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I know that both of the binary number system and complex number system depend on each others with real number system respectively and share some of their conditions and operation properties.

My question is: Are there any famous number system competely independence from the real number system that show its signifance in math history?

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What does independent mean? The integers and rational numbers are real numbers, but they can be defined without using the real numbers. – William Sep 6 '12 at 20:19
@William - Competely independence mean that no operation properties and definitions could be share... – Victor Sep 6 '12 at 20:21
Then how about the $\mathbb{Z} / n \mathbb{Z}$ and the $p$-adic numbers $\mathbb{Q}_p$. – William Sep 6 '12 at 20:24
@William - Explain what that is in your answer, thanks in advance! – Victor Sep 6 '12 at 20:27
Not sure if fits here, but I'll leave it here anyway: ordinal arithmetic. – user2468 Sep 7 '12 at 13:40
up vote 2 down vote accepted

To expand on William's comment:

${\bf Z}/n{\bf Z}$ is the integers modulo $n$. You probably know about modular arithmetic and if you don't you can find tons of information about it on the web and in texts about Number Theory and/or Discrete Mathematics. It doesn't contain the reals and it's not contained in the reals and the operations are not the operations in the reals.

Now pick a prime number $p$. Every non-zero rational number $a/b$ can be written as $(r/s)p^t$ where $r,s,t$ are integers and $p$ divides neither $r$ nor $s$. Define a sort of absolute value on the rationals by $|a/b|_p=p^{-t}$. This extends to a distance on the rationals by defining the distance $d(x,y)$ between $x$ and $y$ to be $|x-y|_p$. Now if you know how to get the reals from the rationals by putting in all the limits of convergent sequences, you can do the same thing but using $|\ |_p$ instead of the usual absolute value, and what you get is the $p$-adic numbers. Again, it's not a subset of the reals, nor does it contain the reals, and its distance structure is very different from that on the reals. Again, tons of info on the web and in (somewhat more advanced) Number Theory texts.

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