Questions tagged [soft-question]

For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

For questions whose answers cannot be objectively evaluated as correct or incorrect, but which are still relevant to mathematics.se.

12450 questions
1673
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92 answers

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve , which are easy to understand and explain but are mathematically beautiful at the same time. Do you know…
RBS
  • 891
919
votes
29 answers

Can I use my powers for good?

I hesitate to ask this question, but I read a lot of the career advice from MathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that I am going to leave academia soon. I do enjoy…
Flounderer
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866
votes
27 answers

How to study math to really understand it and have a healthy lifestyle with free time?

Here's my issue I faced; I worked really hard studying Math, so because of that, I started to realised that I understand things better. However, that comes at a big cost: In the last few years, I had practically zero physical exercise, I've gained…
Leo
  • 11,152
659
votes
164 answers

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found that I, too, lament the uninspiring…
Liz
  • 101
547
votes
35 answers

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to…
Neil Mayhew
  • 1,683
460
votes
23 answers

Proofs that every mathematician should know.

There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge, etc. So I'd like to know what mathematical proofs you've come across that you…
user10389
  • 1,275
456
votes
15 answers

Fourier transform for dummies

What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on Kevin Lin's question , which didn't quite fit in MathOverflow. Answers at any level of sophistication are welcome.
user218
452
votes
22 answers

On "familiarity" (or How to avoid "going down the Math Rabbit Hole"?)

Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole." For example, suppose you come across the novel term vector space , and want to learn more about it. You look up various definitions, and…
kjo
  • 15,004
394
votes
75 answers

'Obvious' theorems that are actually false

It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true". Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem). But…
beep-boop
  • 11,865
383
votes
125 answers

Collection of surprising identities and equations.

What are some surprising equations/identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc. I'd request to avoid 'standard' / well-known…
Calvin Lin
  • 78,054
372
votes
11 answers

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
Ryan
  • 5,689
361
votes
23 answers

Why don't we define "imaginary" numbers for every "impossibility"?

Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such as…
lily
  • 4,802
348
votes
36 answers

A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally…
AgCl
  • 6,632
336
votes
23 answers

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? Every time I search for "Advanced Techniques of…
user3002473
  • 9,395
300
votes
27 answers

Why do mathematicians use single-letter variables?

I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated when I try to follow mathematical notation. Specifically, I get frustrated trying to keep track…
eater
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