Answer
The sum, difference and product of two rational numbers will be rational.
(see step by step for proof).
The product of irrational numbers is not always irrational
(see step by step for proof)
The sum of irrational numbers is not always irrational
(see step by step for proof)
Work Step by Step
-Proof of the sum, difference and product of rational numbers will be rational.
Given $\frac{a}{b}$ and $\frac{c}{d}$ are rational numbers with a,b,c,d integers (b,d$\ne$0), $\frac{a}{b}$+$\frac{c}{b}$=$\frac{a+c}{b}$, $\frac{a}{b}$x$\frac{c}{d}$=$\frac{ac}{bd}$. These fractions have integers in the numerator and denominator making the result rational numbers.
-Proof that the product of irrational numbers are sometimes rational.
$\sqrt 2$($\frac{1}{\sqrt 2})$=1, this is a rational number.
-Proof that the sum of irrational numbers is sometimes rational.
$\pi$+(-$\pi$)=0 zero is a rational number
