Top new questions this week:

Why is 987654321/123456789 = 8.0000000729?

Many years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill since …

(arithmetic) (rational-numbers)

asked by marty cohen 46 votes answered by WimC 59 votes

What is $-i$ exactly?

We all know that $i$ doesn't have any sign: it is neither positive nor negative. Then how can people use $-i$ for anything? Also, we define $i$ a number such that $i^2 = -1$. But it can also be seen …

(soft-question) (complex-numbers)

asked by Πάρτη Κοηλί 29 votes answered by user1620696 27 votes

Why is $\varphi$ called "the most irrational number"?

I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio …

(number-theory) (irrational-numbers) (golden-ratio)

asked by PyRulez 26 votes answered by Hagen von Eitzen 23 votes

What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, …

(logic) (proof-writing) (proof-theory) (big-picture)

asked by user78743 24 votes answered by Ittay Weiss 19 votes

Yitang Zhang: Prime Gaps

Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific. EDIT$^1$: Are there any experts here who can …

(number-theory) (elementary-number-theory) (prime-numbers) (math-history) (prime-twins)

asked by Trancot 21 votes answered by lhf 13 votes

explaining the derivative of $x^x$

You set the following exercise to your calculus class: Q1. Differentiate $y(x) = x^x$. A student submits the following solution: Let $g(a)=a^x$ and $f(x)=x$. Then $y(x) = g(f(x))$, so by …

(calculus) (teaching) (fake-proofs)

asked by mt_ 20 votes answered by Tim 24 votes

How do you respond to "I was always bad at math"?

Here in the U.S., it is my experience that over 75% of adults I meet socially will volunteer that phrase or a variation upon learning that I am a mathematician. I find this frustrating, since almost …

(soft-question)

asked by vadim123 19 votes answered by Steve 11 votes

Greatest hits from previous weeks:

Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?

(geometry) (big-list) (math-software)

asked by Lucky 44 votes answered by Jorge Miranda 31 votes

Results that came out of nowhere.

Most big results in mathematics are built on years and years of groundwork by the author and other mathematicians, such as Wiles' proof of FLT or the classification of finite simple groups. …

(soft-question) (math-history) (big-list)

asked by Alexander Gruber 48 votes answered by Ed Pegg 39 votes

Can you answer these?

Ulm and Frattini Subgroups

Let $A$ be an abelian group. We define $U(A)=\cap (nA), n\in \mathbb N$ be the Ulm subgroup of $A$. The Frattini subgroup of $A$ is $\Phi(A)=\cap(pA)$ ($p\in \mathbb P$). I was trying to show that …

(abstract-algebra) (group-theory)

asked by W4cc0 7 votes

Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?

This question came up in a recent video series of lectures by Mike Freedman available through Max Planck Institut's website. He proves the "difficult" converse direction, that $X\times \mathbb R\cong …

(general-topology)

asked by Grumpy Parsnip 19 votes

A contest question

$p$ is an odd prime,denote $$f(x)=\sum_{k=0}^{p-1}\binom{2k}{k}^2x^k.$$ Prove that for every $x\in Z$,$$(-1)^\frac{p-1}2f(x)\equiv f(\frac{1}{16}-x)\pmod{p^2}.$$ This is a contest question,I do not …

(combinatorics) (number-theory) (elementary-number-theory) (contest-math)

asked by Hecke 9 votes