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1500 questions
143
votes
2 answers
Is there any mathematical reason for this "digit-repetition-show"?
The number $$\sqrt{308642}$$ has a crazy decimal representation : $$555.5555777777773333333511111102222222719999970133335210666544640008\cdots $$
Is there any mathematical reason for so many repetitions of the digits ?
A long block containing only…
Peter
- 84,454
142
votes
17 answers
How do you respond to "I was always bad at math"?
Here in the U.S., it is my experience that over 75% of adults I meet socially will volunteer that phrase or a variation upon learning that I am a mathematician. I find this frustrating, since almost nobody would brag about being bad at history or…
vadim123
- 82,796
142
votes
6 answers
Studying for the Putnam Exam
This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the advisability of participating, the career…
Potato
- 40,171
142
votes
16 answers
Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$
Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$
where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums.
Can someone provide a nice proof that
$$A(1,1) =…
Mike Spivey
- 55,550
141
votes
0 answers
Pullback and Pushforward Isomorphism of Sheaves
Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\mathcal{F}$ is an $\mathcal{O}_X$-module and…
Matt
- 7,358
141
votes
36 answers
Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
b1_
- 1,585
140
votes
24 answers
Visually deceptive "proofs" which are mathematically wrong
Related: Visually stunning math concepts which are easy to explain
Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually wrong. (e.g. missing square puzzle)
Do you know the…
puzzlet
- 785
140
votes
6 answers
Why is integration so much harder than differentiation?
If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product of integrals can't be expressed in general in…
Venge
- 1,581
140
votes
11 answers
What is the difference between regression and classification?
What is the difference between regression and classification, when we try to generate output for a training data set $x$?
Bober02
- 2,546
139
votes
11 answers
Why is $\infty \cdot 0$ not clearly equal to $0$?
I did a bit of math at school and it seems like an easy one - what am I missing?
$$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
- 1,517
139
votes
16 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
- 9,746
138
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
- 4,231
138
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
- 2,425
138
votes
6 answers
Intuition behind Conditional Expectation
I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know.
Let me get more specific. Let…
Stefan
- 6,360
138
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
- 13,573