Mathematics

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1500 questions
110
votes
15 answers

Comparing $\pi^e$ and $e^\pi$ without calculating them

How can I compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?
Mirzodaler
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110
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8 answers

What is the equation for a 3D line?

Just like we have the equation $y=mx+b$ for $\mathbb{R}^{2}$, what would be a equation for $\mathbb{R}^{3}$? Thanks.
Ovi
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110
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8 answers

Is Bayes' Theorem really that interesting?

I have trouble understanding the massive importance that is afforded to Bayes' theorem in undergraduate courses in probability and popular science. From the purely mathematical point of view, I think it would be uncontroversial to say that Bayes'…
user78270
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110
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8 answers

Are calculus and real analysis the same thing?

I guess this may seem stupid, but how calculus and real analysis are different from and related to each other? I tend to think they are the same because all I know is that the objects of both are real-valued functions defined on $\mathbb{R}^n$, and…
Tim
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110
votes
30 answers

What is the single most influential book every mathematician should read?

If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
c4il
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110
votes
9 answers

What are good "math habits" that have improved your mathematical practice?

I currently feel like I am not doing maths the best way I could; that is, I'm not making the most out of my time when I'm working on maths problems. The main thing I feel is that I'm not organizing my mind and my derivations as clear as I could,…
user56834
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110
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2 answers

Connected metric spaces with disjoint open balls

Let $X$ be the $S^1$ or a connected subset thereof, endowed with the standard metric. Then every open set $U\subseteq X$ is a disjoint union of open arcs, hence a disjoint union of open balls. Are there any other metric spaces with this property?…
Hagen von Eitzen
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110
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9 answers

If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, would it still be considered unfair for $50$…
Adam Freymiller
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110
votes
5 answers

How to get a reflection vector?

I'm doing a raytracing exercise. I have a vector representing the normal of a surface at an intersection point, and a vector of the ray to the surface. How can I determine what the reflection will be? In the below image, I have d and n . How can I…
Nick Heiner
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109
votes
19 answers

Chatting about mathematics (with real-time LaTeX rendering)

Do you know about some tools which can be used for online chat about mathematics? In particular, I am interested in software which would be able to render LaTeX formulas. (Since LaTeX is probably the fastest possibility to type mathematics.) Have…
Martin Sleziak
  • 56,568
108
votes
20 answers

What are some examples of mathematics that had unintended useful applications much later?

I would like to know some examples of interesting (to the layman or young student), easy-to-describe examples of mathematics that has had profound unanticipated useful applications in the real world. For my own purposes, the longer the gap between…
Eric Tressler
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108
votes
8 answers

Intuitive meaning of Exact Sequence

I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all. Can anyone explain them for me? Thanks in advance.
finnlim
  • 2,853
108
votes
1 answer

Can we remove any prime number with this strange process?

This is a prime-removal algorithm I made, which may appear to be quite complex so I will start with an example. @Max has since added this sequence to OEIS, number A332198 . The process goes as follows: Start with the first prime number, $$S(1)=2.$$…
TheSimpliFire
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108
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18 answers

What seemingly innocuous results in mathematics require advanced proofs?

I'm interested in finding a collection of basic results in mathematics that require rather advanced methods of proof. In this list we're not interested in basic results that have tedious simple proofs which can be shorted through more advanced…
mark
  • 1,791
108
votes
1 answer

Relations between p norms

The $p$ -norm on $\mathbb R^n$ is given by $\|x\|_{p}=\big(\sum_{k=1}^n |x_{k}|^p\big)^{1/p}$ . For $0 < p < q$ it can be shown that $\|x\|_p\geq\|x\|_q$ ( 1 , 2 ). It appears that in $\mathbb{R}^n$ a number of opposite inequalities can also be…
PianoEntropy
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