Theorem

$n$ is irrational unless $n$ is a perfect square

Arithmetic • Number System • Rational Numbers

Formula

n rational ⟺ n a perfect square

For a positive integer

n

n

is rational if and only if

n

is a perfect square. For every other

n

it is irrational, and then so is any non-zero rational multiple of it and any rational added to it.

How to use it

1Test $n$ for being a perfect square; if it is not, the root is irrational and nothing further need be computed.
2Take out the largest square factor first — $75 = 53$ — and read the verdict off what is left under the root.
3Combine with the closure facts: $53$ , $2 + 3$ and $\frac{3}{4}$ are all irrational because each is a rational operation applied to one irrational.

45

rational or irrational?

Step-by-step

1
$45 = 9 \times 5$ , so $45 = 35$
Why: Pull out the largest square factor.
2
$5$ is not a perfect square, so $5$ is irrational
Why: Apply the theorem to what remains under the root.
3
$35$ is a non-zero rational times an irrational, so it is irrational

Answer

Irrational

Watch out

The test is perfect squares, not primes:

4

and

36

are rational although

4

and

36

are composite. The theorem also says nothing about a combination of two different surds —

3 + 2

is irrational, yet

(3 + 2) (3 - 2) = 1

is rational.

Builds on

ab = a b

a / b = a / b

Also known as

root 2 is irrational, square root of a non-perfect square is irrational, surd irrationality, is sqrt(n) rational, irrational number test