Which of the following is a real number that is not rational?

• A. sqrt(2)
• B. 1/3
• C. 0.25
• D. -5

Answer: (A)

Real numbers can be rational or irrational. A rational number can be written as a fraction p/q with integers p and q (q ≠ 0). An irrational number cannot be written that way. The square root of 2 is irrational because you cannot express it as a fraction of integers; a standard contradiction shows this: if sqrt(2) = p/q in lowest terms, then p^2 = 2 q^2, which implies p is even, so p = 2k, leading to q^2 = 2k^2 and thus q is even, contradicting that p/q was in lowest terms. The other numbers can be written as fractions of integers: 1/3 is already a ratio of integers, 0.25 equals 1/4, and -5 equals -5/1. Therefore, the real number that is not rational is sqrt(2).