The Math of Zeno's Paradox.

[peter]

In Christopher Potters book How to make a Human he says the resolution of the hare and the tortoise paradox

is the understanding that an infinite sum of reducing elements is not always itself infinite

He gives the example of the sum 1 + 1/2 + 1/4 + 1/8........ not being infinity [as the greeks supposed] but being, = 2 [ie equal to 2]

He then says that How to integrate together infinite sequences that contain infinitely small elements is at the heart of all calculus.

Can anyone prove the above sum for me and set me on the path to an understanding that has to date eluded me.

[Only kidding, but it would be fun to see the sum proved.]

[I'm Murrin]

There's a pretty simple way to prove it.

Rather than looking at the sum, let's just take the result of such a thing: An infinitely recurring decimal:

0.9999...

If you subtract 0.9999... from 1, what is the result? It can only be 0.0000... recurring infinitely; if there is an endless number of 9s, then you'll never reach a trailing 1.

So 1 - 0.9999... = 0

Therefore 0.9999... = 1

This is the reason Zeno's Paradox is not actually a paradox.

[Fist and Faith]

He gives the example of the sum 1 + 1/2 + 1/4 + 1/8........ not being infinity [as the greeks supposed] but being, = 2 [ie equal to 2]

It does not equal 2. It always falls short by the amount of the last fraction. If you stop at 1 + 1/2, it is 1/2 less than 2. If you stop at 1 + ... 1/1024, it is 1/1024 less than 2. It will never equal 2, no matter how long you go on. It will only be a smaller and smaller fraction less than 2.

[I'm Murrin]

The remainder shrinks to zero as the number of iterations approaches infinity. At an infinite number of iterations, the remainder is exactly zero - as can be proven with the sumple sum I showed. If 0.999... has an infinite number of 9s, then when you subtract that from 1 you get 0.000... with an infinitely recurring number of 0s, AKA exactly 0.

In other words: If there's always another iteration, then there's never a remainder. Otherwise there would be a real paradox and Achilles would never overtake the Tortoise.

[wayfriend]

Can anyone prove the above sum for me and set me on the path to an understanding that has to date eluded me.

peter, if you look at the total that each term in the series produces - 1, 1.5, 1.75, 1.875, etc. - it's the classic case of iteratively halving the distance to the finish line. Each term is getting half again closer to to 2.00. it will never get there (as Fist correctly states), it will just get closer and closer and closer.

Mathematically, the total value at iteration N is 2-(1/(2^N)). As N approaches infinite, 1/(2^N) approaches zero, and so 2-(1/(2^N)) approaches 2.

Why is it 2-(1/(2^N))? Because

1+1/2+1/4+1/8 ... + 1/(2^N) =

(2^N + 2^(N-1) + 2^(N-2) ... + 1) / (2^N) =

(2^(N+1) - 1) / (2^N) =

(2^(N+1)) / (2^N) - 1/(2^N) =

2 - 1/(2^N)

[wayfriend]

Here's another interesting one. Leibniz formula for pi.

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 .... = π/4

[peter]

Bravo Guys - I like that, I like that a lot!

If I get it the answer is both 2 and not 2 depending as to how you see it. 2 if you accept that the final interval at infinity is zero, not 2 if you try to do it in practice.

[what is n in the last one Wayfriend?] ....and why would a string of additions like this relate to the ratio of a circles diameter to its circumferance {not that I'm doubting it does - it just fascinates me that it should be so!}]

[wayfriend]

[what is n in the last one Wayfriend?] ....and why would a string of additions like this relate to the ratio of a circles diameter to its circumferance {not that I'm doubting it does - it just fascinates me that it should be so!}]

It's supposed to say (pi over 4). Yes, it's absolutely fascinating that it gets you to pi -- that's why I posted it! Anybody can get to 2 ...

[peter]

I read a short section in a book recently that briefly touched on the problems that maths cause philosophers [like why does it work, is it 'real' etc] and I'm guessing that stuff like this would sit well within those problems.

[Vraith]

I read a short section in a book recently that briefly touched on the problems that maths cause philosophers [like why does it work, is it 'real' etc] and I'm guessing that stuff like this would sit well within those problems.

Heh...we have whole threads on that. I think you even started at least one of them.

But I don't think math causes philosophers any more trouble than anything else...and probably significantly less trouble than knowledge, aesthetics, morality, even language/communication.

[peter]

Agreed V., but for some reason I just find it more fascinating that something as 'rational' as maths should at it's far reaches of trying to understand 'what it is', prove to be so nebulous [wrong word - I mean 'difficult to nail down']. With say 'beauty' you can get it. Morality - no problem, but Math! It shouldn't happen should it .

[Vraith]

Agreed V., but for some reason I just find it more fascinating that something as 'rational' as maths should at it's far reaches of trying to understand 'what it is', prove to be so nebulous [wrong word - I mean 'difficult to nail down']. With say 'beauty' you can get it. Morality - no problem, but Math! It shouldn't happen should it .

Heh...it would probably make things easier for us if it didn't happen.
Or maybe not....
If math were nailable, it would probably only apply in universes where intelligence was impossible...