Tell me something cool

Not only that, a good mathematician finds faster ways of doing something instead of drudging through a long problem the slow way

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Genetic algorithms are cool, but they're not directly mathematical. I suppose you could apply statistical analyses to them...

 

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The colours according to how fast they diverge isn't what makes the fractal patterns, they are there with 'simple' black and white. The colours make for the psychadelic posters that hippies will buy, but I'd argue the mandlebrot set would be well known without this hippie support.


If you want something cool, look into Dirichlet's theorem for primes in arithmetic progressions. If c and d are relatively prime, then the arithmetic progression c+d, 2*c+d, 3*c+d, 4*c+d, ... contains infinitely many primes. Furthurmore, for a fixed c, the corresponding admissible d's (those relatively prime to c) will asymptotically have the same distribution of primes, that is to say they are somewhat 'evenly" distributed amongst these arithmetic progressions. For example, the number of primes of the form 4n+1 that are less than x will be roughly x/(2*log(x)) and likewise for primes of the form 4n+3, i.e. they evenly split what the prime number theorem would predict. Also of interest is the so-called "chebyshev bias" or "renyi-shanks prime number race" which assert that even though asymptotically the split is even, 4n+3 is somehow favored to contain more primes than 4n+1.

 

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The coolest thing in Maths I ever learnt about was the Fibonacci number: 1.6180339887.... otherwise known as Phi, the "Golden Number".

Not for how it is generated / the various formulae for deriving it, but for its occurrence in nature:

http://goldennumber.net/


Some of it I find just freaky.
But then to a Biologist (which I am most certainly not) and to other fields it probably seems very normal.

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I like conic sections (Hyperbola, Parabola, Ellipse, Circle).
I particularly like the way that they can be produced in so many different ways:
- Slicing a cone
- Path of an object in free-fall in a 1/r² field
- Pencil and string
- Focus and directrix
- Cartesian equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0

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(Math2.org)

 

Have you heard the Ramanujan, Hardy, and 1729 story?

 

This is off topic (and in saying this, I'm not claiming to currently have a strong grasp of math-- just wanted to toss in my 2 cents)
That was one of the biggest turnoffs for me about math. Not neccesarily the methods used in solving the problems, or learning the material.......just doing 1000 repetitions from variations of the same darn problem.

 

No... care to share, or are you going to make me go look it up?

 

As the story goes, Hardy went to visit Ramanujan in the hospital. Hardy mentioned something about the number of the taxicab that brought him there, 1729, saying it was a dull number. Ramanujan immediately replied that it was the smallest numebr that can be written in two ways as a sum of two positive cubes, 1729=12^3+1^1 and 1729=9^3+10^3, making it rather not dull.

 

This can partially be blamed on a serious dumbing down of the way things are taught. Your standard calculus text will have you graph 500 parabolas. This kind of rote "learning" can be done without much thought, which the average student seems to dislike. [they dislike the repetition as well, but more wil be capable of getting through it, so less complaining]

This defect mostly vanishes in 'serious' math courses, where the problems will be meant to challenge your understanding of the material and perhaps introduce new topics.

 

That reminds me of the "smallest uninteresting number paradox".

Which reminds me of "the smallest positive integer not nameable in under eleven words" paradox.

Fun!