Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $x/y$ is a rational number

Question

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $\frac{x}{y}$ is a rational number.

How do I prove this, and also which proving method would I use? I'm confused between that and also which definition to use in the proof.

Answer 1

If $x$ and $y$ are rational number then $x=p/q$ and $y=m/n$ for some integers $p,q,m,n$. Then $x/y=pn/mq$. So $x/y$ is rational.

Answer 2

I'd start with the definition of what a rational number is. A rational can be expressed as the ratio of two integers $p/q$, where $q \neq 0$.

So, let $x = p_1/q_1, y = p_2/q_2$, where $q_1, q_2 \neq 0$.

• What does $y \neq 0$ imply?
• What do you see when you write out $x/y$?

Answer 3

It is confusing, since it depends on what you are allowed to assume. Chances are good that you will be allowed to use a quite simple proof. Since $x$ is rational, there exist integers $a$ and $b$, with $b\ne 0$, such that $x=\frac{a}{b}$.

Since $y$ is rational and non-zero, there exist integers $c$ and $d$, with neither $c$ nor $d$ equal to $0$, such that $y=\frac{c}{d}$. We have $$\frac{x}{y}=\frac{a/b}{c/d}=\frac{ad}{bc}.\tag{1}$$ The number $ad$ is an integer, and the number $bc$ is a non-zero integer, since neither $b$ nor $c$ is $0$.

Remark: We have assumed that the basic arithmetical rule that gives us (1) can be used without proof, as well as the fact that the product of two non-zero integers is non-zero.