show that every rational number has one and only one multiplicative inverse

Question

I am stumped and have no idea on how I prove this. I don't know what else to say. I am beyond lost.

Answer 1

Let $r$ be rational. Suppose there are two inverses $a$ and $b$. Then $$ ar=br=1. $$ Most importantly $$ ar=br. $$ Multiply both sides by $a$ to get $$ ara=bra. $$ Since $ra=1$, you have $$ a=b. $$

Answer 2

Suppose that $a$ is a nonzero rational number. Suppose that $b, c$ are multiplicitive inverses. Then we have$ab=ac=1$. Multiplying by $b$ on the left we see that $b(ab)=(ba)b=b=b(ac)=(ba)c=c$. So $b=c$. Note thatI used associtativity

Answer 3

Suppose you have a rational number $x$ and that $a$ and $b$ are inverses. Can you show that $a = b$?