Which statement best describes the sum of a rational number and an irrational number?

Which statement best describes the sum of a rational number and an irrational number?

• A. True
• B. False
• C. Sometimes
• D. It depends on numbers

Answer: (A)

The sum of a rational number and an irrational number is always irrational. Reason: Let r be rational and i be irrational. If r + i were rational, then i = (r + i) − r would be the difference of two rational numbers, which is rational. That would make i rational, contradicting that i is irrational. Therefore r + i cannot be rational; it must be irrational. For example, 3 + √2 is irrational, and in general adding any fixed rational amount to an irrational number cannot produce a rational result.

The sum of a rational number and an irrational number is always irrational.

Reason: Let r be rational and i be irrational. If r + i were rational, then i = (r + i) − r would be the difference of two rational numbers, which is rational. That would make i rational, contradicting that i is irrational. Therefore r + i cannot be rational; it must be irrational.

For example, 3 + √2 is irrational, and in general adding any fixed rational amount to an irrational number cannot produce a rational result.