Most Popular
1500 questions
93
votes
9 answers
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$. I already know the logical Proof: $${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$ Hence summation can…
Lance C
- 931
93
votes
9 answers
Why is the construction of the real numbers important?
There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers as a complete ordered field? What's the importance…
user42912
- 24,450
93
votes
3 answers
Do you prove all theorems whilst studying?
When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my studies -- sequence two of Calculus -- but what I'm…
user24383
93
votes
8 answers
Will it become impossible to learn math?
I was thinking about this today and it seems like a good question. Assuming mathematics will keep on expanding, do you think it will ever become impossible for a beginner to learn all the known material on a subject (such as mechanics), simply…
Ninja Boy
- 3,223
92
votes
3 answers
What is the importance of the infinitesimal generator of Brownian motion?
I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have found lack any motivation or intuition. What is…
Potato
- 41,689
92
votes
4 answers
difference between dot product and inner product
I was wondering if a dot product is technically a term used when discussing the product of $2$ vectors is equal to $0$. And would anyone agree that an inner product is a term used when discussing the integral of the product of $2$ functions is equal…
CuriousGeorge119
- 1,287
- 1
- 11
- 13
92
votes
5 answers
Is it possible to represent every huge number in abbreviated form?
Consider the following expression. $1631310734315390891207403279946696528907777175176794464896666909137684785971138$ $2649033004075188224$ This is a $98$ decimal digit number. This can be represented as $424^{37}$ which has just 5 digits. or…
endrendum
- 233
92
votes
6 answers
How deep is the liquid in a half-full hemisphere?
I have a baking recipe that calls for $1/2$ tsp of vanilla extract, but I only have a $1$ tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere. My question is, to what depth (as a…
Holly
- 993
92
votes
6 answers
What is the intuition behind uniform continuity?
There’s another post asking for the motivation behind uniform continuity. I’m not a huge fan of it since the top-rated comment spoke about local and global interactions of information, and frankly I just did not get it. Playing with the definition,…
Spencer Kraisler
- 3,139
92
votes
12 answers
Why determinant of a 2 by 2 matrix is the area of a parallelogram?
Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$. How could we show that $ad-bc$ is the area of a parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+b, c+d)$? Are the areas of the following parallelograms the same? $(1)$ parallelogram with…
user
92
votes
4 answers
Can a row of five equilateral triangles tile a big equilateral triangle?
Can rotations and translations of this shape perfectly tile some equilateral triangle? I've now also asked this question on mathoverflow . Notes: Obviously I'm ignoring the triangle of side $0$ . Because the area of the triangle has to be a multiple…
Oscar Cunningham
- 17,047
92
votes
3 answers
What is the Tor functor?
I'm doing the exercises in "Introduction to commutive algebra" by Atiyah&MacDonald. In chapter two, exercises 24-26 assume knowledge of the Tor functor. I have tried Googling the term, but I don't find any readable sources. Wikipedia's explanation…
Fredrik Meyer
- 20,820
92
votes
3 answers
Cute Determinant Question
I stumbled across the following problem and found it cute. Problem: We are given that $19$ divides $23028$, $31882$, $86469$, $6327$, and $61902$. Show that $19$ divides the following determinant: $$\left| \begin{matrix} 2 & 3&0&2&8 \\ 3 & 1&8&8&2\\…
Potato
- 41,689
91
votes
4 answers
Is there any easy way to understand the definition of Gaussian Curvature?
I am new to differential geometry and I am trying to understand Gaussian curvature. The definitions found at Wikipedia and Wolfram sites are too mathematical. Is there any intuitive way to understand Gaussian curvature?
Shan
- 1,737
91
votes
4 answers
Difference between complete and closed set
What is the difference between a complete metric space and a closed set? Can a set be closed but not complete?
ABC
- 2,033