Which number is irrational?

Which number is irrational?

• A. sqrt(9) = 3
• B. sqrt(5)
• C. 1/3
• D. 0

Answer: (B)

The main idea is how we tell apart rational numbers from irrational ones. A number is rational if it can be written as a fraction of two integers. The square root of a number that isn’t a perfect square usually can’t be written that way, so it’s irrational. For the square root of five, try to express it as p divided by q in lowest terms. Then squaring both sides gives 5 q^2 = p^2, which means p^2 is a multiple of 5, so p must be a multiple of 5. Let p = 5k. Substituting back leads to q^2 = 5 k^2, so q is also a multiple of 5. That means p and q share a factor, contradicting the assumption that the fraction was in lowest terms. Hence sqrt(5) cannot be written as a ratio of integers, so it is irrational. The other options are rational: the square root of nine equals three, an integer; one over three is a simple fraction; and zero is 0/1. Their decimal expansions terminate or repeat, unlike sqrt(5), which goes on without repeating.

The main idea is how we tell apart rational numbers from irrational ones. A number is rational if it can be written as a fraction of two integers. The square root of a number that isn’t a perfect square usually can’t be written that way, so it’s irrational. For the square root of five, try to express it as p divided by q in lowest terms. Then squaring both sides gives 5 q^2 = p^2, which means p^2 is a multiple of 5, so p must be a multiple of 5. Let p = 5k. Substituting back leads to q^2 = 5 k^2, so q is also a multiple of 5. That means p and q share a factor, contradicting the assumption that the fraction was in lowest terms. Hence sqrt(5) cannot be written as a ratio of integers, so it is irrational.

The other options are rational: the square root of nine equals three, an integer; one over three is a simple fraction; and zero is 0/1. Their decimal expansions terminate or repeat, unlike sqrt(5), which goes on without repeating.