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1500 questions
64
votes
8 answers
Graphs for which a calculus student can reasonably compute the arclength
Given a differentiable real-valued function $f$, the arclength of its graph on $[a,b]$ is given by
$$\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,\mathrm{d}x$$
For many choices of $f$ this can be a tricky integral to evaluate, especially for calculus…
Mike Pierce
- 19,406
64
votes
8 answers
Is there a third dimension of numbers?
Is there a third dimension of numbers like real numbers, imaginary numbers, [blank] numbers?
awesomeguy
- 759
64
votes
2 answers
Is there a step by step checklist to check if a multivariable limit exists and find its value?
Do we rely on certain intuition or is there an unofficial general crude checklist I should follow?
I had a friend telling me that if the sum of the powers on the numerator is smaller then the denominator, there is a higher chance that it may not…
Yellow Skies
- 1,750
64
votes
14 answers
Is there a math expression equivalent to the conditional ternary operator?
Is there a math equivalent of the ternary conditional operator as used in programming?
a = b + (c > 0 ? 1 : 2)
The above means that if $c$ is greater than $0$ then $a = b + 1$, otherwise $a = b + 2$.
dataphile
- 822
64
votes
8 answers
Is there a common symbol for concatenating two (finite) sequences?
Say we have two finite sequences $X = (x_0,...,x_n)$ and $Y = (y_0,...,y_n)$.
Is there a more or less common notation for the concatenation of these sequences, like $\sum (X,Y) = (x_0,...,x_n,y_0,...,y_n)$?
chtenb
- 1,491
64
votes
1 answer
Which Algebraic Properties Distinguish Lie Groups from Abstract Groups?
This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group, and wants to be a sort of "converse".
Here I am taking an abstract group $G$ and looking for necessary conditions for it to admit…
Lor
- 5,738
64
votes
4 answers
Why can we use induction when studying metamathematics?
In fact I don't understand the meaning of the word "metamathematics". I just want to know, for example, why can we use mathematical induction in the proof of logical theorems, like The Deduction Theorem, or even some more fundamental proposition…
183orbco3
- 1,965
64
votes
1 answer
Does Fermat's Last Theorem hold for cyclotomic integers in $\mathbb{Q(\zeta_{37})}$?
The first irregular prime is 37. Does FLT(37)
$$x^{37} + y^{37} = z^{37}$$
have any solutions in the ring of integers of $\mathbb Q(\zeta_{37})$, where $\zeta_{37}$ is a primitive 37th root of unity?
Maybe it's not true, but how could I go about…
quanta
- 12,733
64
votes
7 answers
What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?
Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) force us to work very hard to prove the main…
Qiaochu Yuan
- 468,795
64
votes
6 answers
What is the required background for Robin Hartshorne's Algebraic Geometry book?
It seems that Robin Hartshorne's Algebraic Geometry is the place where a whole generation of fresh minds have successfully learned about modern algebraic geometry. But is it possible for someone who is out of academia and has not much background,…
Hooman
- 3,787
64
votes
5 answers
What is the difference between the weak and strong law of large numbers?
I don't really understand exactly what the difference between the weak and strong law of large numbers is. The weak law says
\begin{align*}
\lim_{n \rightarrow \infty} \mathbb{P}[\mid \bar{X}_n - \mu \mid \leq \epsilon ] = 1,
\end{align*}
while the…
Stefan Falk
- 1,317
- 1
- 12
- 26
64
votes
10 answers
How to solve an $n$-th degree polynomial equation
The typical approach of solving
$$
f_2(x):=ax^2+bx+c=0
$$
is to solve for the roots
$$x_{1/2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.$$
Here, the degree of $f$ is given to be $2$.
However, I was wondering on how to generalize this problem.
For example,…
Ayush Khemka
- 1,013
64
votes
12 answers
If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?
A two-sided coin has just been minted with two different sides (heads and tails). It has never been flipped before. Basic understanding of probability suggests that the probability of flipping heads is .5 and tails is .5. Unexpectedly, you flip the…
skurwa
- 777
64
votes
10 answers
Big List of Erdős' elementary proofs
Paul Erdős was one of the greatest mathematicians of all time and he was famous for his elegant proofs from The Book. I posted a question about one of his theorem and got a reference, and I have other questions I want to know the answer to too. But,…
Saikat
- 2,561
64
votes
3 answers
Motivation and methods for self-study
First, a little background: Beginning with calculus in my first semester of college, I fell in love with mathematics. That was the point at which the concepts became interesting to me, and I started reading up, through Wikipedia and various other…
Alex Petzke
- 8,971
- 8
- 70
- 96