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2 answers

Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be easy to read. Thanks
Finance
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74
votes
6 answers

Order of finite fields is $p^n$

Let $F$ be a finite field. How do I prove that the order of $F$ is always of order $p^n$ where $p$ is prime?
Mohan
  • 14,856
74
votes
4 answers

How to calculate $\,(a-b)\bmod n\,$ and $ {-}b \bmod n$

Consider the following expression: (a - b) mod N Which of the following is equivalent to the above expression? 1) ((a mod N) + (-b mod N)) mod N 2) ((a mod N) - (b mod N)) mod N Also, how is (-b mod N) calculated, i.e., how is the mod of a…
J.P.
  • 895
74
votes
13 answers

What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle.

Try to solve this puzzle: The first expedition to Mars found only the ruins of a civilization. From the artifacts and pictures, the explorers deduced that the creatures who produced this civilization were four-legged beings with a tentatcle…
Vishnu Vivek
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74
votes
4 answers

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z...: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer number of them. Is there a unified framework that includes all these transforms as…
kjo
  • 14,334
74
votes
4 answers

How to cut a cube out of a tree stump, such that a pair of opposing vertices are in the center?

I saw this picture of a cube cut out of a tree stump. I've been trying to craft the same thing out of a tree stump, but I found it hard to figure out how to do it. One of the opposing vertices pair is on the center of the tree stump: I've been…
Aaron Cheuk
  • 711
74
votes
2 answers

Are there any valid continuous Sudoku grids?

A standard Sudoku is a $9\times 9$ grid filled with digits such that every row, column, and $3\times 3$ box contains all the integers from $1$ to $9$. I am thinking about a generalization of Sudoku which I call "continuous Sudoku", which consists of…
ZKG
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74
votes
4 answers

What is the difference between a Ring and an Algebra?

In mathematics, I want to know what is indeed the difference between a ring and an algebra?
user70795
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74
votes
8 answers

How come $32.5 = 31.5$? (The "Missing Square" puzzle.)

Below is a visual proof (!) that $32.5 = 31.5$. How could that be? (As noted in a comment and answer, this is known as the "Missing Square" puzzle.)
Mehper C. Palavuzlar
  • 2,096
74
votes
17 answers

Mathematical subjects you wish you learned earlier

I am learning geometric algebra, and it is incredible how much it helps me understand other branches of mathematics. I wish I had been exposed to it earlier. Additionally I feel the same way about enumerative combinatorics. What are some less…
Jonathan Fischoff
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74
votes
22 answers

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
Ssegawa Victor
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74
votes
10 answers

Why can't the second fundamental theorem of calculus be proved in just two lines?

The second fundamental theorem of calculus states that if $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$ on the same interval, then: $$\int_a^b f(x) dx= F(b)-F(a).$$ The proof of this theorem in both my textbook and Wikipedia is…
Newton
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74
votes
3 answers

Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?

The question is written like this: Is it possible to find an infinite set of points in the plane, not all on the same straight line, such that the distance between EVERY pair of points is rational? This would be so easy if these points could be on…
Ahmed Amir
  • 1,013
74
votes
18 answers

Why do we need to learn integration techniques?

After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math. These classes are integral calc and multivariable calc. Integral calc started out okay, learning about Riemann sums and…
mr real lyfe
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74
votes
6 answers

How to generate random points on a sphere?

How do I generate $1000$ points $\left(x, y, z\right)$ and make sure they land on a sphere whose center is $\left(0, 0, 0\right)$ and its diameter is $20$ ?. Simply, how do I manipulate a point's coordinates so that the point lies on the sphere's…
Filip
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