Thread: Math trivia
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Old 07-17-2017, 06:05 PM
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lpetrich lpetrich is offline
Join Date: Jul 2004
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Default Re: Math trivia

One can find 2aleph-0 by considering binary representations of numbers between 0 and 1. Their cardinality turns out to be C. That is likewise true for any other number base b:
baleph-0 = C

The open-ended real line segment (0,1) has the same cardinality as the closed-ended one [0,1] by a pushing-in-the-ends argument. The real numbers are easily mapped onto (0,1) and vice versa:
x <-> (1 + x/sqrt(1+x2))/2

So both (0,1) and [0,1] in the reals have cardinality C.

One can find C*C by considering an ordered pair of numbers from [0,1] and interleaving the digits. One gets another real number, making C*C = C. That is also true of any finite-length ordered n-tuples, by the same argument.

If one finds |all (real number, integer)|, one also gets C, because 1 < aleph-0 < C.

The number of infinite series of rational numbers, (aleph-0)aleph-0 is also C.

The number of permutations of positive integers is the number of their self-bijections: (aleph-0)! = C.

Interestingly, Caleph-0 = C. Thus, the total number of continuous functions from real numbers to real numbers is C, and that is also true of all the infinite sequences of real numbers.
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Dragar (07-18-2017), slimshady2357 (07-18-2017)
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