Question

rational numbers and irrational numbers. ove 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 two nonzero rational numbers is rational. |
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. For rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ (where $a,b,c,d$ are integers, $b,d

eq0$), their sum is $\frac{ad+bc}{bd}$, which is rational.

1. A rational plus irrational number cannot be rational; proof by contradiction shows this is impossible.
2. The product of irrationals can be rational (e.g., $\sqrt{2} \times \sqrt{2}=2$) or irrational (e.g., $\sqrt{2} \times \sqrt{3}=\sqrt{6}$).
3. A nonzero rational times irrational is always irrational; proof by contradiction confirms this.

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