Did you know...?

[robvalue]

Here's another handy everyday practical fact to keep in mind:

If you add up this infinite series (allow some time for this):

1/1 + 1/4 + 1/9 + 1/16 + ...

Then multiply the answer by 6...

Then take the square root...

You get the ratio between the circumference and the diameter of any circle.

[Alex K]

Wow cool

I've not heard that one before. So do you mean adding the terms up in a different order?

Yes, a simple example of an infinite permutation is this: add them by always taking the next two even ones and then one of the odd ones, then again two even ones, then one odd, you get the idea:

1/2 + 1/4 - 1/3 + 1/6 + 1/8 - 1/5 + ....

This series will eventually reach each term of the original one, but it converges to roughly 0.65 instead of 0.31 ! This always happens with series which are only convergent if you have alternating signs, but will diverge if you put "+" everywhere as in this case. Any series which is not still convergent if you put "+"es (is not "absolute convergent"), can be rearranged to yield an arbitrary real number. It's a theorem, I forgot how it is called.

[pocaracas]

ooooOOO math geekiness!

The sum of all infinite positive natural numbers (1 + 2 + 3 + 4 + .... ) is a negative fractional number!! O.o

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-1/12

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[Alex K]

ooooOOO math geekiness!

The sum of all infinite positive natural numbers (1 + 2 + 3 + 4 + .... ) is a negative fractional number!! O.o

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-1/12

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Silly String theorists are to blame. But this is only true if you mean by "1+2+3+4+5+6+" not ordinary naive summation but interpret it as a limiting case of representations of the Riemann zeta function. The ordinary series obtained by summing "1+2+3+..." is simply divergent.

To be precise: The sum over (n^-z) from n=1 to infinity for constant z gives the famous Riemann zeta function zeta(z).

Sending z to -1 naively turns this series into "1+2+3+4+5+...", and the zeta function analytically continued in the complex plane to z = -1 is zeta(-1)=-1/12, which has spawned the meme that "1+2+3+4+5+ ... = -1/12", even though the n^-z series which we used to define the Riemann function at points away from z=-1 has no validity there. Still, it's quite mind-boggling that zeta(z->-1) doesn't simply diverge.

[pocaracas]

ooooOOO math geekiness!

The sum of all infinite positive natural numbers (1 + 2 + 3 + 4 + .... ) is a negative fractional number!! O.o

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-1/12

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Silly String theorists are to blame. But this is only true if you mean by "1+2+3+4+5+6+" not ordinary naive summation but interpret it as a limiting case of representations of the Riemann zeta function. The ordinary series obtained by summing "1+2+3+..." is simply divergent.

Yes... the video does have a link to the Riemann Zeta function explanation...
But the one in there is still mind blowing...
However, the grain of salt starts with that first sum, 1-1+1-1+1-1.... = 1/2.... this is a limiting case, and not the ordinary series summation. The ordinary version would be just undefined.

[Alex K]

See my edit for details. There are many ways to reinterpret 1+2+3+4..., I choose to think of it as a shorthand for the analytic continuation of the Riemann function.

[pocaracas]

See my edit for details. There are many ways to reinterpret 1+2+3+4...,  I choose to think of it as a shorthand for the analytic continuation of the Riemann function.

"I choose to think"?
What is this? a religion?

[Alex K]

See my edit for details. There are many ways to reinterpret 1+2+3+4...,  I choose to think of it as a shorthand for the analytic continuation of the Riemann function.

"I choose to think"?
What is this? a religion?

Yes. Religion is one way to try and come to terms with the absurd

[brewer]

Math geek diversion. Stop it!

[abaris]

Did you know that Jaws from the James Bond movies has been a mathematician teacher before becoming an actor?