two rational numbers

[logistic_guy]

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

 

[pka]

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

What is your definition of a rational number?
How does that apply to this given question?
YOU show some work!

 

[logistic_guy]

What is your definition of a rational number?

\(\displaystyle \mathbb{Q} = \left\{\frac{a}{b} \ \big| \ a, b \in \mathbb{Z}, b \neq 0 \right\}\) are the rational numbers.

How does that apply to this given question?
YOU show some work!

I will start with a star:

*The sum of two integers is another integer. And the product of two integers is another integer.

I will let \(\displaystyle x\) and \(\displaystyle y\) be two rational numbers. It means that \(\displaystyle x = \frac{a}{b}\) and \(\displaystyle y = \frac{c}{d}\), \(\displaystyle b,d \neq 0\).

The sum:
\(\displaystyle x + y = \frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{cb}{db} = \frac{ad + cb}{bd}, \ \ bd \neq 0\)
From *, \(\displaystyle ad, cb, bd, \text{and} \ ad + cb\) are another integer, so \(\displaystyle = \frac{ad + cb}{bd}\) is a rational number.

The product:
\(\displaystyle xy = \frac{a}{b}\frac{c}{d} = \frac{ac}{bd}, \ \ bd \neq 0\)
From *, \(\displaystyle ac \ \text{and} \ bd\) are another integer, so \(\displaystyle \frac{ac}{bd}\) is a rational number.

 

[pka]

\(\displaystyle \mathbb{Q} = \left\{\frac{a}{b} \ \big| \ a, b \in \mathbb{Z}, b \neq 0 \right\}\) are the rational numbers.
I will start with a star:
*The sum of two integers is another integer. And the product of two integers is another integer.
I will let \(\displaystyle x\) and \(\displaystyle y\) be two rational numbers. It means that \(\displaystyle x = \frac{a}{b}\) and \(\displaystyle y = \frac{c}{d}\), \(\displaystyle b,d \neq 0\).

The sum:
\(\displaystyle x + y = \frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{cb}{db} = \frac{ad + cb}{bd}, \ \ bd \neq 0\)
From *, \(\displaystyle ad, cb, bd, \text{and} \ ad + cb\) are another integer, so \(\displaystyle = \frac{ad + cb}{bd}\) is a rational number.

The product:
\(\displaystyle xy = \frac{a}{b}\frac{c}{d} = \frac{ac}{bd}, \ \ bd \neq 0\)
From *, \(\displaystyle ac \ \text{and} \ bd\) are another integer, so \(\displaystyle \frac{ac}{bd}\) is a rational number.

YES. that will suffice for a proof.

 

[logistic_guy]

YES. that will suffice for a proof.