Yly
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26 answers
239 votes
46k views
What are some examples of when Mathematics 'accidentally' discovered something about the world?
87 votes

Arago's spot is a classical (and classic) example of a beautiful mathematical theory anticipating a beautiful physical fact. Briefly, the story goes like this: Back in the 1800's, scientists were ...

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19 answers
167 votes
20k views
Mathematical ideas that took long to define rigorously
72 votes

Natural transformations are a "natural" example of this. Mathematicians knew for a long time that certain maps--e.g. the canonical isomorphism between a finite-dimensional vector space and ...

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16 answers
153 votes
21k views
What's new in higher dimensions?
51 votes

Exotic spheres are a feature only of dimensions higher than 3. These are topological spaces which are homeomorphic to a sphere, but with different differential structure. Informally, you can ...

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7 answers
49 votes
8k views
Is there a known non-euclidean geometry where two concentric circles of different radii can intersect? (as in the novel "The Universe Between")
48 votes

Yes, with the appropriate definition of "circle". Namely, define a circle of radius $R$ centered at $x$ on manifold $M$ to be the set of points which can be reached by a geodesic of length $R$ ...

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4 answers
49 votes
9k views
What exactly is a matrix?
45 votes

1. Definition of a matrix. The question of what a matrix is, precisely, is one I had for a long time as a high school student. It took many tries to get a straight answer, because people tend to ...

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5 answers
68 votes
12k views
Why do engineers use derivatives in discontinuous functions? Is it correct?
39 votes

The main question here seems to be "why can we differentiate a function only defined on integers?". The proper answer, as divined by the OP, is that we can't--there is no unique way to ...

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8 answers
21 votes
4k views
How "messy" can a multivariable function be?
32 votes

The granddaddy of ill-behaved functions is $e^{1/x}$. Even in one dimension it is very strange (having what is called an "essential singularity" at the origin), but in 2D or in the complex ...

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18 answers
60 votes
9k views
Unsolved Problems due to Lack of Computational Power
26 votes

Historically, a very important, computationally intensive problem arising from physics was lattice QCD (LQCD). LQCD is a theoretical framework for computing basic quantities like the mass of the ...

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3 answers
18 votes
2k views
Sum of $4$ dice rolls greater than the product
Accepted answer
24 votes

Case work: If 3 or 4 of the rolls come up 1, it's straightforward to see that the desired inequality holds. If 0 or 1 of the rolls come up 1, then we can show that the inequality never holds. (The ...

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26 answers
239 votes
46k views
What are some examples of when Mathematics 'accidentally' discovered something about the world?
22 votes

Bell's theorem on the foundations of quantum mechanics showed that not all philosophical questions are impervious to experiment, to the extreme surprise of pretty much every physicist on Earth. (It ...

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3 answers
21 votes
640 views
A conference uses $4$ main languages. Prove that there is a language that at least $\dfrac{3}{5}$ of the delegates know.
19 votes

If somebody speaks only one language, then everybody speaks that language, and we're done. If somebody speaks all four languages, then the desired statement follows by induction after removing that ...

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18 answers
95 votes
15k views
What seemingly innocuous results in mathematics require advanced proofs?
17 votes

Fermat's Last Theorem states that there are no positive integer solutions $a,b,c$ to $a^n+b^n=c^n$ for $n>2$. This one irked mathematicians for 350 years until the theory of elliptic curves was ...

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5 answers
19 votes
5k views
Can a chemical engineer become a self-taught mathematician?
16 votes

It depends on what you mean by "self-taught" and "mathematician". What is definitely true is that you probably have the chops to learn any particular branch of math that you are interested in. And ...

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3 answers
6 votes
981 views
Differential equation in optics
Accepted answer
16 votes

Rewrite it as $$\frac{2y'y''}{(y')^2+1} = \frac{2ky'}{y}$$ Integrate to get $$\ln \left((y')^2+1\right) = 2k \ln y + c$$ So $$(y')^2+1 = C_1y^{2k}$$ Rearrange to get $$1=\frac{y'}{\sqrt{C_1y^{2k}-1}}$$...

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21 answers
283 votes
44k views
Really advanced techniques of integration (definite or indefinite)
15 votes

According to Wikipedia, the "tangent half-angle substitution" (a.k.a. the Weierstrauss substitution) is the "world's sneakiest substitution". It consists of subbing $x=\arctan(2t)$, which allows you ...

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1 answers
11 votes
405 views
Median of the set of numbers which consists of all positive integers whose digits strictly increase from left to right
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14 votes

There are 9 of these numbers having 1 digit, $\binom{9}{2}$ having 2 digits, and in general $\binom{9}{k}$ having $k$ digits. The total number of elements in this set is thus $\binom{9}{1} + \binom{9}...

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1 answers
4 votes
2k views
(Graph Theory) Prove that $H_n$ has a Hamiltonian cycle for $n$ ≥ 2.
14 votes

Here's one solution: Your graph $H_n$ is an n-dimensional hypercube. For the induction step, separate the cube into two "faces" by cutting along one dimension. Do parallel Hamiltonian cycles on each ...

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14 answers
74 votes
5k views
"Naturally occurring" non-Hausdorff spaces?
13 votes

In non-standard analysis, to any set $A$ there is an associated set $^*\!A$, which consists of the original set $A$ plus a bunch of new points infinitesimally close to $A$. There are two natural ...

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5 answers
8 votes
1k views
A problem regarding the hands of a clock
12 votes

The easy way: You don't actually need to find the two times when the hour and minute hand are at right angles. Just note that the difference between these two times is how long it takes for the ...

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12 answers
58 votes
7k views
Are there mathematical concepts that exist in dimension $4$, but not in dimension $3$?
12 votes

At the risk of sounding flippant, I'm going to propose the universe as one such example. It is a peculiar, but experimentally well verified fact, that the universe has four dimensions (three spatial ...

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4 answers
18 votes
1k views
Need a *trivial* proof of an "obvious" combinatorial result
9 votes

You requested a proof in three lines, and conveniently the proof actually consists of three computations :) I'll break the proof up along these computations. However, in the interest of clarity, I ...

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4 answers
12 votes
2k views
Prove that $\det(AB - BA) = \frac{1}{3}\left(\mathrm{Trace}(AB - BA)^3\right)$
8 votes

Choose a basis which puts $AB-BA$ in Jordan normal form (both $\det$ and $\text{Tr}$ are invariant under basis change, so this is allowed). Then since a commutator is traceless, the diagonal must ...

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1 answers
7 votes
156 views
Combinatorics and pigeon hole principle
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7 votes

The question seems to be requiring that the subsets be distinct. In this case, certain sums can only be attained in one way, and it is thus impossible to put two pigeons in these pigeonholes, ...

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1 answers
9 votes
250 views
Intuition behind spectrum of an operator
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7 votes

This is not a complete answer, but since you are looking for intuition, it may be helpful. Some background: Historically, the spectrum originated in quantum mechanics, and in particular in the energy ...

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7 answers
24 votes
2k views
What problems have been frequently computationally verified for large values?
7 votes

The Hadamard Conjecture states that a Hadamard matrix of order $4k$ should exist for every positive integer $k$. It has been numerically verified for all orders up to 668. The Circulant Hadamard ...

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4 answers
12 votes
811 views
Evaluate $\int_0^\pi \frac{\sin\frac{21x}{2}}{\sin \frac x2} dx$ (from MIT Integration Bee)
6 votes

Note that if we call $a=e^{ix/2}$, then we have $$\frac{\sin(21x/2)}{\sin(x/2)} = \frac{a^{21}-a^{-21}}{a-a^{-1}} = a^{-20}\frac{a^{42}-1}{a^2-1}=a^{-20}\frac{(a^2-1)(a^{40}+a^{38}+a^{36}+\cdots+1)}{a^...

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4 answers
15 votes
6k views
Dividing a rectangle into 4 parts in the ratio 1:2:3:4, with only 2 lines
6 votes

There is a one-parameter infinite family of solutions. Below is an algorithm to find them all. As a spoiler, I'll point out up front that this algorithm works equally well for any convex subset of $\...

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1 answers
6 votes
436 views
Finding subspaces with trivial intersection
Accepted answer
6 votes

The union of the two spaces $P\cup P'$ is not all of $V$, so there is a vector $v_1\notin P\cup P'$. Now proceed inductively on the spaces $P\oplus \text{span}\{v_1\}$ and $P'\oplus \text{span}\{v_1\}...

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6 answers
46 votes
6k views
Does advanced math "power" more rudimentary math?
6 votes

What you are asking, in some sense, is "Is abstraction useful?" Abstraction is the art of embedding easy-to-understand ideas inside of harder-to-understand ideas. As to the usefulness of this ...

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1 answers
6 votes
216 views
Find the coefficient of $x^{17}$ in $(x^2+x^3+x^4+x^5+x^6+x^7+...)^3.$
Accepted answer
6 votes

Your answer is right, your book is wrong. Here are two alternative ways to prove it: The coefficient of $x^{11}$ is the number of ways to put 11 $x$'s in 3 baskets. This is the "stars and bars" ...

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