Times and divide by a rational

The idea

The product and the quotient of a non-zero rational number and an irrational number are irrational.

3 times root 2

We know √2 is irrational. Multiply it by the non-zero rational 3.

Is 3√2 rational or irrational?

• Rational
• Irrational
• Not sure

Test it by contradiction

Assume 3√2 is rational, then climb.

Product and quotient

The product and the quotient of a non-zero rational number and an irrational number are irrational.

Why “non-zero”? Because 0 × √2 = 0, and 0 is rational.

The same idea works for division: if √5 divided by 2 were a fraction a/b, then √5 would be 2 times that fraction, which is rational.

Notes

The product and quotient of a non-zero rational number and an irrational number are irrational. Example: 3√2 is irrational.

Check yourself

Which of these is irrational?

Why must the rational number be non-zero?

• 5√3 — correct. Yes! A non-zero rational times an irrational is irrational.
• 0 × √3. 0 × √3 = 0, which is rational. The rational must be non-zero.
• √3 × √3. √3 × √3 = 3, which is rational. Both factors are irrational, so the rule does not apply.
• 2 × 6. 12 is rational.
• 0 times an irrational number is 0, which is rational — correct. Yes! That is why the rule says non-zero.
• Zero is irrational. Zero is rational: 0 = 0/1.
• We cannot divide by 1. We can. The problem is multiplying by 0.