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1500 questions
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
- 563
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
141
votes
23 answers
Good Book On Combinatorics
What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions with full proofs. The book should be for a…
Anonymous
- 281
141
votes
8 answers
Prove that $||x|-|y||\le |x-y|$
I've seen the full proof of the Triangle Inequality
\begin{equation*}
|x+y|\le|x|+|y|.
\end{equation*}
However, I haven't seen the proof of the reverse triangle inequality:
\begin{equation*}
||x|-|y||\le|x-y|.
\end{equation*}
Would you please…
Anonymous
- 2,536
140
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
- 4,331
140
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 $(a,c)\ne1$, for…
user7530
- 50,625
140
votes
0 answers
Probability for an $n\times n$ matrix to have only real eigenvalues
Let $A$ be an $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has only real eigenvalues?
The answer cannot be $0$ or $1$, since the set of matrices with distinct real…
Exodd
- 12,241
- 2
- 27
- 44
140
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
- 3,381
140
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 =…
diimension
- 3,570
140
votes
36 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
- 440,053
140
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
- 13,859
140
votes
11 answers
What is the difference between regression and classification?
What is the difference between regression and classification, when we try to generate output for a training data set $x$?
Bober02
- 2,604
140
votes
46 answers
What are some examples of a mathematical result being counterintuitive?
As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition.
My first and favorite experience of this is Gabriel's Horn that you see in intro Calc course, where the figure has finite…
Steven-Owen
- 5,726
139
votes
5 answers
Getting better at proofs
So, I don't like proofs.
To me building a proof feels like constructing a steel trap out of arguments to make true what you're trying to assert.
Oftentimes the proof in the book is something that I get if I study, but hard to come up with on my own.…
bobobobo
- 9,782
139
votes
24 answers
Visually deceptive "proofs" which are mathematically wrong
Related: Visually stunning math concepts which are easy to explain
Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually wrong. (e.g. missing square puzzle)
Do you know the…
puzzlet
- 785