Q is closed under division or not?

1. Sep 10, 2011

flyingpig

1. The problem statement, all variables and given/known data

Q = rational numbers

My professor proved that it is closed under addition yesterday. I kinda understood a bit...

How the heck does it work for when n_1 or n_2 = 0?

2. Sep 10, 2011

micromass

Staff Emeritus
Take a look at the first line. There we take $r_1,r_2\in \mathbb{Q}$. Then we say that

$$r_i=\frac{m_i}{n_i}$$

but obviously this doesn't work for $n_i=0$ since division by zero is not defined in $\mathbb{Q}$.

That is $\frac{m}{n}\in \mathbb{Q}$ if and only if $n\neq 0$...

3. Sep 10, 2011

flyingpig

I thought it is the same logic for integers, I take 1 and 0 and take 0 to be the bottom, can't do that.

If I take 0 and 1 and put 0 on top, that works, but it is an exception when it is the bottom?

4. Sep 10, 2011

micromass

Staff Emeritus
Yes, you can never have 0 in the denominator. By definition of a rational number.

5. Sep 10, 2011

flyingpig

But the whole close division thing i can take two integers (including 0)?

6. Sep 10, 2011

micromass

Staff Emeritus
No, an element of $\mathbb{Q}$ is defined as a fraction $\frac{m}{n}$, where n is nonzero. So you can't take 0 is the denominator, by definition.

You can't divide by 0.

7. Sep 10, 2011

SammyS

Staff Emeritus
No. The integers are not closed under division for reasons other than the fact that 1/0 is undefined.

Is 4/3 an integer?

Perhaps you should review the definition of a 'ring'.

8. Sep 10, 2011

flyingpig

Thank you, you just added more work for me...

9. Sep 10, 2011

SammyS

Staff Emeritus
Anytime! You're welcome.

To help out a bit more:

Whether we're discussing the Integers, the Rationals, the Reals, or Complex Numbers (all with the usual arithmetic operations):
It's always true that 0 times any element is 0, 0 being the identity element for the addition operation. Because of this, there is no multiplicative inverse for 0, and thus division (the operation that is the inverse of multiplication) by 0 is undefined. Therefore, when discussing whether division is closed, we exclude the case of division by zero.​