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1500 questions
75
votes
1 answer
Why is TREE(3) so big? (Explanation for beginners)
I am not a mathematician but I am interested in big numbers. I find them to be really interesting, almost god-like.
I am watching a series of videos from David Metzler on YouTube. I have a basic understanding of some fast growing functions.
David…
Josh Kerr
- 957
75
votes
2 answers
A comprehensive list of binomial identities?
Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do.
user813
75
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
- 2,176
75
votes
9 answers
Intuition on group homomorphisms
So I'm studying for finals now, and came across the idea of homomorphisms again. This is not a new idea for me at all, having seen them in groups, rings, fields ect. However, on reevaluating them I realized suddenly that I really don't understand…
user45793
75
votes
5 answers
Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:
"The rank of a matrix A is the number
of non-zero rows in the reduced
row-echelon form of A".
The…
Vivi
- 1,509
75
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
- 1,352
75
votes
9 answers
Is it bad form to write mysterious proofs without explaining what one intends to do?
Often when doing assignments, I find myself deliberately writing in a "mysterious" way. By this I mean that the reader usually will not understand what exactly is going on and what for, until the very end where all the things come together.
A…
Jaood
- 1,415
75
votes
14 answers
What is $x^y$? How to understand it?
$x+y=z$
I have a pen. He has a pen. Total is two pen. This is plus.
$x-y=z$
I had two pens. A pen was lost. So, I have a pen. Total remaining is one. This is minus.
$x\cdot y=z$
I have two pens. Three other friends have two pens. $4…
Spacez Ly Wang
- 851
75
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
- 1,250
75
votes
6 answers
Strategies to denest nested radicals $\sqrt{a+b\sqrt{c}}$
I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$. This may be able to see through easily, but how…
JSCB
- 13,698
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- 125
75
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
- 1,577
- 3
- 19
- 23
75
votes
9 answers
Why does the symbol for the multiplication operation change shape?
Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing…
user64742
- 2,195
75
votes
5 answers
How to write a good mathematical paper?
I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar.
A good time always goes too fast! Two years are fled. In the third year of PHD, my major is general topology and I'm facing…
Paul
- 21,141
75
votes
2 answers
Is it mathematically valid to separate variables in a differential equation?
I read the following statement in a book on Calculus, as part of my mathematics course:
Technically this separation of $\frac{dy}{dx}$ is not mathematically valid. However, the resulting integration leads to correct answer.
The book also contains…
Devarsh Ruparelia
- 1,140
75
votes
1 answer
Properly discontinuous action: equivalent definitions
Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = e$. I would like to prove that this property is…
Pedro
- 6,936