Mathematics

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1500 questions
65
votes
1 answer

Why are asymptotically one half of the integer compositions gap-free?

Question summary The number of gap-free compositions of $n$ can already for quite small $n$ be very well approximated by the total number of compositions of $n$ divided by $2$. This question seeks to understand why. The details A composition of an…
Daniel R
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65
votes
13 answers

Is there an equation to describe regular polygons?

For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required? Using the Wolfram Alpha site, this input…
Vincent Tan
  • 753
65
votes
14 answers

What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, such that those entities satisfy certain basic…
user78743
65
votes
17 answers

Intuitive understanding of the derivatives of $\sin x$ and $\cos x$

One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) chain of derivatives: sin(x) cos(x) -sin(x) -cos(x)…
Justin L.
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65
votes
2 answers

When are nonintersecting finite degree field extensions linearly disjoint?

Let $F$ be a field, and let $K,L$ be finite degree field extensions of $F$ inside a common algebraic closure. Consider the following two properties: (i) $K$ and $L$ are linearly disjoint over $F$ : the natural map $K \otimes_F L \hookrightarrow KL$…
Pete L. Clark
  • 101,276
65
votes
4 answers

A "new" general formula for the quadratic equation?

Maybe the question is very trivial in a sense. So, it doesn't work for anyone. A few years ago, when I was a seventh-grade student, I had found a quadratic formula for myself. Unfortunately, I didn't have the chance to show it to my teacher at that…
lone student
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65
votes
15 answers

Why can't you flatten a sphere?

It's a well-known fact that you can't flatten a sphere without tearing or deforming it. How can I explain why this is so to a 10 year old? As soon as an explanation starts using terms like "Gaussian curvature", it's going too far for the audience in…
Joe
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65
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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
  • 19,622
65
votes
6 answers

Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$

This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions , and here: List of abstract duplicates . What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty}…
Eric Naslund
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65
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
  • 1,501
65
votes
3 answers

Optimal strategy for cutting a sausage?

You are a student, assigned to work in the cafeteria today, and it is your duty to divide the available food between all students. The food today is a sausage of 1m length, and you need to cut it into as many pieces as students come for lunch,…
Stenzel
  • 633
65
votes
6 answers

Create unique number from 2 numbers

is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result must be unique(or with very high probability unique)…
drizzt
  • 719
65
votes
16 answers

What exactly is calculus?

I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is: What exactly is calculus? I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure…
Hugh Chalmers
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65
votes
9 answers

Riemann hypothesis: is Bender-Brody-Müller Hamiltonian a new line of attack?

There is a beautiful paper in Physical Review Letters [PRL 118, 130201 (2017), DOI:10.1103/PhysRevLett.118.130201] by Carl Bender, Dorje Brody, and Markus Müller (BBM) on a Hamiltonian approach to the Riemann Hypothesis. The paper is surprisingly…
Slava Kashcheyevs
  • 1,732
65
votes
1 answer

How to find a total order with constrained comparisons

There are $25$ horses with different speeds. My goal is to rank all of them, by using only runs with $5$ horses, and taking partial rankings. How many runs do I need, at minimum, to complete my task? As a partial answer, I know that is possible to…
Jack D'Aurizio
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