Cool Things. (1 Viewer)

[wannabesurgeon]

I remember when we were learning calculus a couple of weeks back we were doing differentiation by first principles and then the short method came along and I was ecstatic xD

 

[humphdogg]

Cool, Andrew is my supervisor at the moment. Are you his old student who is currently working with Peter Sarnak?

If by currently, you mean flying over to the US tomorrow to begin, then yes

 

[seanieg89]

If by currently, you mean flying over to the US tomorrow to begin, then yes

Exciting! Best of luck . I may run into you some time in the future.

 

[Shadowdude]

 

[cutemouse]

Lol, yeah that is cool. There's two ways of actually doing that.

Method 1: Use the fact that sin z = [e^(iz)-e^(-iz)]/(2i) and solve the resultant quadratic

Method 2: Use sin(x+iy)=sinx cosh y + i cosx sinh y. But this method only works if the imaginary or real part is zero.

 

[shongaponga]

0! = 1

 

[jet]

0! = 1

You can extend the factorial to all complex numbers except for the non-positive integers using the gamma function


It's not too hard to prove that using integration by parts which means when is a positive integer.

 

[seanieg89]

You can extend the factorial to all complex numbers except for the negative integers using the gamma function


It's not too hard to prove that using integration by parts which means when is a positive integer.

*non-positive integers*

 

[Sy123]

0! = 1

Is this because there is only 1 way to pick nothing? Or is there a deeper reason?

 

[seanieg89]

Is this because there is only 1 way to pick nothing? Or is there a deeper reason?

By convention the product of an empty collection of numbers is 1 and the sum of an empty collection of numbers is 0. I stress that this is just convention, it is convenient for this to be the definition.

 

[Carrotsticks]

Is this because there is only 1 way to pick nothing? Or is there a deeper reason?

Yep. The empty set could be said to have only 1 possible permutation (identity permutation).

Hence 0! = 1.

 

[Trebla]

You can extend the factorial to all complex numbers except for the non-positive integers using the gamma function


It's not too hard to prove that using integration by parts which means when is a positive integer.

Reminds me of this nice result

 

[JINOUGA]

Not really a result, but I absolutely love The Pigeonhole Principle. So intuitive to understand yet oh so effective for many more complex problems

 

[seanieg89]

Not really a result, but I absolutely love The Pigeonhole Principle. So intuitive to understand yet oh so effective for many more complex problems

Big +1. I find things like this much more beautiful than random analysis identities...because they are ideas that are universal. Simple and elegant.

 

[Carrotsticks]

If my sock drawer has 4 pairs of matching socks, and I pick 5 socks at random, what is the probability of me getting at least 1 matching pair?

 

[Timske]

If my sock drawer has 4 pairs of matching socks, and I pick 5 socks at random, what is the probability of me getting at least 1 matching pair?

1

 

[seanieg89]

A cooler application: http://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem . Related to 1989 Q8.

 

[Carrotsticks]

A cooler application: http://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem . Related to 1989 Q8.

Haha we studied that in 1st Year SSP to compute convergents for continued fractions.

 

[seanieg89]

Haha we studied that in 1st Year SSP to compute convergents for continued fractions.

Nice! Did you look at Pell's equation?

 

[Carrotsticks]

Yep was part of it. Real sweet topic.