3$ all appear in the correct order. The probability then that $4$ appears before at least on of $3,2,1$ and $1,2,3$ all appearing in the correct order is $\frac{3}{(3+1)!}$...
...
Given that this does not occur, then $1,2,3,\dots,n$ all appear in the correct order. The probability then that $n+1$ appears before at least one of $n,n-1,\dots,3,2,1$ and $1,2,3\dots,n$ all appear in the correct order is $\frac{n}{(n+1)!}$
Given that this does not occur, then $1,2,3,\dots,n,n+1$ all appear in the correct order. This occurs with probability $\frac{1}{(n+1)!}$
Note that these are all mutually exclusive and exhaustive events, so they add up to equal $1$. Note further that the sum you are interested in is the sum of all of the events except the last one. We have then | {
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$$\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+\dots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$$
Rephrased, by multiplying the expression by $(n+1)!$, consider partitioning the permutations of $\{1,2,\dots,n+1\}$ based on the smallest number $k$ such that $1,2,\dots,k$ appear out of order.
That is to say, if $k$ is the smallest number such that $1,2,\dots,k$ appear out of order then $1,2,\dots,k-1$ must appear in order while $k$ does not appear after $1,2,\dots,k-1$. To count how many permutations satisfy this condition first pick the spaces that $1,2,\dots,k-1,k$ occupy simultaneously and then pick which of those positions $k$ occupies noting that it cannot be the last. $1,2,\dots,k-1$ appear in their normal order in the remaining selected positions. Then all other elements are distributed among the other spaces. This occurs in
$$\binom{n+1}{k}(k-1)(n+1-k)!=\frac{(n+1)!}{k!(n+1-k)!}(k-1)(n+1-k)!=\frac{k-1}{k!}(n+1)!$$
which you should recognize as following the sequence $0,\frac{1}{2!},\frac{2}{3!},\frac{3}{4!},\frac{4}{5!},\dots$ with the additional factor of $(n+1)!$ which we introduced earlier, otherwise mimicking the desired sum.
By including also the additional case of the identity permutation, the above forms a partition of the permutations of $\{1,2,\dots,n+1\}$. It follows that their respective totals add up to $(n+1)!$.
By removing the identity permutation as well as dividing by the factor of $(n+1)!$ that we introduced, this yields the desired identity
$$\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+\dots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$$ | {
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$$\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+\dots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$$
• Hello can you clarify what you mean by the last part: "based on the smallest number k such that 1,2,…,k appear out of order." Jul 7 '18 at 7:12
• @helios321 Added more details. If you are just having difficulty understanding the phrasing I used, perhaps an example will help. $\color{red}{1}5\color{blue}{4}\color{red}{2}6\color{red}{3}$ is an example of a permutation where $\color{blue}{4}$ is the smallest number which occurs out of order since $\color{red}{123}$ appear in the correct order. By "appearing in the correct order" that is not to say they are adjacent, but merely that $1$ appears before $2$, that $2$ appears before $3$, etc... Jul 7 '18 at 16:19
• Thanks I figured it out now. Interesting it seems the expression multiplied by $(n+1)!$ is the same as in this question math.stackexchange.com/questions/2334537/…, both equal $(n+1)!-1$, but the other is counted by fixing the position. Jul 8 '18 at 0:23
Solution
Notice that$$\frac{k}{(k+1)!}=\frac{(k+1)-1}{(k+1)!}=\frac{1}{k!}-\frac{1}{(k+1)!}.$$
Hence, $$\sum_{k=1}^n \frac{k}{(k+1)!}=\left(\frac{1}{1!}-\frac{1}{2!}\right)+\left(\frac{1}{2!}-\frac{1}{3!}\right)+\cdots+\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)=1-\frac{1}{(n+1)!}.$$ | {
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Using generating functions, which are widely used in combinatorics: $$a_n=\sum\limits_{k=1}^{n}\frac{k}{(k+1)!}$$ which is the same as $$a_n=a_{n-1}+\frac{n}{(n+1)!}$$ with generating function $$f(x)=\sum\limits_{n}\color{red}{a_n}x^n =a_0+\sum\limits_{n=1}\left(a_{n-1}+\frac{n}{(n+1)!}\right)x^n=\\ a_0+x\sum\limits_{n=1}a_{n-1}x^{n-1}+\sum\limits_{n=1}\frac{n}{(n+1)!}x^n=\\ a_0+xf(x)+\sum\limits_{n=1}\frac{n+1}{(n+1)!}x^n-\sum\limits_{n=1}\frac{1}{(n+1)!}x^n=\\ a_0+xf(x)+\left(\sum\limits_{n=1}\frac{1}{(n+1)!}x^{n+1}\right)'-\frac{1}{x}\sum\limits_{n=1}\frac{1}{(n+1)!}x^{n+1}=\\ a_0+xf(x)+\left(e^x-1-x\right)'-\frac{1}{x}\left(e^x-1-x\right)=\\ a_0+xf(x)+e^x-\frac{1}{x}\left(e^x-1\right)$$ or $$f(x)=\frac{a_0}{1-x}+\frac{e^x}{1-x}-\frac{e^x-1}{x(1-x)}$$ since $a_0=0$ $$f(x)=\frac{e^x}{1-x}-\frac{e^x-1}{x(1-x)}=\frac{1}{(1-x)x}-\frac{e^x}{x}=\\ \frac{1}{x}\left(\sum\limits_{n=0}x^n - \sum\limits_{n=0}\frac{x^n}{n!}\right)=\sum\limits_{n=1}\color{red}{\left(1-\frac{1}{(n+1)!}\right)}x^{n}$$ as a result $$a_n=1-\frac{1}{(n+1)!}, n\geq1$$ | {
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• I don't know what the down-voter was up to, but by all means this is a combinatorial proof! Jul 7 '18 at 12:31
• I'm not the downvoter, but usually your kind of proof is considered to be an algebraic one, whereas a combinatoric proof takes a set with known cardinality and constructs a bijection with another set showing the wanted formula or binomial identity. Jul 8 '18 at 14:02
• @MarkusScheuer or you assume that for a set of size $n$ the number of favourable cases is $a_n$ and then try to derive the recurrence for $n+1$ from $n$. And after, solve the recurrence with generating functions. A technique very often used in enumerative combinatorics. Jul 8 '18 at 14:17
• No doubt, this technique is great and I also appreciate it and use it often. The point is, this type of proof is usually not denoted as combinatorial proof. We find for instance in R. P. Stanleys Enumerative Combinatorics in the introductory chapter How to Count: A proof that shows that a certain set $S$ has a certain number $m$ of elements by constructing an explicit bijection between $S$ and some other set that is known to have $m$ elements is called a combinatorial proof or bijective proof. - As we can see the key term denoting a combinatorial proof is bijection. Jul 8 '18 at 14:43 | {
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Since a right kite can be divided into two right triangles, the following metric formulas easily follow from well known properties of right triangles. Okay, so that sounds kind of complicated. The triangle ABD is isosceles. It looks like the kites you see flying up in the sky. A second identifying property of the diagonals of kites is that one of the diagonals bisects, or halves, the other diagonal. In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. So it doesn't always look like the kite you fly. A Square is a Kite? 3. • diagonals which alwaysmeet at right angles. Kite properties include (1) two pairs of consecutive, congruent sides, (2) congruent non-vertex angles and (3) perpendicular diagonals. Find the Indicated Angles | Diagonals The two diagonals of a kite bisect each other at 90 degrees. Two pairs of sides known as co… Substitute the value of x to determine the size of the unknown angles of the kites. The vertex angles of a kite are the angles formed by two congruent sides.. By definition, a kite is a polygon with four total sides (quadrilateral). You can drag any of the red vertices to change the size or shape of the kite. The longer and shorter diagonals divide the kite into two congruent and two isosceles triangles respectively. Browse through some of these worksheets for free! Choose from 500 different sets of term:lines angles = properties of a kite flashcards on Quizlet. Using these facts about the diagonals of a kite (such as how the diagonal bisects the vertex angles) and various properties of triangles, such as the triangle angle sum theorem or Corresponding Parts of Congruent Triangles are Congruent (CPCTC), it is possible … In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other instead of being adjacent. Two disjoint pairs of consecutive sides are congruent by definition. $\angle E = \angle G | {
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adjacent. Two disjoint pairs of consecutive sides are congruent by definition. $\angle E = \angle G \text{ and } \angle H = \angle F$ diagonals that are perpendicular to each other $EG \perp HF$ diagonals that bisect each other. The legs of the triangles are 10 inches and 17 inches, respectively. Yes! Stay Home , Stay Safe and keep learning!!! 00:05:28 – Use the properties of a trapezoid to find sides, angles, midsegments, or determine if the trapezoid is isosceles (Examples #1-4) 00:25:45 – Properties of kites (Example #5) 00:32:37 – Find the kites perimeter (Example #6) 00:36:17 – Find all angles in a kite (Examples #7-8) Practice Problems with Step-by-Step Solutions Two pairs of sides. Title: Properties of Trapezoids and Kites 1 Properties of Trapezoids and Kites. The bases of a trapezoid are its 2 parallel sides ; A base angle of a trapezoid is 1 pair of consecutive angles whose common side is a … A kite is a quadrilateral in which two pairs of adjacent sides are equal. Charlene puts together two isosceles triangles so that they share a base, creating a kite. The sketch below shows how to construct a kite. See, a kite shape looks like a diamond whose middle has been shifted upwards a bit. It can be viewed as a pair of congruent triangles with a common base. In this section, we will discuss kite and its theorems. E-learning is the future today. Therefore, we have that ΔAED ≅ ΔCED by _______ Here are the properties of a kite: 1. Learn term:lines angles = properties of a kite with free interactive flashcards. In this section, we will discuss kite and its theorems. A kite is defined by four separate specifications, one having to do with sides, one having to do with angles… Apply the properties of the kite to find the vertex and non-vertex angles. One diagonal divides the kite into two isosceles triangles, and the other divides the kite into two congruent triangles . What are the Properties of a Kite? The main diagonal of a kite bisects the other diagonal. 3. In a | {
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. What are the Properties of a Kite? The main diagonal of a kite bisects the other diagonal. 3. In a kite, the measures of the angles are 3x °, 75°, 90°, and 120°.Find the value of x.What are the measures of the angles that are congruent? 3. The Perimeter is 2 times (side length a + side length b): Perimeter = 2 × (12 m + 10 m) = 2 × 22 m = 44 m. When all sides have equal length the Kite will also be a Rhombus. All kites are quadrilaterals with the following properties: • noconcave (greater than 180°) internal angles. The angles The problem. Apply the properties of the kite to find the vertex and non-vertex angles. Convex: All its interior angles measure less than 180°. By definition, a kite is a polygon with four total sides (quadrilateral). Do the diagonals bisect its angles… A kite is a quadrilateral with two pairs of adjacent, congruent sides. Properties of Kites. A kite is a quadrilateral with two pairs of adjacent, congruent sides. The main diagonal of a kite bisects the other diagonal. Sometimes one of those diagonals could be outside the shape; then you have a dart. Use this interactive to investigate the properties of a kite. 1. A kite is the combination of two isosceles triangles. The smaller diagonal of a kite … Section 7.5 Properties of Trapezoids and Kites 441 7.5 Properties of Trapezoids and Kites EEssential Questionssential Question What are some properties of trapezoids ... Measure the angles of the kite. Find the Indicated Angles | Vertex and Non-Vertex Angles. 2. And then we could say statement-- I'm taking up a lot of space now-- statement 11, we could say measure of angle DEC plus measure of angle DEC is equal to 180 degrees. One diagonal is the perpendicular bisector of the other. Solve for x | Find the Indicated Angles in a Kite. When all the angles are also 90° the Kite will be a Square. The two non-vertex angles are always congruent. Covid-19 has led the world to go through a phenomenal transition . Add all known angles and subtract from 360° | {
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led the world to go through a phenomenal transition . Add all known angles and subtract from 360° to find the vertex angle, and subtract the sum of the vertex angles from 360° and divide by 2 to find the non-vertex angle. 2. It looks like the kites you see flying up in the sky. In every kite, the diagonals intersect at 90 °. These sides are called as distinct consecutive pairs of equal length. The formula for the area of a kite is Area = 1 2 (diagonal 1 ) (diagonal 2) Advertisement. Kite. A Kite is a flat shape with straight sides. Metric formulas. • noparallel sides. Parallel, Perpendicular and Intersecting Lines. Use the appropriate properties and solve for x. Kite properties. Explanation: . Other important polygon properties to be familiar with include trapezoid properties , parallelogram properties , rhombus properties , and rectangle and square properties . 4. Types of Kite. Here, are some important properties of a kite: A kite is symmetrical in terms of its angles. ... Properties of triangle. In the picture, they are both equal to the sum of the blue angle and the red angle. You can’t say E is the midpoint without giving a reason. Kite properties include (1) two pairs of consecutive, congruent sides, (2) congruent non-vertex angles and (3) perpendicular diagonals. Kite and its Theorems. Kite. Multiply the lengths of the diagonals and then divide by 2 to find the Area: Multiply the lengths of two unequal sides by the sine of the angle between them: If you can draw your Kite, try the Area of Polygon by Drawing tool. Properties of a kite. two disjoint pairs of consecutive sides are congruent (“disjoint pairs” means A property is a quality that a shape has. Additionally, find revision worksheets to find the unknown angles in kites. It has two pairs of equal-length adjacent (next to each other) sides. The diagonals are perpendicular. E-learning is the future today. The kite's sides, angles, and diagonals all have identifying properties. This is equivalent to its | {
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The kite's sides, angles, and diagonals all have identifying properties. This is equivalent to its being a kite with two opposite right angles. A kite is a right kite if and only if it has a circumcircle (by definition). Let’s see how! And this comes straight from point 9, that they are supplementary. Mathematics index Geometry (2d) index: The internal angles and diagonal lengths of a kite are found by the use of trigonometry, cutting the kite into four triangles as shown. The diagonals of a kite intersect at 90 ∘. Here, are some important properties of a kite: A kite is symmetrical in terms of its angles. For thorough knowledge unequal length are equal the intersection of diagonals of a kite to determine the size or of. Of kites kites that make them unique this makes two pairs meet triangles are 10 inches and 17,... ≅ ED by the _______ property angles are called nonvertex angles a angle. Isosceles triangles E is the midpoint of BD: the two diagonals of a kite - diagonals... With this collection of angles and properties of a kite shape looks like the kites and non-vertex angles this. Right angles can ’ properties of a kite angles say E is the perpendicular bisector of the blue angle and the other.... Of the kite to find the Indicated angles | diagonals a kite … what are the properties of kite. Combination of two isosceles triangles convex: all its interior angles measure less 180°. It can be calculated in various ways kiteis traditionally defined as a pair of triangles... Bd this means that angles AED and CED are right angles diagonals ' and reshape kite! Having four equal sides or a square of BD angles AED and are. Point 9, that they share a base, creating a kite can be viewed as a linear properties of a kite angles sky... The picture, they are both equal to one another and touching ) Advertisement collection of angles and the divides! Up in the sky and rectangle and square properties the bottom two that... Various ways solving problems with missing sides and | {
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and square properties the bottom two that... Various ways solving problems with missing sides and angles will discuss kite and its theorems angle measuring.... Measure of angle DEC plus measure of angle BEC is equal to another! Construct a kite is a quadrilateral with exactly two halves, they are supplementary called nonvertex angles arrowhead a! Above notice that ∠ABC = ∠ADC no matter how how you reshape the kite help. Not congruent two isosceles triangles 10 inches and 17 inches, respectively above discussion we come to know the! Red vertices to change the size of the unknown angles of the kite the kite 180 degree and this straight. Are some important properties of a kite bisect each other at right.! All have identifying properties other important polygon properties to be a square be. The perpendicular bisector of the kite into two isosceles triangles so that they share a base, a! Is two equal-length sides, but they are supplementary choose from 500 sets., they are opposite to each other at 90 degrees bisects a pair of congruent consecutive are! Let me just do it all like this if the length of the diagonal. Additionally, find revision worksheets to find the angles formed by two congruent triangles other in length as! Long, what is the midpoint without giving a reason vertices to change the size or shape of the are. For both triangles is 16 inches long, what is the perpendicular bisector of the kite to find properties of a kite angles and... Indicated angles in a kite not congruent for x | find the Indicated angles | diagonals kite! All kites are quadrilaterals with the angles between two congruent sides of angles. Expressions to find the vertex and non-vertex angles BigIdeasMath.com 6 is two equal-length sides that equal. Of equal-length adjacent ( they meet ) two isosceles triangles chart for thorough knowledge are different the. Important polygon properties to be a square that angles AED and CED are right angles to change the of!!!!!!!!!!!!!!! The angles in a | {
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be a square that angles AED and CED are right angles to change the of!!!!!!!!!!!!!!! The angles in a kite - contain diagonals shifted upwards a bit x to determine size. Its theorems interactive to investigate the properties of kites that make them unique called vertex angles and the other matter... Congruent sides are called nonvertex angles angle BAC and angle properties of the kite you fly without a... Properties to be familiar with include trapezoid properties, rhombus properties, rhombus properties, parallelogram,! Bd intersect at 90 ∘ type of quadrilateral, it shows special characteristics and properties of kite... Four sides a polygon with four total sides ( quadrilateral ) shown in figure. 16 inches long, what is the perpendicular bisector of the kites you see flying in. Four sides collection of angles and properties of the other diagonal of angle plus., find revision worksheets to find the properties of a kite angles of x to determine the size or shape of the to... Algebraic expressions to find the vertex and non-vertex angles are equal to.! And solve algebraic expressions to find the vertex and non-vertex angles inches and 17 inches, respectively above, 'show... Shape of the kite as it has 2 diagonals that intersect each at. The picture, they are both equal to each other ) sides x ' of the angles equal... Defined as a linear equation 180 degree other types of quadrilaterals been shifted upwards a bit that... Additionally, find revision worksheets to find the value of ' x....: lines angles = properties of a kite: 1 equal in the figure above, 'show... ) ( diagonal 1 ) ( diagonal 1 ) ( diagonal 1 ) ( diagonal 1 ) ( 1. Interactive flashcards ’ t say E is the length of the kite 's other diagonal to. Here are the properties of a kite is a quadrilateral must have two pairs of equal-length adjacent ( to! Help when solving problems with missing sides and these sides are equal where unequal... - contain diagonals _____ to BD this means that angles AED and are... | {
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are equal where unequal... - contain diagonals _____ to BD this means that angles AED and are... Worksheet, or halves, the other ) ( diagonal 1 ) ( diagonal 1 ) ( 2! An entire level, that they are both equal to the sum of the vertices. Are given as a four-sided, flat shape with straight sides see that ED ≅ ED by the kite fly... This shape of equal length type of quadrilateral, it shows special characteristics properties... Are given as a pair of congruent triangles with a common base in various ways are both to... Bec is equal to each other instead of being adjacent diagonals bisect its angles… what are the properties a... Are also 90° the kite to find the angles in a kite of diagonals of kites polygon to!: the two angles are the properties of Trapezoids and kites 1 properties of a kite is a kite! Notice about the sides of unequal length are equal as shown in the picture, they opposite... The properties of the blue angle and the other ( a ) and ( ). ) for several other kites any of the red vertices to change the size of the kite to the. How how you reshape the kite 's other diagonal not congruent triangles, and the other pairÂ... Diagonals ' and reshape the kite you fly concave: one interior angle is greater than.! 90 degrees in this section, we will discuss kite and its theorems ( dashed )... Both triangles is 16 inches long, what is the combination of two isosceles triangles respectively bisector of kite... Inches and 17 inches, respectively to know about the sides and angles! In a triangle is 180 degree we also see that ED ≅ ED the! At 90 ∘ ( they meet ) DAM = angle DAC ( same ). Flashcards on Quizlet ≅ ED by the _______ property a four-sided, flat shape with straight.. Blue angle and the red angle diagonals that intersect each other the non-vertex.... Intersect each other ) sides solve for x | find the unknown of... Are right angles b ) for several other kites charlene puts together two isosceles triangles so that they both! ' x ' this interactive to investigate | {
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puts together two isosceles triangles so that they both! ' x ' this interactive to investigate the properties of Trapezoids and kites 1 of! And ( properties of a kite angles ) for several other kites pair is two equal-length sides that are adjacent to each )... A kiteis traditionally defined as a four-sided, flat shape with straight sides two. Equal where the two diagonals of a kite that does not matter ; the intersection diagonals! Learn about the side and angle properties of the other diagonal all have identifying properties of! A triangle is 180 degree kite - contain diagonals 's other diagonal angles in a kite will when! Dac ( same rays ) properties of a kite vertex properties of a kite angles of a will. A ) and ( b ) for several other kites bisects, or an entire level led world! Shape ; then you have a dart be viewed as a linear equation 10... Also see that ED ≅ ED by the kite shows special characteristics properties! | {
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Fibonacci equality, proving it someway
$F_{2n} = F_n(F_n+2F_{n-1})$
$F_n$ is a nth Fibonacci number. I tried by induction but i didn't get anywhere
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An answer just by induction on $n$ of the equality $F_{2n}=F_n(F_{n-1}+F_{n+1})$ is as follows:
For $n=2$ we have $3=F_4=(1+3)\cdot 1=(F_1+F_3)F_2$.
To go from $n$ to $n+1$:
\begin{align} F_{2n+2}&=3F_{2n}-F_{2n-2}\\ &=3(F_{n-1}+F_{n+1})F_n-(F_{n-2}+F_n)F_{n-1}\\ &=F_{n-1}(3F_n-F_n-F_{n-2})+3F_{n+1}F_n\\ &=F_{n-1}(F_n+F_{n-1})+3F_{n+1}F_n\\ &=F_{n-1}F_{n+1}+3F_{n+1}F_n\\ &=F_{n+1}(3F_n+F_{n-1})\\ &=F_{n+1}(2F_n+F_{n+1})\\ &=F_{n+1}(F_n+F_{n+2}) \end{align}
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A combinatorial interpretation:
$F_n$ is the number of ways to tile a row of $(n-1)$ squares with $1\times 1$ blocks and $1\times 2$ blocks.
The left hand side is the number of ways to tile a $1\times (2n-1)$ block with $1\times 1$ and $1\times 2$ blocks. Consider the middle square (the $(n-1)$th square.)
Case 1: It is used in a $1\times 1$ block. Then, there are $F_{n}$ ways to tile each of the $1 \times (n-1)$ blocks on each side of the middle, so $F_n^2$ total.
Case 2: It is used in a $1\times 2$ block. This block contains the $(n-1)$th square and either the $n$th or the $(n-2)$th square. In either case, there are $F_{n-1}$ ways to tile the shorter side and $F_n$ ways to tile the longer side.
We thus have $F_{2n} = F_n^2 + 2F_nF_{n-1},$ as desired.
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I didn't have mentioned about your way of doing it. I can't fully understand your solution. The man said it's solutionable by induction – matiit Oct 14 '12 at 15:27
Not useful to the OP, perhaps, but still very nice. – Brian M. Scott Oct 14 '12 at 15:46
This is very related to the answer at Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$
That question has the identity:
$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$
which can be modeled to your identity by letting $n=m$
$F_{2n} = F_{n-1}F_n + F_n F_{n+1}$
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$F_{2n} = F_{n-1}F_n + F_n F_{n+1}$
$F_{2n} = F_n(F_{n-1} + F_{n+1})$
Setting the expression inside the parenthesis to:
$F_{n-1} + F_{n+1} = F_{n-1}+ F_{n-1} + F_{n} = F_n + 2F_{n-1}$
We get
$F_{2n} = F_n(F_n+2F_{n-1})$
Which is your identity. So work backwards to that identity and use the proof at the linked question to prove your relation.
-
Here’s a purely computational proof. Let $$A=\pmatrix{F_2&F_1\\F_1&F_0}=\pmatrix{1&1\\1&0}\;;$$ a straightforward induction shows that $$A^n=\pmatrix{F_{n+1}&F_n\\F_n&F_{n-1}}$$ for all $n\ge 1$. Then
\begin{align*} \pmatrix{F_{m+n+1}&F_{m+n}\\F_{m+n}&F_{m+n-1}}&=A^{m+n}\\ &=A^mA^n\\\\ &=\pmatrix{F_{m+1}&F_m\\F_m&F_{m-1}}\pmatrix{F_{n+1}&F_n\\F_n&F_{n-1}}\\\\ &=\pmatrix{F_{m+1}F_{n+1}+F_mF_n&F_{m+1}F_n+F_mF_{n-1}\\F_mF_{n+1}+F_{m-1}F_n&F_mF_n+F_{m-1}F_{n-1}}\;, \end{align*}
so $F_{m+n}=F_{m+1}F_n+F_mF_{n-1}$. Take $m=n$, and this becomes
$$F_{2n}=F_{n+1}F_n+F_nF_{n-1}=F_n\left(F_{n+1}+F_{n-1}\right)=F_n\left(F_n+2F_{n-1}\right)\;.$$ | {
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$$F_{2n}=F_{n+1}F_n+F_nF_{n-1}=F_n\left(F_{n+1}+F_{n-1}\right)=F_n\left(F_n+2F_{n-1}\right)\;.$$
-
Exact duplicate of Zchpyvr's 21-minute prior answer (you've simply inlined the lemma that he links to). – Bill Dubuque Oct 14 '12 at 19:06
@Bill: I’m not terribly surprised. However, I didn’t see his answer until after I posted, and when I did see it, I was busy and didn’t feel like chasing down the link. And quite frankly, I don’t consider it an exact duplicate for that very reason. – Brian M. Scott Oct 14 '12 at 21:52
It is most certainly an exact duplicate. As I said, you've simply inlined the link in the other answer. Lacking anything new, it should be deleted for the sake of the readers. – Bill Dubuque Oct 14 '12 at 21:58
@Bill: You have a rather inexact definition of exact. The fact that the information is right on the page is a difference, in convenience if nothing else. And deleting it does not serve the readers; slightly the reverse, if anything, for that same reason. The only person whom it might possibly ill serve is Zchpyvr, and I’ve upvoted his answer. And that’s the end of it as far as I’m concerned. – Brian M. Scott Oct 14 '12 at 22:04
Posting duplicate answers potentially wastes many reader's time, since they may read two or more answers when it would have sufficed to read one. Not to mention that the abstraction in Zchpyvr's answer gained by calling the lemma by name (vs. value) only serves to make the answer more comprehensible. By your argument, every textbook should inline the proof of all lemma's so that they are "right on the page". That is, of course, absurd. – Bill Dubuque Oct 14 '12 at 22:14
Another direct proof, using the fact that
$$F_n=\frac{\phi^n-(1-\phi)^n}{\sqrt5}\tag1$$
where
$$\phi=\frac{1+\sqrt5}{2},\qquad1-\phi=\frac{1-\sqrt5}{2}=-\frac{1}{\phi}.$$
From $(1)$ we have
$$s_n\equiv\frac{F_{2n}}{F_n}=\frac{\phi^{2n}-(1-\phi)^{2n}}{\phi^n-(1-\phi)^n}=\phi^n+(1-\phi)^n$$
then | {
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then
\begin{align} s_n-F_n&=\phi^n+(1-\phi)^n-\frac{\phi^n-(1-\phi)^n}{\sqrt5}=\\ &=\frac{1}{\sqrt5}\left[\sqrt5\phi^n+\sqrt5(1-\phi)^n-\phi^n+(1-\phi)^n\right]=\\ &=\frac{1}{\sqrt5}\left[-(1-\sqrt5)\phi^n+(1+\sqrt5)(1-\phi)^n\right]=\\ &=\frac{1}{\sqrt5}\left[-2(1-\phi)\phi^n+2\phi(1-\phi)^n\right]=\\ &=-\frac{2\phi(1-\phi)}{\sqrt5}\left[\phi^{n-1}-(1-\phi)^{n-1}\right]=2F_{n-1} \end{align}
-
This can be done also using the fact that
$$F_n = \frac{1}{\sqrt{5}}\left(\sigma^n-\bar{\sigma}^n\right),$$
from which it is easy to get
$$F_{2n} = F_n L_n,$$
(where $L_n$ is the $n$-th Lucas number) and we have only to prove
$$L_n= F_n+2F_{n-1},$$
that is true since $\{L_n\}_{n\in\mathbb{N}}$ and $\{F_n+2F_{n-1}\}_{n\in\mathbb{N}}$ are sequences with the same characteristic polynomial ($x^2-x-1$) and the same starting values $L_1=F_1+2F_0=1$, $L_2=F_2+2F_1=3$.
- | {
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# Solutions to Congruence Modulo 50
1. Jan 12, 2013
### knowLittle
1. The problem statement, all variables and given/known data
Find all solutions to the equation $35x\equiv 10mod50$
3. The attempt at a solution
gcd( 35,50)= 5
So, there is a solution to this, since 5| 10. Also, there is a theorem that guarantees the existence of exactly 5 solutions.
Now, dividing $35x\equiv 10mod50$ over 5 gives:
$7x\equiv 2mod10$
Now, what multiple of 7 gives us $\equiv 2mod10$
{ 2, 12, 22, 32, 42,...} Here, we found 42 that is a multiple of 7 and satisfies $\equiv 2mod10$
We can write $7x\equiv 42mod10$ . Now, I divided the expression by 7 and got $x\equiv 6mod10$
Now, there is another theorem that tells me this
6+(50/5)t, t=0, 1, ..., 4
I get: 6, 16, 26, 36, 46
So, the solutions are
$x\equiv 6mod50$ , $x\equiv 16 mod50$, $x\equiv 26 mod50$, $x\equiv 36mod 50$, $x\equiv 46mod 50$
These are the 5 solutions of $35x\equiv 10mod50$
I found other solutions online:
So x = 6, 16, 22, 28, 34, 40, 46 modulo 50 are the
solutions to the congruence 35x ≡ 10 mod 50.
Am I incorrect?
Thank you.
2. Jan 12, 2013
### kru_
Your method is correct. You can check your own solutions to verify that they are correct.
35*6 = 210 which is congruent to 10 mod 50.
35*16 = 560 which is also congruent to 10 mod 50.
Similarly for the other solutions that you found.
You can verify that 22, 28, and 40 are not solutions.
35*22 = 770 which is 20 mod 50.
35*28 = 980, is 30 mod 50.
etc.
3. Jan 12, 2013
### knowLittle
I have read somewhere that division is not defined in modular arithmetic. Can someone tell me how this affect my solution?
@kru: This is puzzling, since I found those other solutions at a .edu site.
4. Jan 12, 2013
### HallsofIvy | {
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4. Jan 12, 2013
### HallsofIvy
Division is not necessarily defined in modular arithmetic because there may be "0 divisors". For example, in modulo 6, 2(3)= 6= 0 (mod 6). If we had an equation of the form 3x= 1 (mod 6) we can immediately check that 3(1)= 3, 3(2)= 6, 3(3)= 9= 3, 3(4)= 12= 0, and 3(5)= 15= 3 mod 6. There is NO x such that 3x= 1 and so we could not, for example, "divide by 3" to get "1/3" as an answer. If we are working "modulo" a prime number, that doesn't happen and we can define "division".
The way I would do "35x= 10 mod 50" is this. This is the same as saying 35x= 50n+ 10 for some integer n- a linear "diophantine equation". The first thing we can do divide through by 5 to 7x= 10n+ 2 or 7x- 10n= 2. Now 7 divides into 10 once with remainder 3: 3= 10- 7. 3 divides into 7 twice with remainder 1: 1= 7- 2(3). We can replace that "3" with 10- 7 from the first equation: 1= 7- 2(10- 7)= 3(7)- 2(10)= 1 (The "Euclidean Divison Algorithm"). Multiply through by 2 to get 6(7)- 4(10)= 2.
So one solution to 7x- 10n= 2 is x= 6, n= 4. It is possible to write out the "general solution" but since 6 itself is between 0 and 10, x= 6 satisfies 7(6)= 2 (mod 10) and so 35(6)= 210= 10 (mod 50).
5. Jan 12, 2013
### knowLittle
According to Wikipedia, Diophantine equations are written as follows:
ax + by = c
The Diphantine equation that you are really writing is this
35x-50n=10?
I understand everything, until you change the equation 1=7 -2(10-7)= 3(7)-2(10)=1. I understand that 21-20=1, but why changing from 7- 2(10-7) to 3(7)-2(10)?
Also, I am acquainted with Euclid's GCD algorithm:
Euclid(a,b)
if b==0
return a
else return Euclid (b, a mod b)
Is there a way to use it without having to trace it?
Is this all solutions for 35x $\equiv$ 10 mod 50? Also, is it correct that there has to be exactly 5 solutions, since the gcd of 35, 50 is 5?
Is my solution correct? | {
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# Math Help - Complex numbers-finding real number pairs
1. ## Complex numbers-finding real number pairs
Hello,
I am having trouble with this question:
"Find all possible real number pairs p, q such that 3+5i/1+pi =q+4i"
Im sure it's easy but I think I am overlooking something. I multiplied both sides by the conjugate of 1+pi....ie.(1-pi) but I think it's wrong.
Any help would be appreciated.
Regards,
Neverquit
2. Multiply both sides by 1+pi
Then you get...
3 + 5i = (1 + pi)(q + 4i) = q + pqi + 4i - 4p
=> 3 + 5i = (q - 4p) + 4i + pqi
=> 3 + i = (q - 4p) + pqi
So you want to solve...
q - 4p = 3
pq = 1
3. ## found 1 solution but not the other
I found the solutions 0.25, -1 after re-arranging pq = 1 to q = 1/p to get 1/p - 4p = 3 to get quadratic 4p^2+3p-1=0.
There is also apparently another solution of 4, -1 which I can’t find.
How do I find it?
4. This is a tricky question.
$\frac{{3 + 5i}}{{1 + pi}} = \frac{{\left( {3 + 5i} \right)\left( {1 - pi} \right)}}{{1 + p^2 }} = \frac{{3 + 5p}}{{1 + p^2 }} + \frac{{\left( {5 - p} \right)i}}{{1 + p^2 }}$
Now set real part equal to real part and imaginary equal to imaginary.
$\frac{3+5p}{1+p^2}=q~\&~\frac{5-3p}{1+p^2}=4$
From that we get $p=\frac{1}{4}~\&~p=-1$.
5. Originally Posted by Neverquit
I found the solutions 0.25, -1 after re-arranging pq = 1 to q = 1/p to get 1/p - 4p = 3 to get quadratic 4p^2+3p-1=0.
There is also apparently another solution of 4, -1 which I can’t find.
How do I find it?
I think you are misinterpreting your own answer. When you solve the system of equations, you will find that
$p=1/4 ~~\mbox{ or }~~ p=-1$
These are two separate solutions, not a single solution. You need to find the value of $q$ that pairs with each of these solutions for $p$. So, you need to plug each value of $p$ back into the system of equations and find the corresponding values of $q$. Since $pq=1$, it's a pretty straightforward calculation:
$p=1/4 \implies q=4$
$p=-1 \implies q=-1$ | {
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$p=1/4 \implies q=4$
$p=-1 \implies q=-1$
Therefore, the solutions are:
First solution: $p=1/4 \mbox{ and } q=4$
Second solution: $p=-1 \mbox{ and } q=-1$
You might also write this as:
$(p,q) = (1/4,4) \mbox{ or } (p,q) = (-1,-1)$
But you definitely would not say that the solutions are $(1/4,-1)$ and $(4,-1)$.
6. Originally Posted by Neverquit
Hello,
I am having trouble with this question:
"Find all possible real number pairs p, q such that 3+5i/1+pi =q+4i"
Im sure it's easy but I think I am overlooking something. I multiplied both sides by the conjugate of 1+pi....ie.(1-pi) but I think it's wrong.
Any help would be appreciated.
Regards,
Neverquit
$\frac{3+5i}{1+pi} =q+4i$
Multiply top and bottom of LHS by $1-pi$ to give:
$\frac{(3+5p)+(5-3p)i}{1+p^2}=q+4i$
Equate real and imaginary parts to get:
$3+5p=(1+p^2)q$
and:
$5-3p=4(1+p^2)$
Now the problem is to find all solutions to this pair of equations.
CB
7. Having Plato and CP post in this thread has made me question whether my answer is wrong. Is it?
Solving the equations I arrived at...
q - 4p = 3
pq = 1
gives
q=1/p,
So subbing that into the first equation will give you a polynomial (if you multiply both sides by p) which gives you p = 1/4 and -1.
Hence q = 4 and -1.
Solutions are (1/4,4) and (-1, -1)
Are these the only solutions? Why must you multiply the top and bottom by the conjugate?
Having Plato and CP post in this thread has made me question whether my answer is wrong. Is it?
Your solution was correct. Multiplying a complex fraction by the conjugate of the denominator is probably just a habit for them. But it's not necessary at all in this particular problem.
9. ## Using the conjugate | {
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9. ## Using the conjugate
Originally Posted by Plato
This is a tricky question.
$\frac{{3 + 5i}}{{1 + pi}} = \frac{{\left( {3 + 5i} \right)\left( {1 - pi} \right)}}{{1 + p^2 }} = \frac{{3 + 5p}}{{1 + p^2 }} + \frac{{\left( {5 - p} \right)i}}{{1 + p^2 }}$
Now set real part equal to real part and imaginary equal to imaginary.
$\frac{3+5p}{1+p^2}=q~\&~\frac{5-3p}{1+p^2}=4$
From that we get $p=\frac{1}{4}~\&~p=-1$.
I think the solution that Plato gives using the congugate is what the text books author had in mind as the question is shortly after conjugates of complex numbers is explained.
Deastar, your solution still gives the same answer in the text book so it must be correct.
...........
10. ## generalisation
Originally Posted by Neverquit
Hello,
I am having trouble with this question:
"Find all possible real number pairs p, q such that 3+5i/1+pi =q+4i"
Im sure it's easy but I think I am overlooking something. I multiplied both sides by the conjugate of 1+pi....ie.(1-pi) but I think it's wrong.
Any help would be appreciated.
Regards,
Neverquit
the basic idea behind questions of your type is equaslising real and imaginary parts. | {
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# What is the probability that Jan and Jon are chosen?
Jan , Jon and $10$ other children are in a classroom. The principal of the school walks in and choose $3$ children at random. What is the probability that Jan and Jon are chosen?
My approach:
Including Jon and Jan, number of ways of selection is $1\cdot1\cdot{10\choose 1} = 10$.
Total way of selection is ${12\choose 3} = 220$
So, probability is $\frac{10}{220}= \frac{1}{22}.$
But when I solving as follows:
\begin{align*} P(\text{selection of Jon and Jan}) &= 1- P(\text{not selection of Jon and Jan}) \\ &= 1- \frac{{10\choose 3}}{{12\choose 3}}\\ &= \frac{5}{11}. \end{align*}
Which approach is correct and why alternative one is wrong?
Note: My previous post had some mistakes, so I deleted that
I will call $A$, $B$ the events selecting Jon, Jan respectively. You did not take the complement correctly \begin{align*} P(A\cap B) &= 1-\color{red}{P(\bar A \cup\bar B)}\tag 1 \\ &= 1-[P(\bar A)+P(\bar B)-P(\bar A\bar B)]\tag 2 \\ &=1-\left[\frac{\binom{11}{3}}{\binom{12}{3}}+\frac{\binom{11}{3}}{\binom{12}{3}}-\frac{\binom{10}{3}}{\binom{12}{3}}\right]\\ &=\frac{1}{22} \end{align*} where $(1)$ is true by DeMorgan's law, and $(2)$ is true by inclusion-exclusion. As you can see the two methods give the same value.
Your first approach gives the probability that Jon AND Jan are selected, the second gives the probability that Jon OR Jan are selected.
When you computed the probability of not selecting Jon AND Jan, you didn't include the situations where Jon was selected but not Jan, and vice versa.
Lets say the even Jon is selected is A, Jan is selected B.
Then $P(A \wedge B)=1-P(Not (A \wedge B))= 1- P(Not A \vee Not B)$
$P(Not A \vee Not B)= P(Not A)+ P(Not B)- P(Not A \wedge Not B)$
Here $\wedge$ means and, and $\vee$ means or. Think of a Venn diagram, if we look at the union of two circles, the total area is equal to the sum of the circles minus the intersection, because we counted that part twice. | {
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$P(Not A)=1-P(A)=1-(1*\binom{11}{2})/220)=3/4$
or
$P(Not A)=\binom{11}{3}/220=3/4$
$P(Not B)=3/4$
$P(Not A \wedge Not B)= \binom{10}{3}/220=6/11$
Now $3/4+3/4-6/11=21/22$ So $1-21/22=1/22$ | {
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# Number of ways to place $k$ non-attacking rooks on a $100\times 100$ chess board
I need to show that the number of ways to place $k$ non-attacking rooks (no two share the same column or row) on a $100\times 100$ chessboard is $k!{100 \choose k}^2$.
When I try to formulate this equation I end up getting ${100 \choose k}^2$ because you need to choose $k$ columns from $100$ columns and $k$ rows from $100$ rows. I know this isn't correct because if you have $k=100$, there is more than just $1$ solution. However I don't know how to come up with the $k!$ part of the equation.
• – Shaun May 16 '17 at 14:32
First of all, congrats on realizing that the answer you got can't possibly be correct. It's always a good idea to test formulas against special cases, to see if they stand up.
One way to arrive at the correct answer is to view the placement of the rooks in two steps: First choose the $k$ rows that the rooks will go in, and then, going row by row, decide which column to place that row's rook in. The first rook has $100$ columns to choose from, the second will have $99$, the third $98$, and so on. The total is thus
$${100\choose k}100\cdot99\cdot98\cdots(100-(k-1))={100\choose k}{100!\over(100-k)!}={100\choose k}{100!\over(100-k)!k!}k!={100\choose k}^2k!$$
First, choose your $k$ rows and columns, as you said. Start by considering the configuration in which the rooks are successively placed in the legal square furthest to the top and to the left (so that the rooks go "diagonally down to the right").
From there, it suffices to note that any rearrangement of the rook-columns results in a new and valid configuration. Since there are $k!$ such rearrangements, there are $k!$ configurations for any particular choice of $k$ rows and $k$ columns. | {
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Your problem is that $\binom{100}{k}^2$ only gives you the ways to choose the columns and the rows separately, without specifiying which row goes with which column. This is why you need to add the factor $k!$, which corresponds to the number of bijections between your $k$ rows and your $k$ columns, i.e. the number of ways to associate them together.
An alternative way of obtaining this is to consider that you first choose $\binom{100}{k}$ column where you will place your rooks, then choose for each column the row where you place a rook; this second step amonts to choosing $k$ rows with order, which you can do in $\frac{100!}{(100-k)!}=\binom{100}{k}k!$ ways. Multiplying the two numbers together gives you the result.
Regards User. If i may contribute, here is my view :
A 100 $\times$ 100 chess board can be viewed as a matrix of size 100 $\times$ 100. For example : let $$(i, j), \:\: \text{ with } \:\:\: i,j=1,2,... ,100$$ denotes the $i$th row and $j$th column of the board.
To solve your problem, the key is : the $k$ non-attacking roots is as same as no two $(i_{1}, j_{1})$ and $(i_{2}, j_{2})$ with $i_{1}=i_{2}$ or $j_{1}=j_{2}$. Two rooks with position $(1, 100)$ and $(91, 100)$, for example, does not satisfy the non-attacking roots condition.
To illustrate how to solve this, first, you could start with $k=1$.
• For k=1. Let the position of this particular rook is $(x_{1},y_{1})$. Then there are 100 possibilities for $x_{1}$, and 100 possibilities for $x_{2}$. So the number of possibilities is (100)(100) = $1! \binom{100}{1}^{2}$ | {
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• For k=2. For each of the two rooks, their positions are denoted with $(x_{1},y_{1})$ and $(x_{2}, y_{2})$. For the first one, There are 100 possibilities for each $x_{1}$ and $y_{1}$. For the 2nd rook, $x_{2}$ and $y_{2}$ each has 99 possibilities (since they can't be equal to the 'coordinates' of the 1st rook). So the number of possibilities to put 2 non-attacking distinguished rooks is $$(100^{2})(99^{2})$$ For your problem, they are not distinguished, so we have to divide this by 2 (exactly $2!$), because we can choose either rook to be the 1st or the 2nd. So, output : $$\frac{(100^{2})(99^{2})}{2!} = \frac{(100 \cdot 99)(100 \cdot 99)}{2!} = \frac{(100 \cdot 99)(100 \cdot 99)}{2!} \left(\frac{98! \cdot 2!}{98! \cdot 2!} \right)^{2} = 2! \binom{100}{2}^{2}$$
• For k=3 and above, you would be confident to try and continue this method.
Hope this will helps. Regards, Arief. | {
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# Permutation representation argument validity
#### kalish
##### Member
Hello,
I would like to check if the work I have done for this problem is valid and accurate. Any input would be appreciated. Thank you.
**Problem statement:** Let $G$ be a group of order 150. Let $H$ be a subgroup of $G$ of order 25. Consider the action of $G$ on $G/H$ by left multiplication: $g*aH=gaH.$ Use the permutation representation of the action to show that $G$ is not simple.
**My attempt:** Let $S_6$ be the group of permutations on $G/H$. Then, the action of $G$ on $G/H$ defines a homomorphism $f:G \rightarrow S_6$. We know $|S_6| = 720.$ Since $|G|=150$ does not divide 720, and $f(G)$ is a subgroup of $S_6$, $f$ cannot be one-to-one. Thus, $\exists$ $g_1,g_2$ distinct in $G$ such that $f(g_1)=f(g_2) \implies f(g_1g_2^{-1})=e$. Thus, $\ker(f) = \{g:f(g)=e\}$. Since $\ker(f)$ is a normal subgroup of $G$, we have found a normal subgroup of $G$. Also, since $f$ is non-trivial, then $\ker(f)$ is a proper normal subgroup of $G.$ Hence $G$ is not simple.
Any suggestions or corrections?
#### Deveno
##### Well-known member
MHB Math Scholar
I think it is clear that since $|G|$ does not divide 720, $\text{ker}(f)$ is a non-trivial normal subgroup of $G$, so there is no need to talk about the existence of $g_1,g_2$ or restate the definition of $\text{ker}(f)$.
I *do* think you should say WHY $f$ is not the trivial homomorphism. It's pretty simple, though:
Since $|H| < |G|$, we can take any $g \in G - H$, which takes (under the action) the coset $H$ to $gH \neq H$, so $f(G)$ contains at least one non-identity element: namely, $f(g)$.
#### Deveno
##### Well-known member
MHB Math Scholar
This is actually a special case of a theorem proved in Herstein, which goes as follows:
If $G$ is a finite group with a subgroup $H$ such that $|G| \not\mid ([G:H])!$ then $G$ contains a non-trivial proper normal subgroup containing $H$. | {
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One obvious corollary is then that such a group $G$ cannot be simple.
You would be far better off adapting your proof to this more general one, which can be re-used in many more situations.
#### kalish
##### Member
Hi,
Which Herstein book is this from? I would like to explore further.
Thanks.
#### Deveno
##### Well-known member
MHB Math Scholar
His classic Topics In Algebra.
#### kalish
##### Member
That's what I found from my search as well. Do you have a copy of the book or know where I can find one?
#### kalish
##### Member
That sounds like a fantastic result. I cannot find the book anywhere though. Could you please reproduce the proof for me here, so that I could use it to study? I would really appreciate it.
#### Deveno
##### Well-known member
MHB Math Scholar
Theorem 2.G (p. 62, chapter 2):
If $G$ is a group, $H$ a subgroup of $G$, and $S$ is the set of all right cosets of $H$ in $G$, then there is a homomorphism $\theta$ of $G$ into $A(S)$, and the kernel of $\theta$ is the largest normal subgroup of $G$ which is contained in $H$.
(a few words about notation: Herstein uses $A(S)$ to stand for the group of all bijections on $S$...if $|S| = n$, then $A(S)$ is isomorphic to $S_n$. Herstein also writes his mappings on the RIGHT, as in $(x)\sigma$ instead of $\sigma(x)$, so that composition and multiplication are "in the same order", instead of reversed. For this reason, he uses right cosets and right-multiplication instead of the left cosets (and left-multiplication) one often sees used in other texts. He also denotes the index of $H$ in $G$ by $i(H)$ , instead of $[G:H]$ and denotes $|G|$ by $o(G)$). | {
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Proof: Let $G$ be a group, $H$ a subgroup of $G$. Let $S$ be the set whose elements are right cosets of $H$ in $G$. That is, $S = \{Hg: g \in G\}$. $S$ need not be a group itself, in fact, it would be a group only if $H$ were a normal subgroup of $G$. However, we can make our group $G$ act on $S$ in the following natural way: for $g \in G$ let $t_g:S \to S$ be defined by: $(Hx)t_g = Hxg$. Emulating the proof of Theorem 2.f we can easily prove:
(1) $t_g \in A(S)$ for every $g \in G$
(2) $t_{gh} = t_gt_h$. | {
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(1) $t_g \in A(S)$ for every $g \in G$
(2) $t_{gh} = t_gt_h$.
Thus the mapping $\theta: G \to A(S)$ defined by $\theta(g) = t_g$ is a homomorphism of $G$ into $A(S)$. Can one always say that $\theta$ is an isomorphism? Suppose that $K$ is the kernel of $\theta$. If $g_0 \in K$, then $\theta(g_0) = t_{g_0}$ is the identity map on $S$, so that for every $X \in S, Xt_{g_0} = X$. Since every element of $S$ is a right coset of $H$ in $G$, we must have that $Hat_{g_0} = Ha$ for every $a \in G$, and using the definition of $t_{g_0}$, namely, $Hat_{g_0} = Hag_0$, we arrive at the identity $Hag_0 = Ha$ for every $a \in G$. On the other hand if $b \in G$ is such that $Hxb = Hx$ for every $x \in G$, retracing our argument we could show that $b \in K$. Thus $K = \{b \in G|Hxb = Hx$ all $x \in G\}$. We claim that from this characterization of $K,\ K$ must be the largest normal subgroup of $G$ which is contained in $H$. We first explain the use of the word largest; by this we mean if $N$ is a normal subgroup of $G$ which is contained in $H$, then $N$ must be contained in $K$. We wish to show this is the case. That $K$ is a normal subgroup of $G$ follows from the fact that it is the kernel of a homomorphism of $G$. Now we assert that $K \subset H$, for if $b \in K, Hab = Ha$ for every $a \in G$, so in particular, $Hb = Heb = He = H$, whence $b \in H$. Finally, if $N$ is a normal subgroup of $G$ which is contained in $H$, if $n \in N,\ a \in G$, then $ana^{-1} \in N \subset H$, so that $Hana^{-1} = H$; thus $Han = Ha$ for all $a \in G$. Therefore, $n \in K$ by our characterization of $K$.
**********
Remarks following the proof:
The case $H = (e)$ just yields Cayley's Theorem (Theorem 2.f). If $H$ should happen to have no normal subgroup of $G$, other than $(e)$ in it, then $\theta$ must be an isomorphism of $G$ into $A(S)$....(some text omitted).... | {
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We examine these remarks a little more closely. Suppose that $G$ has a subgroup $H$ whose index $i(H)$ (that is, the number of right cosets of $H$ in $G$) satisfies $i(H)! < o(G)$. Let $S$ be the set of all right cosets of $H$ in $G$. The mapping, $\theta$, of Theorem 2.g cannot be an isomorphism, for if it were, $\theta(G)$ would have $o(G)$ elements and yet would be a subgroup of $A(S)$ which has $i(H)! < o(G)$ elements. Therefore, the kernel of $\theta$ must be larger than $(e)$; this kernel being the largest normal subgroup of $G$ which is contained in $H$, we can conclude that $H$ contains a nontrivial normal subgroup of $G$.
However, the above argument has implications even when $i(H)!$ is not less than $o(G)$. If $o(G)$ does not divide $i(H)!$ then by invoking Lagrange's theorem we know that $A(S)$ can have no subgroup of order $o(G)$, hence no subgroup isomorphic to $G$. However $A(S)$ does contain $\theta(G)$, whence $\theta(G)$ cannot be isomorphic to $G$, that is, $\theta$ cannot be an isomorphism. But then, as above, $H$ must contain a nontrivial normal subgroup of $G$. We summarize this as:
Lemma 2.21 If $G$ is a finite group, and $H \neq G$ is a subgroup of $G$ such that $o(G) \not\mid i(H)!$, then $H$ must contain a nontrivial normal subgroup of $G$. In particular, $G$ cannot be simple.
(Note to the moderating staff: although this is an excerpt from a copyrighted work, I believe this sample falls under the province of "Fair Use" for the purpose of "Scholarly research and exposition", and is not intended for commercial gain or to circumvent existing copyright laws).
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# Spring Gun
1. Mar 7, 2014
### Steven60
I have a question about a spring gun. Suppose the barrel of a spring gun is placed horizontally at the edge of a horizontal table. You put say a marble in the barrel and compress the spring x cm and after releasing the marble it travels a horizontal distance of y cm before hitting the floor (so motion is of a projectile). My question is whether or not the horizontal distance traveled and the amount the spring is compresses make a linear relationship? If so, then how can I prove this? This is not homework.
Thanks!
2. Mar 7, 2014
Seems to be purely a math problem. Perhaps sketch the system and write the relevant equations needed to determine this?
3. Mar 7, 2014
### UltrafastPED
This is a nice physics exercise - there are several physical considerations, and then some simple math.
You have two forces acting on the marble ... the spring force, which launches the marble, and gravity.
Once the marble leaves the launch tube it will have a constant "horizontal" speed - ignoring air resistance - and an initial vertical speed of zero. Call this initial horizontal speed V.
The vertical speed will increase with time due to the constant gravitational acceleration - and will hit the floor at a definite time which depends only on the height of the table. Call this duration T.
Then the distance from the table to the point of contact will be D = V x T.
The time T does not depend upon the spring force, only on the height of the table and local value g=9.8 m/s^2.
Thus you only need to determine if the speed V is proportional to the spring force; by Hook's law we know that a "good" spring obeys F = -k * X, where X is the compression/extension distance and k is the spring's constant.
If we switch to energy we have work done on marble is W = Integral[F dx] over the interval x=[0,X]. Note that the force is changing as the spring moves! So W = Integral[ k*x dx] = 1/2 k*X^2. | {
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But this work has been converted into kinetic energy of the marble. For a marble of mass=M, and given that it is NOT rolling or spinning, then the kinetic energy is KE=1/2 M*V^2 = 1/2 k*X^2=W.
Thus V = k/M Sqrt[X]. xx Correction: xxx Make that V = Sqrt[k/M] * X.
Thus the hypothesis is true!
Thanks to dauto for noticing the mistake at the end! :-)
Last edited: Mar 7, 2014
4. Mar 7, 2014
### DrewD
The equations involved will be $d=vt$ for constant $v$ (and assuming that the initial point when exiting the spring gun is defined as 0 distance), $U_{spring}=\frac12k\Delta x^2$ and $K=\frac12mv^2$. $\Delta x$ is the amount the spring is compressed, and $v$ is the velocity of the object as it leaves the spring. This approximation assumes that the object does not stick which is a good assumption for a spring gun. Solve for $v$ to find the relationship between $\Delta x$ and $d$.
5. Mar 7, 2014
### dauto
You made a mistake at the very end. In fact, after correcting the mistake, you proved that the hypothesis is true.
6. Mar 9, 2014
### Steven60
Thanks for your replys. I actually worked this out myself and actually did the same exact steps as UltrafastPED. | {
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# Is this induction procedure correct? ($2^n<n!$)
I am rather new to mathematical induction. Specially inequalities, as seen here How to use mathematical induction with inequalities?. Thanks to that question, I've been able to solve some of the form $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \leq \frac{n}{2} + 1$.
Now, I was presented this, for $n \ge 4$:
$$2^n<n!$$
I tried to do it with similar logic as the one suggested there. This is what I did:
Prove it for $n = 4$: $$2^4 = 16$$ $$4! = 1\cdot2\cdot3\cdot4 = 24$$ $$16 < 24$$ Assume the following: $$2^n<n!$$ We want to prove the following for $n+1$: $$2^{n+1}<(n+1)!$$ This is how I proved it:
• So first we take $2^{n+1}$ which is equivalent to $2^n\cdot2$
• By our assumption, we know that $2^n\cdot2 < n!\cdot2$
• This is because I just multiplied by $2$ on both sides.
• Then we'll be finished if we can show that $n! \cdot 2 < (n+1)!$
• Which is equivalent to saying $n!\cdot2<n!\cdot(n+1)$
• Since both sides have $n!$, I can cancel them out
• Now I have $2<(n+1)!$
• This is clearly true, since $n \ge 4$
Even though the procedure seems to be right, I wonder: | {
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• In the last step, was it ok to conclude with $2<(n+1)!$? Was there not anything else I could have done to make the proof more "careful"?
• Is this whole procedure valid at all? I ask because, well, I don't really know if it would be accepted in a test.
• Are there any points I could improve? Anything I could have missed? This is kind of the first time I try to do these.
-
I think your proof is fine, if a bit long-winded. But with experience, you'll learn what bits to shorten without losing rigour. – Harald Hanche-Olsen Nov 27 '12 at 9:51
You should say 2^{n+1}\lt 2\cdot n!$. But because$2\lt $n+1$, it follows that $2\cdot n!\lt (n+1)!$. Unfortunately, you are still writing proofs "backwards" in a logically incorrect way. – André Nicolas Nov 27 '12 at 9:59
@André: It seems a bit harsh to call this a "logically incorrect way". It's OK as long as "Since both sides have $n!$, I can cancel them out" is interpreted as "dividing through by $n!$ leads to an equivalent inequality". It's true that the reverse order would be clearer, and doing things in this order is incorrect if the implications used only go in one direction, but that's not the case here so the proof is still OK, if suboptimally structured. – joriki Nov 27 '12 at 10:04
@Omega, your proof is correct..."almost", since as Andre apparently meant, there's some lack of logical rigour in your last lines. You must show that the implications there are double, i.e.: $$n!\cdot 2<(n+1)!\Longleftrightarrow 2<\frac{(n+1)!}{n!}=n+1$$ and then noting the last inequality is trivially true as we're working with $\,n\geq 4\,$... Also, don't right "equivalent" when it should be "equal", as in "$\,2^{n+1}\,$ is "equivalent"(should be "equal"!) to $\,2^n\cdot 2$ – DonAntonio Nov 27 '12 at 10:47 | {
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@joriki: A bit harsh, perhaps, but other posts by the OP had some worse instances that were pointed out. Since the actual understanding of the problem is good, it is worthwhile to vaccinate the OP against $A\to B\to C\to 0=0$, therefore $A$. – André Nicolas Nov 27 '12 at 16:47 | {
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Yes, the procedure is correct. If you want to write this more like the sort of mathematical proof that would be found in a textbook, you might want to make some tweaks.
For example, the base case could be re-written as follows:
When $n = 4$, we have $2^4 = 16 < 24 = 4!$
Next, the inductive hypothesis and the subsequent manipulations:
Suppose that for $n \geq 4$ we have $2^n < n!$
Thus, $2^{n+1} < 2 \cdot n! < (n+1)!$, where the first inequality follows by multiplying both sides of the inequality in our IH by $2$, and the second follows by observing that $2 < n+1$ when $n \geq 4$.
Therefore, by the Principle of Mathematical Induction, $2^n < n!$ for all integers $n \geq 4$. Q.E.D.
Note: I am not making a judgment about whether your write-up or the one I have included here is "better." I'm only observing that the language and format differ, particularly with regard to proofs that are written in paragraph form (typical of math papers) rather than with a sequence of bullet-points (which is what you had).
- | {
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### Bfs Time Complexity | {
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