What Are Irrational Numbers?
Numbers can be split into two major families: rational and irrational. You already know rational numbers well — they are any numbers that can be written as a fraction where and are integers and . This includes whole numbers, terminating decimals, and repeating decimals.
Irrational numbers, however, cannot be expressed as a simple fraction. Their decimal expansions go on forever without repeating or terminating.
A number that cannot be written in the form where and are integers and . Its decimal expansion is infinite and non-repeating.
Some famous examples:
- — never ends, never repeats
- — the base of natural logarithms
- — never ends, never repeats
Think of rational numbers as a perfectly tiled floor — neat, predictable, repeating patterns. Irrational numbers are like the jagged edge of a coastline — no matter how closely you zoom in, the pattern never repeats exactly.
Together, rational and irrational numbers form the set of real numbers, which covers every point on the number line.
Introducing Surds
A surd is a specific type of irrational number — it is a root (square root, cube root, etc.) that cannot be simplified to a rational number.
An irrational root of a rational number. For example, , , are surds because they cannot be expressed as exact fractions or terminating decimals.
Not every square root is a surd! If the answer comes out as a whole number (or a rational number), it is not a surd:
| Expression | Value | Surd? |
|---|---|---|
| ✗ No — it's rational | ||
| ✗ No — it's rational | ||
| ✓ Yes | ||
| ✓ Yes | ||
| ✗ No — it's rational | ||
| ✓ Yes |
A quick test: if the number under the root sign (the radicand) is a perfect square (or perfect cube for cube roots), the result is rational — not a surd.
Perfect squares to memorise:
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