The difference of a rational and irrational number is

• A. Irrational
• B. Rational
• C. Zero
• D. Undefined

Answer: (A)

The main idea is that subtracting an irrational number from a rational number always gives an irrational result.

Why this is true: if you take a rational number r and subtract an irrational number i, and suppose the result r − i were rational, then you could solve for i as i = r − (r − i). Since r and (r − i) would both be rational, their difference would be rational, which would make i rational—a contradiction because i is irrational. Therefore the result cannot be rational; it must be irrational. In particular, it can’t be zero, since that would require the irrational to equal the rational, which isn’t possible. Subtraction of real numbers is always defined, so the result isn’t undefined either.

A quick example: 3 − √2 is irrational, illustrating the general rule.