Prove that both the sum and the product of two rational numbers are rational.

Proof:

Let $A=a/b$ and $B=c/d$ where $a,b,c,d\in\mathbb{Z}$ and both $b$ and $d\neq0$.

It follows that$$A\cdot B=\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$$Since $ab,bd\in\mathbb{Z}$, $A\cdot B$ is rational.

Similarly,$$A+B=\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$Since $ad+bc,bd\in\mathbb{Z}$, $A+B$ is rational. $\blacksquare$

Am I right?

Question

Prove that both the sum and the product of two rational numbers are rational.

Proof:

Let $A=a/b$ and $B=c/d$ where $a,b,c,d\in\mathbb{Z}$ and both $b$ and $d\neq0$.

It follows that$$A\cdot B=\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$$Since $ab,bd\in\mathbb{Z}$, $A\cdot B$ is rational.

Similarly,$$A+B=\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$Since $ad+bc,bd\in\mathbb{Z}$, $A+B$ is rational. $\blacksquare$

Am I right?

jamesj · Answer

Exactly.  How old are you again?  14 right?

anonymous · Answer

I'm almost 15, Mr. James.

jamesj · Answer

You should feel good about yourself. 

What you're doing is quite a bit more sophisticated than what I was doing when I was 14 or 15.

anonymous · Answer

here they say that the school system is more rigorous than it is in the west.. but i really feel like i know nothing still; i have much to learn. thank you for your response