Mathematics

Most Popular

more tags
1500 questions
199
votes
14 answers

Intuition behind Matrix Multiplication

If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times. But If I multiply two matrices, what does it mean ? I mean I can't think it in terms of repetitive addition. What is the…
Happy Mittal
  • 3,347
198
votes
14 answers

Convergence of the series $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ for $p > 1$

I am trying to prove the convergence of the $p$ -series $$\sum_{n=1}^{\infty} \frac1{n^p}.$$ for $p > 1$ . I am wondering if there is a proof that this series converges, either directly or by applying some test for convergence.
admchrch
  • 2,942
197
votes
22 answers

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…
Giuseppe Accaputo
  • 101
197
votes
5 answers

A Topology such that the continuous functions are exactly the polynomials

I was wondering which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous if and only if it is a polynomial function $f(x)=a_nx^n+\cdots+a_0$. Obviously, the finite fields with the discrete topology have this…
Dominik
  • 14,740
  • 5
  • 49
  • 98
196
votes
16 answers

What's the intuition behind Pythagoras' theorem?

Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it. $A^{2} + B^{2} = C^{2}$ $C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$ $C^{2} = 25 \text{ cm}^2 + 49 \text{ cm}^2$ $C^{2} = 74 \text{ cm}^2$ ${x} = +\sqrt{74}…
user123399
  • 289
195
votes
9 answers

How to define a bijection between $(0,1)$ and $(0,1]$?

How to define a bijection between $(0,1)$ and $(0,1]$? Or any other open and closed intervals? If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it): I make a function…
user1411893
  • 2,173
195
votes
7 answers

What is the difference between "singular value" and "eigenvalue"?

I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is. Is "singular value" just another name for eigenvalue?
Ramon
  • 1,959
194
votes
25 answers

Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but only had an SD card: Given that I don't know the bias of this SD card, would flipping it be considered a "fair toss"? I thought if I'm just as likely to assign an outcome to one side as to…
Andrew Cheong
  • 2,525
194
votes
31 answers

Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

I recently proved that $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$ using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any…
Fernando Martin
  • 6,097
194
votes
27 answers

Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof on Wikipedia and I understand the proof, but I don't "get it". Can someone help me out with this ? I find it hard to wrap my head around…
hari_sree
  • 2,201
194
votes
2 answers

What are examples of functions with "very" discontinuous derivative?

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval.
user58273
  • 2,057
194
votes
11 answers

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…
bryn
  • 10,135
193
votes
7 answers

Is the product of two Gaussian random variables also a Gaussian?

Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$. Is $XY$ also normally distributed? Is the answer any different if we know that $X$ and $Y$ are independent?
jamaicanworm
  • 4,714
191
votes
29 answers

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan. While it's a good book,…
CritChamp
  • 229
189
votes
8 answers

Intuition of the meaning of homology groups

I am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible I would prefer it if this could be kept…
Spyam
  • 4,575
1 2 3
…
99 100