Rational or irrational?
Fractions, decimals, roots and expressions with roots: which are rational and which are not. Simplify first — the answer often hides behind the notation.
How to solve it
A rational number is one that can be written as a fraction $\frac{p}{q}$ with integers $p$ and $q \ne 0$. Its decimal expansion either terminates or repeats. The square root of a natural number that is not a perfect square is irrational, and so is almost any expression with it, unless the root cancels.
Step by step
- A terminating or repeating decimal, a fraction, an integer: rational.
- Simplify a root first: take out squares, $\sqrt{48} = 4\sqrt{3}$, and take the root of an exact square: $\sqrt{1.96} = 1.4$.
- Expand the brackets and collect like terms. If the roots cancel, $(1 + \sqrt{5})(1 - \sqrt{5}) = 1 - 5 = -4$, the number is rational.
- If $\sqrt{n}$ remains with a non-zero rational coefficient and $n$ is not a square, the number is irrational.
Common mistakes
- Judging by the look: $\frac{\sqrt{80}}{\sqrt{45}}$ “has roots” but equals $\frac{4}{3}$.
- Calling an infinite repeating decimal irrational: $0.(3) = \frac{1}{3}$ is rational.
- Thinking the sum of two irrationals is always irrational: $(2 + \sqrt{3}) + (2 - \sqrt{3}) = 4$.
- Confusing $\frac{22}{7}$ with $\pi$: $\frac{22}{7}$ is a fraction, so rational, while $\pi$ is not.