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1500 questions
143
votes
22 answers
List of interesting math podcasts?
mathfactor is one I listen to. Does anyone else have a recommendation?
Tim
- 119
143
votes
10 answers
What is the Jacobian matrix?
What is the Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone please explain with examples?
Pratik Deoghare
- 13,919
143
votes
15 answers
Why is $1$ not a prime number?
Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$?
bryn
- 10,135
142
votes
6 answers
What's the difference between predicate and propositional logic?
I'd heard of propositional logic for years, but until I came across this question , I'd never heard of predicate logic. Moreover, the fact that Introduction to Logic: Predicate Logic and Introduction to Logic: Propositional Logic (both by Howard…
Alex Basson
- 4,401
142
votes
5 answers
Difference between "≈", "≃", and "≅"
In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"? The Unicode standard lists all of them inside the Mathematical Operators Block. ≈ : ALMOST EQUAL TO (U+2248) ≃ : ASYMPTOTICALLY…
GOTO 0
- 2,120
142
votes
10 answers
What's the difference between theorem, lemma and corollary?
Can anybody explain me what is the basic difference between theorem, lemma and corollary? We have been using it for a long time but I never paid any attention. I am just curious to know.
HPS
- 4,738
142
votes
13 answers
Can someone explain Gödel's incompleteness theorems in layman terms?
Can you explain it in a fathomable way at high school level?
Hyperbola
- 2,485
142
votes
11 answers
Why is $\infty \cdot 0$ not clearly equal to $0$?
I did a bit of math at school and it seems like an easy one - what am I missing? $$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$ $$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ 0}_{n\text{ times}} = 0$$ (i.e add $0$ to $0$ as many times…
Ashley Schroder
- 1,547
142
votes
16 answers
Why did mathematicians take Russell's paradox seriously?
Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most of them mark its formulation as an epochal moment…
Uticensis
- 3,401
142
votes
37 answers
Unexpected examples of natural logarithm
Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common logarithm. Because of that, I have been gathering…
José Carlos Santos
- 443,185
142
votes
6 answers
Why are the solutions of polynomial equations so unconstrained over the quaternions?
An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb C$, but in $\mathbb H$, ${\bf i}\cos x + {\bf j}\sin…
MJD
- 68,126
- 44
- 311
- 628
141
votes
12 answers
Modular exponentiation by hand ($a^b\bmod c$)
How do I efficiently compute $a^b\bmod c$ : When $b$ is huge, for instance $5^{844325}\bmod 21$ ? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$ ? When $(a,c)\ne1$ ,…
user7530
- 50,865
141
votes
16 answers
What are good books to learn graph theory?
What are some of the best books on graph theory, particularly directed towards an upper division undergraduate student who has taken most the standard undergraduate courses? I'm learning graph theory as part of a combinatorics course, and would like…
Dom
- 573
141
votes
1 answer
Prove that simultaneously diagonalizable matrices commute
Two $n\times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices. a) Show that simultaneously diagonalizable matrices commute: $AB = BA$…
diimension
- 3,590
141
votes
36 answers
Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
b1_
- 1,593