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.

Response area with 4 blank spaces
Statement
Always,

Sometimes, or

Never True

The sum of two nonzero rational numbers is rational.
Blank space 4 empty
​

The sum of a nonzero rational number and an irrational number is rational.
Blank space 5 empty

The product of two irrational numbers is irrational.
Blank space 6 empty

The product of a nonzero rational number and an irrational number is rational.
Blank space 7 empty
​


Answer options with 3 options.

Answer 1

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
(If r is rational and x irrational, r + x would be rational ⇒ x = (r + x) − r rational, contradiction.)

The product of two irrational numbers is irrational. — Sometimes True
(e.g. √2·√3 = √6 irrational, but √2·√2 = 2 rational.)

The product of a nonzero rational number and an irrational number is rational. — Never True
(If r ≠ 0 rational and x irrational, r·x rational ⇒ x = (r·x)/r rational, contradiction.)