Mathematics

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1500 questions
129
votes
3 answers

All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$

I would like some reference about this infinitely nested radical expansion for all real numbers between $0$ and $2$ . I'll use a shorthand for this expansion, as a string of signs, $+$ or $-$ , with infinite periods denoted by brackets. $$2=\sqrt{2…
Yuriy S
  • 32,998
128
votes
0 answers

Ring structure on the absolute Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$ . It follows that the absolute Galois group carries the structure of a topological ring isomorphic to…
Martin Brandenburg
  • 183,887
128
votes
8 answers

Are the "proofs by contradiction" weaker than other proofs?

I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by…
AgCl
  • 6,602
128
votes
13 answers

Why can't calculus be done on the rational numbers?

I was once told that one must have a notion of the reals to take limits of functions. I don't see how this is true since it can be written for all functions from the rationals to the rationals, which I will denote $f$ , that $$\forall…
Praise Existence
  • 1,515
128
votes
2 answers

Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful to any student what kind ideas already used but…
users31526
  • 2,077
128
votes
4 answers

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or what the motivation for it is?
Nick Alger
  • 20,322
127
votes
8 answers

derivative of cost function for Logistic Regression

I am going over the lectures on Machine Learning at Coursera. I am struggling with the following. How can the partial derivative of $$J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}y^{i}\log(h_\theta(x^{i}))+(1-y^{i})\log(1-h_\theta(x^{i}))$$ where…
dreamwalker
  • 1,415
127
votes
7 answers

Is infinity a number?

Is infinity a number? Why or why not? Some commentary: I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school — but a difficult one to answer in an intelligent manner.…
Pops
  • 1,405
127
votes
13 answers

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
Billy Thompson
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127
votes
3 answers

More than 99% of groups of order less than 2000 are of order 1024?

In Algebra: Chapter 0 , the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024. Is this for real? How can one deduce this result? Is there a nice way or do we just check all…
Hui Yu
  • 15,589
127
votes
3 answers

Does convergence in $L^p$ imply convergence almost everywhere?

If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$ , do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$ ?
187239
  • 1,345
127
votes
19 answers

What parts of a pure mathematics undergraduate curriculum have been discovered since $1964?$

What parts of an undergraduate curriculum in pure mathematics have been discovered since, say, $1964?$ (I'm choosing this because it's $50$ years ago). Pure mathematics textbooks from before $1964$ seem to contain everything in pure maths that is…
Suzu Hirose
  • 11,999
127
votes
1 answer

Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's equation. Kaufman's article describes algebraic methods…
bolbteppa
  • 4,669
126
votes
2 answers

What makes a theorem "fundamental"?

I've studied three so-called "fundamental" theorems so far (FT of Algebra, Arithmetic and Calculus) and I'm still unsure about what precisely makes them fundamental (or moreso than other theorems). Wikipedia claims: The fundamental theorem of a…
beep-boop
  • 11,865
126
votes
17 answers

Calculating the integral $\int_0^\infty \frac{\cos x}{1+x^2}\, \mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$
Martin Gales
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