# Most Popular

1500 questions

**426**

votes

**22**answers

### On "familiarity" (or How to avoid "going down the Math Rabbit Hole"?)

Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole."
For example, suppose you come across the novel term vector space, and want to learn more about it. You look up various definitions, and…

kjo

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**394**

votes

**1**answer

### The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative Noetherian ring $R$. Player one mods out a nonzero…

Alex Becker

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**388**

votes

**32**answers

### Pedagogy: How to cure students of the "law of universal linearity"?

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”:
$$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$
$$ 2^{-3}…

Peter LeFanu Lumsdaine

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**369**

votes

**33**answers

### If $AB = I$ then $BA = I$

If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.
I do not understand anything more than the following.
Elementary row operations.
Linear dependence.
Row reduced forms and their…

Dilawar

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**369**

votes

**15**answers

### Can every proof by contradiction also be shown without contradiction?

Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantages/disadvantages of proving by contradiction?
As an aside, how is proving by…

sonicboom

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**367**

votes

**70**answers

### 'Obvious' theorems that are actually false

It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true".
Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem).
But…

beep-boop

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**367**

votes

**20**answers

### Find five positive integers whose reciprocals sum to $1$

Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Is your answer the only solution? If so, show why.
I was surprised that a teacher would assign this kind of…

Low Scores

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**364**

votes

**24**answers

### Zero to the zero power – is $0^0=1$?

Could someone provide me with a good explanation of why $0^0=1$?
My train of thought:
$x>0$
$0^x=0^{x-0}=0^x/0^0$, so
$0^0=0^x/0^x=\,?$
Possible answers:
$0^0\cdot0^x=1\cdot0^0$, so $0^0=1$
$0^0=0^x/0^x=0/0$, which is undefined
PS. I've read the…

Stas

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**347**

votes

**8**answers

### Calculating the length of the paper on a toilet paper roll

Fun with Math time.
My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with small error) the total paper length of a toilet…

Laplacian

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**340**

votes

**7**answers

### How can you prove that a function has no closed form integral?

In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations:
addition/subtraction
multiplication/division
raising to powers and…

Simon Nickerson

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**339**

votes

**106**answers

### Surprising identities / equations

What are some surprising equations/identities that you have seen, which you would not have expected?
This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc.
I'd request to avoid 'standard' / well-known…

Calvin Lin

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**339**

votes

**30**answers

### Is it true that $0.999999999\ldots=1$?

I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?

Michael Haren

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**338**

votes

**11**answers

### What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?

Ryan

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**334**

votes

**36**answers

### 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…

AgCl

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**329**

votes

**23**answers

### Why don't we define "imaginary" numbers for every "impossibility"?

Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such…

Lily Chung

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