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If we only consider the natural numbers, is 3/2 undefined? If not what is the answer?

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3 being an odd number, any attempt to split up three objects into two groups will leave you with a remainder of 1, and since there's no natural number that gives 1 when doubled... – J. M. Dec 27 '10 at 15:27
up vote 13 down vote accepted

It depends on what you mean by $3/2$, of course.

But in the usual intepretation where $3/2$ denotes the number such that when it is multiplied by $2$ gives you $3$, then, if you are restricting 'number' to mean 'natural number', then there is no such number.

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I was sure this was the answer, but there are many holes in my maths knowledge that I wanted confirmation. I'll accept when the time limit allows. – ArturPhilibin Dec 27 '10 at 15:32

One never says that the rational number 3/2 is "undefined" as an integer. Rather one simply says that it is nonintegral, or not an integer. On the other hand if 3/2 denotes the value of the (partial) division function on naturals $\rm\:\mathbb N^2\to \mathbb N$ then one may say that this function is undefined at (3,2).

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Kind'a similar analogy... $\sqrt{-1}$ is undefined when considering only real numbers. So mathematicians derived the whole complex number system to give that some meaning.

Its really defining the undefined that has progressed mathematics. So the answer to your question is, 3/2 is defined but only if you extend the natural numbers to a more generalized number system, like positive reals. :)

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Or the rationals ... – Dario Dec 27 '10 at 18:02
or the dyadic numbers ($p/2^q$)... – Yuval Filmus Dec 28 '10 at 6:04

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