title string | question_body string | answer_body string | tags string | accepted int64 |
|---|---|---|---|---|
Real world uses of hyperbolic trigonometric functions | I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses outside academia? | On a map using the Mercator projection, the relationship between the latitude L of a point and its y coordinate on the map is given by $y = \operatorname{arctanh}(\sin(L))$, where $\operatorname{arctanh}$ is the inverse of the hyperbolic tangent function. | |soft-question|big-list|applications|hyperbolic-functions| | 0 |
Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | To the degree that anything actually "exists" in math, yes complex number exist. Once you accept that groups, rings and fields exist, and that isomorphism of rings makes sense, complex numbers can be recognized as (isomorphic to) the subring (which happens to be a field) of the ring of $2 \times 2$ real matrices. Gener... | |soft-question|complex-numbers|education|philosophy| | 0 |
How do you determine if a point sits inside a polygon? | Given the coordinates of a point $(x, y)$, what is a procedure for determining if it lies within a polygon whose vertices are $(x_1, y_1), (x_2, y_2), \ldots , (x_n,y_n)$? | I will assume the polygon has no intersections between the edges except at corners. Call the point $(x_0, y_0)$. First we determine whether we are on a line - this is simple using substitution and range checking. For the range checking, suppose we have a segment $(x_1, y_1)$, $(x_2, y_2)$. We check that $x_1\leq x_0\le... | |geometry|algorithms| | 1 |
Are the "proofs by contradiction" weaker than other proofs? | I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one ... | Nearly always the direct proof is easier to understand, shorter, and more helpful! | |logic|proof-theory| | 0 |
Are the "proofs by contradiction" weaker than other proofs? | I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one ... | Sometimes you might want to know not just that there exists something, you might want to know how to actually go about finding it (and related questions like how quickly you can find it). Proofs by contradiction are non-constructive, while direct proofs are typically constructive in the sense that they actually constru... | |logic|proof-theory| | 0 |
Are the "proofs by contradiction" weaker than other proofs? | I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one ... | Most logicians consider proofs by contradiction to be equally valid, however some people are constructivists/intuitionists and don't consider them valid. ( Edit: This is not strictly true, as explained in comments. Only certain proofs by contradiction are problematic from the constructivist point of view, namely those ... | |logic|proof-theory| | 0 |
Are the "proofs by contradiction" weaker than other proofs? | I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one ... | At first this seems like a silly question - after all isn't the point of a mathematical proof to be a proof and hence to be beyond question. But of course, to prove anything we need assumptions and some people do disagree with the axioms commonly used by mathematicians. I don't have much knowledge of this view, but I a... | |logic|proof-theory| | 0 |
Are there any functions that are (always) continuous yet not differentiable? Or vice-versa? | It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable? The other way around seems a bit simpler -- a differentiable function is obviously always going to be continuous. But are... | Actually, in some sense, almost all of the continuous functions are nowhere differentiable: http://en.wikipedia.org/wiki/Weierstrass_function#Density_of_nowhere-differentiable_functions | |real-analysis|continuity| | 0 |
What exactly does it mean for a function to be "well-behaved"? | Often in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of the requirements), nor do I know why this is necessary (though I imagine the why will depend on each case). Can someone explain it t... | In the sciences (as opposed to in mathematics) people are often a bit vague about exactly what assumptions they are making about how "well-behaved" things are. The reason for this is that ultimately these theories are made to be put to the test, so why bother worrying about exactly which properties you're assuming when... | |terminology|analysis|economics| | 1 |
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language | The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently ... | Now that Wiles has done the job, I think that Fermat's Last Theorem may suffice. I find it a bit surprising still. | |soft-question|big-list|examples-counterexamples| | 0 |
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language | The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently ... | An infinite amount of coaches, each containing an infinite amount of people can be accommodated at Hilbert's Grand Hotel . Visual demonstration here . | |soft-question|big-list|examples-counterexamples| | 0 |
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language | The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently ... | All of three dimensional space can be filled up with an infinite curve . | |soft-question|big-list|examples-counterexamples| | 0 |
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language | The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently ... | My first thought is the ham sandwich theorem --given a sandwich formed by two pieces of bread and one piece of ham (these pieces can be of any reasonable/well-behaved shape) in any positions you choose, it is possible to cut this "sandwich" exactly in half, that is divide each of the three objects exactly in half by vo... | |soft-question|big-list|examples-counterexamples| | 0 |
What are some classic fallacious proofs? | If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already: Let $a = b$. Then $a^2 = ab$ $a^2 - b^2 = ab - b^2$ Factor to $(a-b)(a+b) = b(a-b)$ Then divide out $(a-b)$ to get $a+b = b$ Since $a = b$, then $b+b = b$ Therefore $... | The two envelopes problem is a good one. See also: Card doubling paradox and: https://mathoverflow.net/questions/9037 | |soft-question|big-list|fake-proofs| | 0 |
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language | The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently ... | The Monty Hall problem fits the bill pretty well. Almost everyone, including most mathematicians, answered it wrong on their first try, and some took a lot of convincing before they agreed with the correct answer. It's also very easy to explain it to people. | |soft-question|big-list|examples-counterexamples| | 0 |
Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? | Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly? I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks. | Let's group the terms as follows: Group $1$ : $\displaystyle\frac11\qquad$ ($1$ term) Group $2$ : $\displaystyle\frac12+\frac13\qquad$($2$ terms) Group $3$ : $\displaystyle\frac14+\frac15+\frac16+\frac17\qquad$($4$ terms) Group $4$ : $\displaystyle\frac18+\frac19+\cdots+\frac1{15}\qquad$ ($8$ terms) $\quad\vdots$ In ge... | |calculus|sequences-and-series|harmonic-numbers| | 1 |
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language | The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently ... | There are true statements in arithmetic which are unprovable. Even more remarkably there are explicit polynomial equations where it's unprovable whether or not they have integer solutions with ZFC ! (We need ZFC + consistency of ZFC) | |soft-question|big-list|examples-counterexamples| | 0 |
Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? | Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly? I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks. | This is not as good an answer as AgCl's, nonetheless people may find it interesting. If you're used to calculus then you might notice that the sum $$ 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$$ is very close to the integral from $1$ to $n$ of $\frac{1}{x}$. This definite integral is ln(n), so you should expect $1+\fr... | |calculus|sequences-and-series|harmonic-numbers| | 0 |
What is the single most influential book every mathematician should read? | If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be? | William Dunham's "Journey through Genius." Well, rather that is the book I read that made me want to be a mathematician. | |soft-question|big-list|reference-request| | 0 |
What is the single most influential book every mathematician should read? | If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be? | Following Noah's lead I will mention; "The Man Who Loved Only Numbers" and "How to Read and Do Proofs" | |soft-question|big-list|reference-request| | 0 |
What is the single most influential book every mathematician should read? | If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be? | Probability Theory: the Logic of Science. Or anything by Edwin T Jaynes . | |soft-question|big-list|reference-request| | 0 |
What is the single most influential book every mathematician should read? | If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be? | This is an extremely broad question, especially given the wide variety of mathy people here, but I'll bite. HSM Coxeter's Introduction to Geometry is a book that was very important to the development of my interest in mathematics and inclination towards its geometric aspects. | |soft-question|big-list|reference-request| | 0 |
What is the single most influential book every mathematician should read? | If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be? | Nicolas Bourbaki's Éléments de mathématique (specifically Topologie Générale and Algèbre). | |soft-question|big-list|reference-request| | 0 |
Is there a real number lookup algorithm or service? | Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it? I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers. The intended use would be: write ... | Try Wolfram Alpha . It actually does sequences as well. | |math-software|experimental-mathematics| | 1 |
Can you recommend a decent online or software calculator? | I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. Can you recommend any? | For desktop software, I use SpeedCrunch . Has a history, lots of mathematical functions, supports variable, etc. | |soft-question|big-list|math-software|computer-algebra-systems| | 0 |
Best Maths Books for Non-Mathematicians | I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which tea... | I've been successful in using Courant and Robbins' What Is Mathematics? An Elementary Approach to Ideas and Methods for adults who have not had a math class for a few decades, but are open to the idea of learning more about mathematics. Some sections are too advanced for someone with only high school mathematics, and m... | |reference-request|soft-question|big-list|book-recommendation| | 0 |
Is there a real number lookup algorithm or service? | Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it? I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers. The intended use would be: write ... | Sometimes the decimal digits of numbers will appear in Sloane's On-Line Encyclopedia of Integer Sequences OIES . E.g. here is the decimal expansion of pi. | |math-software|experimental-mathematics| | 0 |
Online resources for learning Mathematics | Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, mos... | Purplemath has a list for math lessons and tutoring. It's a list with various links and short reviews referring to tutoring and instructional resources. | |reference-request|online-resources| | 0 |
List of Interesting Math Blogs | I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills. I'll start with my entries: Division By Zero Tanya Khovanova’s Math Blog | It isn't quite a blog, but Steven Strogatz's 15 part series for the New York Times was excellent. | |soft-question|big-list|online-resources| | 0 |
How come $32.5 = 31.5$? (The "Missing Square" puzzle.) | Below is a visual proof (!) that $32.5 = 31.5$ . How could that be? (As noted in a comment and answer, this is known as the "Missing Square" puzzle .) | The red and blue triangles are not similar (the ratios of the sides are 3/8 = 0.375 and 2/5 = 0.4 respectively), so the "hypotenuse" of your big triangle is not a straight line. | |geometry|recreational-mathematics|fake-proofs| | 0 |
How come $32.5 = 31.5$? (The "Missing Square" puzzle.) | Below is a visual proof (!) that $32.5 = 31.5$ . How could that be? (As noted in a comment and answer, this is known as the "Missing Square" puzzle .) | It's an optical illusion - neither the first nor second set of blocks actually describes a triangle. The diagonal edge of the first is slightly concave and that of the second is slightly convex. To see clearly, look at the gradients of the hypotenuses of the red and blue triangles - they're not 'similar'. gradient of b... | |geometry|recreational-mathematics|fake-proofs| | 1 |
List of Interesting Math Blogs | I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills. I'll start with my entries: Division By Zero Tanya Khovanova’s Math Blog | John D Cook writes The Endeavor One of the MathWorks blogs: Loren on the Art of Matlab ... a few more: eon Peter Cameron's Blog Walking Randomly Todd and Vishal's Blog (Check their blogrolls for more) | |soft-question|big-list|online-resources| | 0 |
Is $0$ a natural number? | Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely ... | Simple answer: sometimes yes, sometimes no, it's usually stated (or implied by notation). From the Wikipedia article : In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers $\{1, 2, 3, \dots\}$ according to the traditional definition; or the set of non-negat... | |terminology|natural-numbers| | 1 |
Is $0$ a natural number? | Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely ... | There is no "official rule", it depends from what you want to do with natural numbers. Originally they started from $1$ because $0$ was not given the status of number. Nowadays if you see $\mathbb{N}^+$ you may be assured we are talking about numbers from $1$ above; $\mathbb{N}$ is usually for numbers from $0$ above. [... | |terminology|natural-numbers| | 0 |
Why is $1$ not a prime number? | Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$? | actually 1 was considered a prime number until the beginning of 20th century. Unique factorization was a driving force beneath its changing of status, since it's formulation is quickier if 1 is not considered a prime; but I think that group theory was the other force. Indeed I prefer to describe numbers as primes, comp... | |abstract-algebra|elementary-number-theory|ring-theory|prime-numbers|terminology| | 0 |
Aren't constructive math proofs more "sound"? | Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox without a proof construction? | when dealing with infinite sets, many propositions cannot be proved mithout using non-constructive proofs. Axiom of Choice (AC, in brief) or an equivalent proposition is required. Russell's paradox is not a problem per se, you just rule out certain collections of things as sets; Banach-Tarski paradox (you may take a ba... | |proof-theory|constructive-mathematics| | 0 |
Is $0$ a natural number? | Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely ... | I remember all of my courses at University using only positive integers (not including $0$) for the Natural Numbers. It's possible that they had come to an agreement amongst the Maths Faculty, but during at least two courses we generated the set of natural numbers in ways that wouldn't make sense if $0$ was included. O... | |terminology|natural-numbers| | 0 |
Online resources for learning Mathematics | Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning. As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, mos... | For basic mathematics mathpage has a long course, including the list below. Don't be put off by the elementary school style lesson names, it does go into some depth with each one. Lesson 1 Reading and Writing Whole Numbers Lesson 2 The Meaning of Decimals Lesson 3 Multiplying and Dividing Lesson 4 More Elementary Addit... | |reference-request|online-resources| | 0 |
How does the wheel paradox work? | I keep looking at this picture and its driving me crazy. How can the smaller circle travel the same distance when its circumference is less than the entire wheel? | the smaller wheel does not just rotate, but also slides. If you had cogwheels instead of smooth wheels, you'd notice that movement is not possible. | |geometry| | 0 |
Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem] | How can I show that $(n-1)!\equiv-1 \pmod{n}$ if and only if $n$ is prime? Thanks. | $$n\text{ is prime if }(n-1)! \equiv -1 \pmod n$$ This direction is easy. If $n$ is composite, then there exists $k|n$ and $k\lt n$ . So $k|(n-1)!$ and $k \equiv 1 \pmod n$ . This means $k$ needs to divide $1$ . So $n$ must be prime (or $1$ , but we can eliminate this by substitution). $$(n-1)! \equiv -1\text{ if }n\te... | |group-theory|elementary-number-theory|modular-arithmetic| | 1 |
What are all the homomorphisms between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$? | Any homomorphism $φ$ between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$ is completely defined by $φ(1)$. So from $$0 = φ(0) = φ(18) = φ(18 \cdot 1) = 18 \cdot φ(1) = 15 \cdot φ(1) + 3 \cdot φ(1) = 3 \cdot φ(1)$$ we get that $φ(1)$ is either $5$ or $10$. But how can I prove or disprove that these two are valid ho... | If one has a homomorphism of two rings $R, S$ , and $R~$ has an identity, then the identity must be mapped to an idempotent element of $S$ , because the equation $x^2=x$ is preserved under homomorphisms. Now $5$ is not an idempotent element in $\Bbb Z_{15}$ , so the map generated by $1 \to 5$ is not a homomorphism. How... | |ring-theory|abstract-algebra| | 1 |
Why are $\Delta_1$ sentences of arithmetic called recursive? | The arithmetic hierarchy defines the $\Pi_1$ formulae of arithmetic to be formulae that are provably equivalent to a formula in prenex normal form that only has universal quantifiers, and $\Sigma_1$ if it is provably equivalent to a prenex normal form with only existential quantifiers. A formula is $\Delta_1$ if it is ... | If a formula $\phi(x_1, ... x_n)$ is in both $\Sigma_1, \Pi_1$, then one can define a Turing machine to determine whether it is true or false. Namely, in parallel, search for a collection of parameters that makes true the existential formula, and search for a collection of formulas that makes false the universal formul... | |logic|computability|proof-theory| | 1 |
Is there a real number lookup algorithm or service? | Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it? I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers. The intended use would be: write ... | I've long used Simon Plouffe's inverse symbolic calculator for this purpose. It is essentially a searchable list of "interesting" numbers. Edit: link updated (Mar 2022). | |math-software|experimental-mathematics| | 0 |
How are we able to calculate specific numbers in the Fibonacci Sequence? | I was reading up on the Fibonacci Sequence, $1,1,2,3,5,8,13,\ldots $ when I noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's p... | A lot of people have mentioned Binet's formula. But I suspect this is not the most practical way to compute the nth Fibonacci number for large n, because it requires either having a very accurate value of $\sqrt{5}$ and carrying around lots of decimal places (if you want to do floating-point arithmetic) or expanding la... | |combinatorics|generating-functions|fibonacci-numbers| | 0 |
Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem] | How can I show that $(n-1)!\equiv-1 \pmod{n}$ if and only if $n$ is prime? Thanks. | [NOTE: it seems that there is some difference between preview and actual output, so instead if using (mod p) I stick with (p)] to show that $(p-1)! \equiv -1 (p)$ without explicitly use group theory, maybe the simplest path is: (the following assumes $p$ is odd, but if $p=2$ then the result is immediate) given $n \ne 0... | |group-theory|elementary-number-theory|modular-arithmetic| | 0 |
Calculating the probability of two dice getting at least a $1$ or a $5$ | So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question. This question comes from the game farkle. | Go backwards: Calculate the probability that neither of them shows a 1 or 5. That means both show a 2, 3, 4, or 6. Thats $(4/6)^2$. Hence the probability that at least one shows a 1 or 5 is $1-(2/3)^2=5/9$. | |probability-theory| | 1 |
Unital homomorphism | What is a unital homomorphism? Why are they important? | A unital homomorphism between rings R and S is a ring homomorphism that sends the identity element of R to the identity element of S. Homomorphisms (between objects in any algebraic category like groups, rings, vector spaces, etc.) preserve the algebraic structure, and if you want a map between rings with an identity e... | |abstract-algebra|terminology|definition| | 1 |
Books on Number Theory for Layman | Books on Number Theory for anyone who loves Mathematics? (Beginner to Advanced & just for someone who has a basic grasp of math) | I would still stick with Hardy and Wright, even if it is quite old. | |number-theory|reference-request|soft-question|book-recommendation| | 0 |
Unital homomorphism | What is a unital homomorphism? Why are they important? | A lot of results about rings just won't work otherwise: for instance, a unital homomorphism of rings sends units to units. A nonunital homomorphism doesn't have to do that. Nonunital homomorphisms can be very degenerate, e.g. the zero homomorphism. Another reason you want homomorphisms to preserve the unit is that this... | |abstract-algebra|terminology|definition| | 0 |
Calculating the probability of two dice getting at least a $1$ or a $5$ | So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question. This question comes from the game farkle. | To visually see the answer given by balpha above, you could write out the entire set of dice rolls [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6] [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [2, 6] [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [3, 6] [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [4, 6] [5, 1], [5, 2], [5, 3], [5, 4], [... | |probability-theory| | 0 |
Best Maths Books for Non-Mathematicians | I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which tea... | I think any book by John Allen Paulos would be something any Math enthusiast could enjoy and learn from. | |reference-request|soft-question|big-list|book-recommendation| | 0 |
Books on Number Theory for Layman | Books on Number Theory for anyone who loves Mathematics? (Beginner to Advanced & just for someone who has a basic grasp of math) | A concise introduction to the theory of numbers by Alan Baker (1970 Fields medalist) covers a lot of ground in less than 100 pages, and does so in a fluid way that never feels rushed. I love this little book. | |number-theory|reference-request|soft-question|book-recommendation| | 0 |
Best Maths Books for Non-Mathematicians | I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which tea... | John Derbyshire's Prime Obsession is about Riemann's hypothesis. One of the stated goals of the author is to explain what "all non-trivial zeros of the zeta function have real part one-half" means to readers who have no background in calculus. Odd-numbered chapters tell the story of how Riemann came to his hypothesis, ... | |reference-request|soft-question|big-list|book-recommendation| | 0 |
Why are $3D$ transformation matrices $4 \times 4$ instead of $3 \times 3$? | Background: Many (if not all) of the transformation matrices used in $3D$ computer graphics are $4\times 4$, including the three values for $x$, $y$ and $z$, plus an additional term which usually has a value of $1$. Given the extra computing effort required to multiply $4\times 4$ matrices instead of $3\times 3$ matric... | To follow up user80's answer, you want to get transformations of the form v --> Av + b, where A is a 3 by 3 matrix (the linear part of transformation) and b is a 3-vector. We can encode this transformation in a 4 x 4 matrix by putting A in the top left with three 0's below it and making the last column be (b,1). Multip... | |linear-algebra|matrices|geometry| | 0 |
Books on Number Theory for Layman | Books on Number Theory for anyone who loves Mathematics? (Beginner to Advanced & just for someone who has a basic grasp of math) | Serre's "A course in Arithmetic" is pretty phenomenal. | |number-theory|reference-request|soft-question|book-recommendation| | 0 |
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language | The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently ... | Similar to the Monty Hall problem, but trickier: at the latest Gathering 4 Gardner, Gary Foshee asked I have two children. One is a boy born on a Tuesday. What is the probability I have two boys? We are assuming that births are equally distributed during the week, that every child is a boy or girl with probability 1/2,... | |soft-question|big-list|examples-counterexamples| | 0 |
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language | The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Apparently ... | You have two identical pieces of paper with the same picture printed on them. You put one flat on a table and the other one you crumple up (without tearing it) and place it on top of the first one. Brouwer's fixed point theorem states that there is some point in the picture on the crumpled-up page that is directly abov... | |soft-question|big-list|examples-counterexamples| | 0 |
List of Interesting Math Blogs | I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills. I'll start with my entries: Division By Zero Tanya Khovanova’s Math Blog | Math-blog is one I have in my google reader and I just found this one through google reader but it's a little over my head: The Unapologetic Mathematician | |soft-question|big-list|online-resources| | 0 |
Why is $x^0 = 1$ except when $x = 0$? | Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined. | If $a$ and $b$ are natural numbers, then $a^b$ is the number of ways you can make a sequence of length $b$ where each element in the sequence is chosen from a set of size $a$ . You're allowed replacements. For example $2^3$ is the number of 3 digit sequences where each digit is zero or $1$ : $000, 001, 010, \ldots, 111... | |definition|exponentiation| | 0 |
Simple numerical methods for calculating the digits of $\pi$ | Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that can be computed by hand in order to compute the first few digits? | Some of the easiest algorithms to use are the spigot algorithms. They allow you to jump rapidly to the nth digit without computing the n-1 digits before. The catch is that they only really work in binary. Here are some implementations in Python. You can see how short they are. | |approximation|numerical-methods|pi| | 0 |
Where can I find a review of discrete math | I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago. | This is the discrete math course one in my school. It contain lecture notes, homework and previous exams. http://www.cs.sunysb.edu/~cse547/ | |soft-question|reference-request|education|discrete-mathematics| | 1 |
Best Maths Books for Non-Mathematicians | I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which tea... | One's that were suggested to me by my Calculus teacher in High School. Even my wife liked them and she hates math now: The Education of T.C. Mits: What modern mathematics means to you Infinity: Beyond the Beyond the Beyond Written and illustrated(Pictures are great ;p) by a couple: Lillian R. Lieber, and Hugh Gray Lieb... | |reference-request|soft-question|big-list|book-recommendation| | 0 |
Best Maths Books for Non-Mathematicians | I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which tea... | Possibly this may not really qualify as presenting much interesting maths, but I think Hardy's A Mathematician's Apology should be on the must-read list. | |reference-request|soft-question|big-list|book-recommendation| | 0 |
What is the single most influential book every mathematician should read? | If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be? | A Mathematician's Apology by G H Hardy. I did in fact read this in high school, and it raised my view of mathematics from a thing of utility to a thing of beauty and wonder. It inspired me to go on to study mathematics at Cambridge myself. It's a pity that the "introduction" by C P Snow is longer than the original and ... | |soft-question|big-list|reference-request| | 0 |
History of the Concept of a Ring | I am vaguely familiar with the broad strokes of the development of group theory, first when ideas of geometric symmetries were studied in concrete settings without the abstract notion of a group available, and later as it was formalized by Cayley, Lagrange, etc (and later, infinite groups being well-developed). In any ... | Edit: Bill Dubuque has pointed out that much of this answer (specifically, the part about FLT) is essentially a mathematical urban legend, albeit a pervasive one. I cannot delete an accepted answer, so here is a link to an answer of his on MO explaining it. Here is also a link to a related question . There's some of th... | |math-history| | 0 |
Best Maths Books for Non-Mathematicians | I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked". So the question: What are good books, for laymen, which tea... | Paul Nahin has a number of accessible mathematics books written for non-mathematicians, the most famous being An Imaginary Tale: The Story of $\sqrt{-1}$ Dr. Euler's Fabulous Formula (Cures Many Mathematical Ills!) Professor Ian Stewart also has many books which each give laymen overviews of various fields or surprisin... | |reference-request|soft-question|big-list|book-recommendation| | 0 |
Where can I find a review of discrete math | I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago. | When wanting to know about a particular mathematics subject, I often find that starting with the " further reading " section of the relevant wikipedia page is a good way in. | |soft-question|reference-request|education|discrete-mathematics| | 0 |
Tiling a $3 \times 2n$ rectangle with dominoes | I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences f(0) = g(0) = 1 f(n) = f(n-1) + 2g(n-1) g(n) = f(n) + g(n-1) where $f(n)$ is the actual answer and $g(n)$ is a helper function that re... | Here's my best shot at the sort of explanation you're asking for, although it's not nearly as clear as the $2 \times n$ case. The negative sign makes combinatorial proofs difficult, so let's rearrange this as: $$f(n) + f(n-2) = 4f(n-1)$$ Then you want to show that the number of $n$-tilings, plus the number of $(n-2)$-t... | |combinatorics|discrete-mathematics|recurrence-relations|tiling| | 0 |
Is there a closed-form equation for $n!$? If not, why not? | I know that the Fibonacci sequence can be described via the Binet's formula. However, I was wondering if there was a similar formula for $n!$. Is this possible? If not, why not? | The relative error of Stirling's approximation gets arbitrarily small as n gets larger. $$n!\sim\sqrt{2\pi n} \left(\frac{n}{e}\right)^n$$ However, it is only an approximation, not a closed-form of $n!$ | |combinatorics|algorithms|factorial|closed-form| | 0 |
Simple numerical methods for calculating the digits of $\pi$ | Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that can be computed by hand in order to compute the first few digits? | There is also the option of approximating $\pi$ using Monte Carlo integration . The idea is this: If we agree that the area of a circle is $\pi r^2$, for simplicity we build a circle of area $\pi$ by setting $r=1$. Placing this circle wholly inside of another region of known area, preferably by inscribing it in a squar... | |approximation|numerical-methods|pi| | 0 |
Good books and lecture notes about category theory. | What are the best books and lecture notes on category theory? | Categories for the Working mathematician by Mac Lane Categories and Sheaves by Kashiwara and Schapira | |reference-request|soft-question|category-theory|big-list|book-recommendation| | 0 |
Is there a closed-form equation for $n!$? If not, why not? | I know that the Fibonacci sequence can be described via the Binet's formula. However, I was wondering if there was a similar formula for $n!$. Is this possible? If not, why not? | If you're willing to accept an integral as an answer, then $n! = \int_0^\infty t^n e^{-t} \: dt$. | |combinatorics|algorithms|factorial|closed-form| | 1 |
Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms? | Context: Rings. Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms? Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck... | I will assume this is in the context of rings (e.g., real numbers, integers, etc). In this case, the axiom defining $0$ is that $x + 0 = x$ for all $x$. $x*0 = 0$ is a result of this since we have $x*0 = x*(0+0) = x*0 + x*0$ which implies $x*0 = 0$ (canceling one of the $x*0$'s). I am guessing that for the second one y... | |abstract-algebra|definition|axioms| | 1 |
Good books and lecture notes about category theory. | What are the best books and lecture notes on category theory? | Lang's Algebra contains a lot of introductory material on categories, which is really nice since it's done with constant motivation from algebra (e.g. coproducts are introduced right before the free product of groups is discussed). | |reference-request|soft-question|category-theory|big-list|book-recommendation| | 1 |
Where can I find a review of discrete math | I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago. | When it comes to textbooks, the Kenneth Rosen text Discrete Mathematics and its Applications is highly recommended. I was first introduced to it at my university, but I've seen it cited in several places. | |soft-question|reference-request|education|discrete-mathematics| | 0 |
Good books and lecture notes about category theory. | What are the best books and lecture notes on category theory? | Arbib, Arrows, Structures, and Functors: The Categorical Imperative More elementary than MacLane. I don't know very much about this, but some stripes of computer scientist have taken an interest in category theory recently, and there are lecture notes floating around with that orientation. | |reference-request|soft-question|category-theory|big-list|book-recommendation| | 0 |
Applications of the Fibonacci sequence | The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence? | They are sometimes used or occur in financial applications, e.g. Elliot Wave Theory . This seems more like magic than math to me, but I am not a trader. | |combinatorics|big-list|applications|fibonacci-numbers| | 0 |
Applications of the Fibonacci sequence | The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence? | It isn't exactly an application as such, but the upper bound of the size of a subtree in a Fibonacci heap whose root is a node with degree $k$ is $F_{k+2}$ where $F_n$ is the $n^{th}$ Fibonacci number. | |combinatorics|big-list|applications|fibonacci-numbers| | 0 |
Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms? | Context: Rings. Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms? Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck... | Answering what you asked, the question is poorly formed because you didn't specify the theory you are talking about. To do that, well, you must define symbols, deduction rules... and axioms. Answering what you probably meant to ask, no. Those are properties that $-$ and $\times$ have in the $\mathbb Z$ set, not axioms.... | |abstract-algebra|definition|axioms| | 0 |
Applications of the Fibonacci sequence | The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence? | They're useful in determining the amortized run time of the appropriately named Fibonacci Heap in computer science. | |combinatorics|big-list|applications|fibonacci-numbers| | 0 |
Sum of reciprocals of numbers with certain terms omitted | I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots$ diverges too, even if really slowly since it's $O(\log \log n)$. But I think I read that if we consider the numbers ... | EDIT: This might be what you're looking for. Found it from looking at the source below. They're called Kempner series. An article here (and cited below) says that one Dr. Kempner proved in 1914 that the series 1+ 1/2 + 1/3 + ..., with any term that has a 9 in the denominator removed, is convergent (though he doesn't sa... | |sequences-and-series|convergence-divergence| | 1 |
Intuitive understanding of the derivatives of $\sin x$ and $\cos x$ | One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) chain of derivatives: sin(x) cos(x) -sin(x) -cos(x) sin(x) ... An analysis of the shape of their graphs confirms some point... | This isn't exactly what you asked, but look at the Taylor series for the polynomials: $$ \sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\text{ for all } x\!$$ $$\cos x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \c... | |calculus|trigonometry| | 0 |
Sum of reciprocals of numbers with certain terms omitted | I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots$ diverges too, even if really slowly since it's $O(\log \log n)$. But I think I read that if we consider the numbers ... | It is not very surprising that the sum is finite, since numbers without a 7 (or any other digit) get rarer and rarer as the number of digits increases. Here's a proof. Let $S$ be the harmonic series with all terms whose denominator contains the digit $k$ removed. We can write $S =S_1 + S_2 + S_3 + \ldots$, where $S_i$ ... | |sequences-and-series|convergence-divergence| | 0 |
Intuitive understanding of the derivatives of $\sin x$ and $\cos x$ | One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) chain of derivatives: sin(x) cos(x) -sin(x) -cos(x) sin(x) ... An analysis of the shape of their graphs confirms some point... | I don't think you can get an intuitive feel for the derivatives without looking at the plots personally. When you consider that a derivative is a rate of change, you need to be looking at a function that is varying, which implies you are looking at the plot/graph of the function. When you further consider that a deriva... | |calculus|trigonometry| | 0 |
What is larger -- the set of all positive even numbers, or the set of all positive integers? | We will call the set of all positive even numbers E and the set of all positive integers N . At first glance, it seems obvious that E is smaller than N , because for E is basically N with half of its terms taken out. The size of E is the size of N divided by two. You could see this as, for every item in E , two items i... | They are both the same size, the size being 'countable infinity' or 'aleph-null'. The reasoning behind it is exactly that which you have already identified - you can assign each item in E to a single value in N. This is true for the Natural numbers, the Integers, the Rationals but not the Reals (see the Diagonal Slash ... | |elementary-set-theory|infinity| | 0 |
What is larger -- the set of all positive even numbers, or the set of all positive integers? | We will call the set of all positive even numbers E and the set of all positive integers N . At first glance, it seems obvious that E is smaller than N , because for E is basically N with half of its terms taken out. The size of E is the size of N divided by two. You could see this as, for every item in E , two items i... | Each item in N could be mapped with one item in E (the item x*2). Yes. Both sets have cardinality aleph-0. | |elementary-set-theory|infinity| | 0 |
Intuitive understanding of the derivatives of $\sin x$ and $\cos x$ | One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) chain of derivatives: sin(x) cos(x) -sin(x) -cos(x) sin(x) ... An analysis of the shape of their graphs confirms some point... | One of the main ways that sine and cosine come up is as the fundamental solutions to the differential equation $y'' = -y$, known as the wave equation. Why is this an important differential equation? Well, interpreting it using Newton's second law it says "the force is proportional and opposite to the position." For exa... | |calculus|trigonometry| | 0 |
What is larger -- the set of all positive even numbers, or the set of all positive integers? | We will call the set of all positive even numbers E and the set of all positive integers N . At first glance, it seems obvious that E is smaller than N , because for E is basically N with half of its terms taken out. The size of E is the size of N divided by two. You could see this as, for every item in E , two items i... | Mathematics is the art of clever forgetting. The first mathematical breakthrough, numbers, came about when people realized that if you just forgot about whether it was 5 cows + 3 cows, or 5 rocks + 3 rocks or whatever you always got 8. Numbers are what you get when you look at collections of objects and forget what kin... | |elementary-set-theory|infinity| | 1 |
Good books and lecture notes about category theory. | What are the best books and lecture notes on category theory? | Another book that is more elementary, not requiring any algebraic topology for motivation, and formulating the basics through a question and answer approach is: Conceptual Mathematics An added benefit is that it is written by an expert! | |reference-request|soft-question|category-theory|big-list|book-recommendation| | 0 |
Why do complex functions have a finite radius of convergence? | Say we have a function $\displaystyle f(z)=\sum_{n=0}^\infty a_n z^n$ with radius of convergence $R>0$. Why is the radius of convergence only $R$? Can we conclude that there must be a pole, branch cut or discontinuity for some $z_0$ with $|z_0|=R$? What does that mean for functions like $$f(z)=\begin{cases} 0 & \text{f... | If the radius of convergence is $R$, that means there is a singular point on the circle $|z| = R$. In other words, there is a point $\xi$ on the circle of radius $R$ such that the function cannot be extended via "analytic continuation" in a neighborhood of $\xi$. This is a straightforward application of compactness of ... | |sequences-and-series|complex-analysis| | 1 |
What property of certain regular polygons allows them to be faces of the Platonic Solids? | It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes. What property of those regular polygons themselves allow them to faces of regular convex polyhedron? Is it something in their angles? Their number of si... | The regular polygons that form the Platonic solids are those for which the measure of the interior angles, say α for convenience, is such that $3\alpha Regular (equilateral) triangles have interior angles of measure $\frac{\pi}{3}$ (60°), so they can be assembled 3, 4, or 5 at a vertex ($3\cdot\frac{\pi}{3} Regular qua... | |geometry|polyhedra|platonic-solids| | 1 |
Construct a bijection from $\mathbb{R}$ to $\mathbb{R}\setminus S$, where $S$ is countable | Two questions: Find a bijective function from $(0,1)$ to $[0,1]$. I haven't found the solution to this since I saw it a few days ago. It strikes me as odd--mapping a open set into a closed set. $S$ is countable. It's trivial to find a bijective function $f:\mathbb{N}\to\mathbb{N}\setminus S$ when $|\mathbb{N}| = |\math... | The proof of the Schroeder-Bernstein theorem allows you to get a bijection for 1 , since we have an injection $(0,1) \to [0,1]$ and a bijection $f: [0,1] \to [1/4, 3/4] \subset (0,1)$ (say $x \to x/2 +1/4$ ). The function's definition will be somewhat messy (basically, it depends on how many times you can lift a point ... | |elementary-set-theory| | 1 |
What can we conclude from correlation? | I just got my statistics test back and I am totally confused about one of the questions! A study was done that took a simple random sample of 40 people and measured whether the subjects were right-handed or left-handed, as well as their ages. The study showed that the proportion of left-handed people and the ages had a... | This is wrong: "We can conclude that many people become right-handed as they grow older." We cannot conclude this at all from the given data. For one, the study only takes a sample at one point in time, rather than selecting a sample and monitoring their progress through many decades. This is what would be needed for u... | |statistics| | 1 |
What can we conclude from correlation? | I just got my statistics test back and I am totally confused about one of the questions! A study was done that took a simple random sample of 40 people and measured whether the subjects were right-handed or left-handed, as well as their ages. The study showed that the proportion of left-handed people and the ages had a... | We can only conclude that the result is interesting and it deserves more research. Without further study we can not say if right-handedness causes age (ie. being left handed causes biological alterations that shorten the life span), age causes right-handedness (ie. as people age they become right handed), they are corr... | |statistics| | 0 |
Intuitive reasoning behind the Chain Rule in multiple variables? | I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour. But I'm having a bit more trouble understanding the Chain Rule as applied to... | Changing time is the same as changing x and changing y. If the changes each one caused in z didn't interact, then the total change would be the sum of both changes. If the function is well behaved (differentiatable) then the interaction caused by an infinitesimal change in x with an infinitesimal change in y will be do... | |calculus|linear-algebra| | 0 |
Intuitive reasoning behind the Chain Rule in multiple variables? | I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour. But I'm having a bit more trouble understanding the Chain Rule as applied to... | The correct context of the chain rule is that taking the tangent bundle is functorial. A more down-to-earth answer is provided by working coordinate free using linear algebra. Suppose $f:X\to Y$ and $g:Y\to Z$ are functions between Banach spaces (these are a generalized version of R^n) such that $f$ is differentiable a... | |calculus|linear-algebra| | 0 |
Are the "proofs by contradiction" weaker than other proofs? | I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? Is there any reason that one ... | Proof by contradiction is just as logically valid as any other type of proof. If you are unsure, I think it might help to consider exactly what a proof by contradiction entails. Say we have a set of statements $\Gamma$, and that $\Gamma\cup\{(\neg\phi)\}$ is not consistent. That is, the statement $\neg\phi$ contradicts... | |logic|proof-theory| | 0 |
Intuitive reasoning behind the Chain Rule in multiple variables? | I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour. But I'm having a bit more trouble understanding the Chain Rule as applied to... | Think of it in terms of causality & superposition. $$z = f(x,y)$$ If you keep $y$ fixed then $\frac{dz}{dt} = \frac{df}{dx} * \frac{dx}{dt}$ If you keep $x$ fixed then $\frac{dz}{dt} = \frac{df}{fy} * \frac{dy}{dt}$. Superposition says you can just add the two together. | |calculus|linear-algebra| | 0 |
Intuitive reasoning behind the Chain Rule in multiple variables? | I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour. But I'm having a bit more trouble understanding the Chain Rule as applied to... | The basic reason is that one is simply composing the derivatives just as one composes the functions. Derivatives are linear approximations to functions. When you compose the functions, you compose the linear approximations---not a surprise. I'm going to try to expand on Harry Gindi's answer, because that was the only w... | |calculus|linear-algebra| | 1 |
Why does the discriminant of a cubic polynomial being less than $0$ indicate complex roots? | The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes , but also that there are three distinct, real roots if $\Delta > 0$, and that there is one real root and two complex roots (complex... | The discriminant of any monic polynomial is the product $\prod_{i \neq j} (x_i - x_j)^2$ of the squares of the differences of the roots (in an algebraic closure, e.g. $C$). Cf. the Wikipedia article on this. Consequently, if the roots are all real and distinct, this must be positive. (If the polynomial is not monic, th... | |algebra-precalculus|polynomials|roots|symmetric-polynomials| | 0 |
Subsets and Splits
SQL Console for glopezas/math_stackexchange_qa
Filters and displays accepted answers excluding questions with specific tags, providing a focused view of less common or specialized topics.