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420chan is Getting Overhauled - Changelog/Bug Report/Request Thread (Updated July 10)
Question about real numbers Ignore Report Reply
Doris Blimmerladge - Mon, 15 Dec 2014 11:40:37 EST ID:akf5zfsA No.14524
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Is there a function that can project the entire set of real numbers onto an arbitrarily sized interval within the real numbers?
Charlotte Brazzledock - Mon, 15 Dec 2014 14:33:34 EST ID:MTIV7/tU No.14526 Ignore Report Reply
Do you mean a continuous function? Because the answer is yes either way.
Doris Hurringnon - Mon, 15 Dec 2014 20:16:39 EST ID:Dk8yywxc No.14529 Ignore Report Reply

It doesn't even have to be continuous. Any interval within the real numbers has the same cardinality as the entire real line, so a bijection (one to one and onto function between the reals and the interval) can be made to demonstrate that they have the same number of elements. It's counterintuitive, but even a tiny interval has the same "amount" of numbers as the entire real line.
Cedric Crebbercocke - Tue, 16 Dec 2014 10:09:36 EST ID:xrV+VzTJ No.14530 Ignore Report Reply
Ah thanks, that's exactly what I meant
Isabella Blidgeville - Tue, 16 Dec 2014 13:07:43 EST ID:sPd/0oB/ No.14531 Ignore Report Reply
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Beatrice Wommergold - Tue, 16 Dec 2014 15:53:58 EST ID:uPru0qmD No.14533 Ignore Report Reply
Hyperbolic tangent :
tanh(x)=(e^x-e^-x)/(e^x+e^-x) maps R->(-1,1)
atanh(x)=1/2 ln((1+x)/(1-x)) maps (-1,1)->R
a*tanh(x)+b maps R->(-a+b,a+b)
atanh((x-b)/a) maps (-a+b,a+b)->R

The nicest functions you're going to find
Esther Fabberhall - Tue, 23 Dec 2014 20:47:35 EST ID:K8qJv5EF No.14542 Ignore Report Reply
^This guy's post is correct if you're looking for continuous bijections. If you're looking for just a continuous map from the reals to [a,b], you can just choose f(x) = [(b-a)/2]sin(x) + (b+a)/2
Thomas Bangerham - Fri, 26 Dec 2014 18:20:04 EST ID:SlKfpVpP No.14546 Ignore Report Reply
another interesting question is whether you can project an arbitrarily sized interval within the real numbers onto the set of real numbers

answer is yes

also you can map a line of length 1 onto a cube of volume 1 etc.

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