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Question about real numbers

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- Mon, 15 Dec 2014 11:40:37 EST akf5zfsA No.14524
File: 1418661637375.png -(3241B / 3.17KB, 120x119) Thumbnail displayed, click image for full size. Question about real numbers
Is there a function that can project the entire set of real numbers onto an arbitrarily sized interval within the real numbers?
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Charlotte Brazzledock - Mon, 15 Dec 2014 14:33:34 EST MTIV7/tU No.14526 Reply
Do you mean a continuous function? Because the answer is yes either way.
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Doris Hurringnon - Mon, 15 Dec 2014 20:16:39 EST Dk8yywxc No.14529 Reply
>>14524


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.
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Cedric Crebbercocke - Tue, 16 Dec 2014 10:09:36 EST xrV+VzTJ No.14530 Reply
Ah thanks, that's exactly what I meant
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Beatrice Wommergold - Tue, 16 Dec 2014 15:53:58 EST uPru0qmD No.14533 Reply
>>14524
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
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Esther Fabberhall - Tue, 23 Dec 2014 20:47:35 EST K8qJv5EF No.14542 Reply
>>14533
^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
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Thomas Bangerham - Fri, 26 Dec 2014 18:20:04 EST SlKfpVpP No.14546 Reply
>>14524
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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