Question

consider the sums and products of nonzero rational numbers and irrational numbers.
move the options to the spaces to show whether each statement is always true, sometimes true, or never true.

statement	always, sometimes, or never true
the sum of a nonzero rational number and an irrational number is rational.
the product of two irrational numbers is irrational.
the product of a nonzero rational number and an irrational number is rational.

always true
sometimes true
never true

Explanation:

1. Analyze "The sum of two nonzero rational numbers is rational."

• Recall the definition of rational numbers: A rational number can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q

eq0$. Let the two nonzero rational numbers be $\frac{a}{b}$ and $\frac{c}{d}$ (where $a,b,c,d$ are integers, $b
eq0$, $d
eq0$, $a
eq0$, $c
eq0$).

• The sum is $\frac{a}{b}+\frac{c}{d}=\frac{ad + bc}{bd}$. Since $a,b,c,d$ are integers, $ad+bc$ and $bd$ are also integers. Also, $bd

eq0$ (because $b
eq0$ and $d
eq0$). So the sum is a rational number. Thus, this statement is always true.

2. Analyze "The sum of a nonzero rational number and an irrational number is rational."

• Let the nonzero rational number be $r=\frac{p}{q}$ ( $p,q$ integers, $q

eq0$, $p
eq0$) and the irrational number be $s$ (which cannot be written as a fraction of two integers).

• Assume $r + s$ is rational, so $r + s=\frac{m}{n}$ ( $m,n$ integers, $n

eq0$). Then $s=\frac{m}{n}-r=\frac{m}{n}-\frac{p}{q}=\frac{mq - pn}{nq}$. But this would mean $s$ is rational (since $mq - pn$ and $nq$ are integers and $nq
eq0$), which contradicts the definition of an irrational number. So the sum of a nonzero rational number and an irrational number is always irrational. Thus, this statement is never true.

3. Analyze "The product of two irrational numbers is irrational."

• Consider examples:
• Example 1: $\sqrt{2}\times\sqrt{2}=2$, where $\sqrt{2}$ is irrational and their product is rational.
• Example 2: $\sqrt{2}\times\sqrt{3}=\sqrt{6}$, where both $\sqrt{2},\sqrt{3}$ are irrational and their product is also irrational.
• So the product of two irrational numbers can be either rational or irrational. Thus, this statement is sometimes true.

4. Analyze "The product of a nonzero rational number and an irrational number is rational."

• Let the nonzero rational number be $r=\frac{p}{q}$ ( $p,q$ integers, $q

eq0$, $p
eq0$) and the irrational number be $s$.

• Assume $r\times s$ is rational, so $r\times s=\frac{m}{n}$ ( $m,n$ integers, $n

eq0$). Then $s=\frac{m}{n}\div r=\frac{m}{n}\times\frac{q}{p}=\frac{mq}{np}$. But this would mean $s$ is rational (since $mq$ and $np$ are integers and $np
eq0$), which contradicts the definition of an irrational number (unless we have a miscalculation, but in general, multiplying a nonzero rational with an irrational gives an irrational). Let's take an example: $2\times\sqrt{2}=2\sqrt{2}$ (irrational). So the product of a nonzero rational number and an irrational number is always irrational. Thus, this statement is never true.

Final Matching:

• "The sum of two nonzero rational numbers is rational." → always true
• "The sum of a nonzero rational number and an irrational number is rational." → never true
• "The product of two irrational numbers is irrational." → sometimes true
• "The product of a nonzero rational number and an irrational number is rational." → never true

Answer:

• The sum of two nonzero rational numbers is rational. : always true
• The sum of a nonzero rational number and an irrational number is rational. : never true
• The product of two irrational numbers is irrational. : sometimes true
• The product of a nonzero rational number and an irrational number is rational. : never true