Question

question
complete each hypothesis about the sum of rational and irrational numbers based on your answers to part a.
select the correct answer from each drop - down menu.
the sum of two rational numbers is
the sum of a rational number and an irrational number is

Explanation:

1. For the sum of two rational numbers: By definition, a rational number can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q

eq0$. Let two rational numbers be $\frac{a}{b}$ and $\frac{c}{d}$ ($b,d
eq0$). Their sum is $\frac{ad + bc}{bd}$, and since $ad + bc$ and $bd$ are integers (and $bd
eq0$), the sum is rational. So the sum of two rational numbers is a rational number.

1. For the sum of a rational and an irrational number: Assume a rational number $r=\frac{p}{q}$ and an irrational number $s$. Suppose $r + s$ is rational, say $r + s=\frac{m}{n}$. Then $s=\frac{m}{n}-\frac{p}{q}=\frac{mq - pn}{nq}$, which would make $s$ rational (since $mq - pn$ and $nq$ are integers, $nq

eq0$), contradicting that $s$ is irrational. So the sum of a rational and an irrational number is irrational.

Answer:

The sum of two rational numbers is a rational number.
The sum of a rational number and an irrational number is an irrational number.