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Comment by chris_wot | Hacker News Reader

can you really always divide two rational numbers to get another rational number?

I'll not give a formal proof, but intuitively the answer is surely "yes"!

Yes, and it's pretty easy to prove. Multiplying any rational number by the inverse of another gives you a new rational number. (Zero doesn't have an inverse).

A rational number is a number that can be represented as one integer divided by another. Take two rational numbers, A = a/b, and B = c/d. If B has an inverse, then A/B = a/b * d/c = ad/bc. ad and bc are both integers, so A/B is rational.

My point was that you can only divide one rational by another to produce a new rational if the denominator is nonzero.

This is not what it says in the book, so there's potential for a newbie to get confused (and anyone reading the book in order to learn from it is, by definition, a newbie).

Oh, then that seems like a pretty pedantic worry. You can't divide by zero in any context; even children know that. It's such an obvious exception that it didn't even occur to me that's what you thought was wrong with the statement.

What if one of them is zero?

The definition of a rational number is a number that can be expressed as the quotient of two integers where the denominator is not 0.

0 \in N, 5 \in N => (0/5) \in Q. Therefore 0 \in Q. QED.

\in?

\in means "in the set of"

Given the definition "The definition of a rational number is a number that can be expressed as the quotient of two integers where the denominator is not 0."

Let N be the set of all integers and Q be the set of all rational numbers.

Trivially 0 is in the set N, 5 is in the set N

by the definition 0/5 is in Q. Therefore 0 is in Q. QED.

Ah, I see! Deleted my old comment. Nice :-)

Watch out for your proof. Q is constructed as a field with + and x only. The availability of the inverse operator is not well documented :-D

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