Proof Infinity is equal to 1 and -1 and 1/2
If like me you thought that infinity was a huge number then think again.
what is 1 + 2 + 4 + 8 + 16 + 32 + ... ?
where ... means carry on forever
let this number be I (Infinity), so:
eq (1) I = 1+2+4+8+16+32+...
taking eq (1) and multiplying both sides by 2 we get
eq (2) 2xI = 2 (1+2+4+8+16+32+...)
2I = 2+4+8+16+32+64....
now if we say eq (3) = eq (1) - eq (1)
2I - I = (2+4+8+16+32+64....) - (1+2+4+8+16+32...)
if you look carefully you will see that eq (2) repeats in eq (1) with the exception of the leading 1, so solving this you get
I = -1
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So if you continue to add bigger and bigger numbers, ad infinitum, then you bend around the time qunatum continium, proving time travel is possible as you end up before you start i.e -1. This is how the flux capacitor in Back to the Future works.
Now lets try this:
1-1+1-1+1-1+1-1....
the above can be written as
(1-1)+(1-1)+(1-1)+(1-1)...
=> 0+0+0+0+0...
so what is the sum of of infinite bunch of zeros ? Zero you say... well lets see....
is 1-1+1 not the same as 1 - (1 -1)
so if we rewrite our original equation as :
1-(1-1)-(1-1)-(1-1)... = 1 - 0 - 0 - 0 ...
mmm... this demands that it is equal to 1 so which is it 0 or 1
lets T be the answer
eq (4) T = 1-1+1-1+1-1+1-1....
taking the -ve of each
eq (5) -T = -1+1-1+1-1+1-1....
this is is the same as eq (4) without the first 1 so if take eq (4) and subtract 1 you get -T
so eq (6) -T = T-1
therefore -2T= 1
T = 1/2
So T is 0, 1 and 1/2
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i
credit/reference :
1. How Not to be Wrong. The power of Mathematical Thinking by Jordan Ellenberg
