165
votes
22answers
13k views

Zero to the zero power - Is $0^0=1$?

Could someone provide me with 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: $0^0 * 0^x = 1 * 0^x$, so ...
165
votes
13answers
21k views

Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
165
votes
5answers
7k views

Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?

My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question: I am thinking of a number... It ...
164
votes
2answers
14k views

Proving you *can't* make $2011$ out of $1,2,3,4$: nice twist on the usual

An undergraduate was telling me about a puzzle he'd found: the idea was to make $2011$ out of the numbers $1, 2, 3, 4, \ldots, n$ with the following rules/constraints: the numbers must stay in order, ...
162
votes
16answers
11k views

In the history of mathematics, has there ever been a mistake?

I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of ...
162
votes
37answers
15k views

Fun but serious mathematics books to gift advanced undergraduates.

I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but ...
161
votes
22answers
9k views

Why do mathematicians use single-letter variables?

I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated trying to follow mathematical notation. ...
160
votes
14answers
10k views

Identification of a quadrilateral as a trapezoid, rectangle, or square

Yesterday I was tutoring a student, and the following question arose (number 76): My student believed the answer to be J: square. I reasoned with her that the information given only allows us to ...
160
votes
4answers
12k views

The Integral that Stumped Feynman?

In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
151
votes
3answers
64k views

What is the intuitive relationship between SVD and PCA

Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional dataset into fewer dimensions while retaining important ...
146
votes
25answers
11k views

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

This morning, I wanted to flip a coin to make a decision but no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...
146
votes
7answers
9k views

Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ coming from ...
146
votes
2answers
4k views

Can we ascertain that there exists an epimorphism $G\rightarrow H$?

Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?
144
votes
91answers
40k views

Which one result in mathematics has surprised you the most? [closed]

A large part of my fascination in mathematics is because of some very surprising results that I have seen there. I remember one I found very hard to swallow when I first encountered it, was what is ...
144
votes
11answers
59k views

What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
144
votes
1answer
4k views

Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...
142
votes
18answers
15k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
142
votes
14answers
32k views

Is computer science a branch of mathematics?

I have been wondering, is computer science a branch of mathematics? No one has ever adequately described it to me. It all seems very math-like to me. My second question is, are there any books about ...
139
votes
28answers
25k views

List of Interesting Math Videos/ Documentaries

This is an offshoot of the question on Fun math outreach/social activities. I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list. Story of ...
137
votes
27answers
22k views

Best Fake Proofs? (A M.SE April Fools Day collection) [closed]

In honor of April Fools Day 2013, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen. I've posted one as an answer below. I'm also thinking of a ...
137
votes
15answers
9k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
137
votes
1answer
4k views

Rational roots of polynomials

Can one construct a sequence $(a_k)_{k\geqslant 0}$ of rational numbers such that, for every positive integer $n$ the polynomial $a_nX^n+a_{n-1}X^{n-1}+\cdots +a_0$ has exactly $n$ distinct rational ...
136
votes
0answers
4k views

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 ...
130
votes
6answers
24k views

Deleting any digit yields a prime… is there a name for this?

My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood. His mother won't indulge his requests, but I often will. Here is the day he wanted 100: But ...
129
votes
3answers
4k views

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 ...
129
votes
1answer
7k views

A variation of Fermat's little theorem

Fermat's little theorem states that for $n$ prime, $$ a^n \equiv a \pmod{n}. $$ The values of $n$ for which this holds are the primes and the Carmichael numbers. If we modify the congruence ...
128
votes
31answers
12k views

Stopping the “Will I need this for the test” question [closed]

I am a college professor in the American education system and find that the major concern of my students is trying to determine the specific techniques or problems which I will ask on the exam. This ...
127
votes
2answers
8k views

Evaluate $ \int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\:\mathrm dx$

Evaluate the following integral $$ \tag1\int_{0}^{\pi/2}\frac1{(1+x^2)(1+\tan x)}\,dx $$ My Attempt: Letting $x=\frac{\pi}{2}-x$ and using the property that $$ \int_{0}^{a}f(x)\,dx = ...
126
votes
12answers
19k views

How can a piece of A4 paper be folded in exactly three equal parts?

This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on ...
126
votes
14answers
14k views

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
125
votes
6answers
11k views

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

I've come across statements in the past 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 ...
125
votes
3answers
19k views

Why can a Venn diagram for 4+ sets not be constructed using circles?

This page gives a few examples of Venn diagrams for 4 sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete ...
123
votes
3answers
5k views

How to show $e^{e^{e^{79}}}$ is not an integer

In this question, I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this? Much later addendum: ...
122
votes
14answers
19k views

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 ...
120
votes
20answers
8k views

Are there any open mathematical puzzles?

Are there any (mathematical) puzzles that are still unresolved? I only mean questions that are accessible to and understandable by the complete layman and which have not been solved, despite serious ...
120
votes
22answers
18k views

Examples of mathematical discoveries which were kept as a secret

There could be several personal, social, philosophical and even political reasons to keep a mathematical discovery as a secret. For example it is completely expected that if some mathematician find ...
117
votes
20answers
9k views

Mental Calculations

This is the famous picture "Mental Arithmetic. In the Public School of S. Rachinsky." by the Russian artist Nikolay Bogdanov-Belsky. The problem presented on a blackboard requires computing the ...
116
votes
18answers
14k views

How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?

So I'm tutoring at the library and an elementary or pre K student shows me a sheet with one problem on it: Put 9 pigs into 4 pens so that there are an odd number of pigs in each pen. I tried to ...
115
votes
9answers
5k views

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
114
votes
29answers
40k views

Best book ever on Number Theory

Which is the single best book for Number Theory that everyone who loves Mathematics should read?
114
votes
33answers
13k views

Can you provide me historical examples of pure mathematics becoming “useful”?

I'm trying to think/know about something but I don't know if my basis premise is plausible, here we go. Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because ...
114
votes
6answers
8k views

In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
114
votes
3answers
9k views

Is 2048 the highest power of 2 with all even digits (base ten)?

I have a friend who turned 32 recently. She has an obsessive compulsive disdain for odd numbers, so I pointed out that being 32 was pretty good since not only is it even, it also has no odd factors. ...
114
votes
5answers
6k views

Symmetry of function defined by integral

Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as $$ f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$ One can use, for ...
113
votes
7answers
165k views

How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first ...
111
votes
11answers
12k views

Do we know if there exist true mathematical statements that can not be proven?

Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven ...
110
votes
17answers
23k views

Is $.\overline{9} = 1$?

I'm told by smart people that $.\overline{9} = 1$ and I believe them, but is there a proof that explains why this is?
110
votes
3answers
6k views

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
110
votes
1answer
6k views

How discontinuous can a derivative be?

There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many ...
110
votes
5answers
8k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...

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