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Question about real numbers

- 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?
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.
Doris Hurringnon - Mon, 15 Dec 2014 20:16:39 EST Dk8yywxc No.14529 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 xrV+VzTJ No.14530 Reply
Ah thanks, that's exactly what I meant
Beatrice Wommergold - Tue, 16 Dec 2014 15:53:58 EST uPru0qmD No.14533 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 K8qJv5EF No.14542 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 SlKfpVpP No.14546 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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