Surd

A surd is a radical expression whose value cannot be simplified to a rational number. In common mathematical usage, surds refer to irrational roots, especially square roots of integers that are not perfect squares. The term originated in early modern algebra and is used to distinguish irrational radical expressions from rational numbers.

For square roots, a surd is typically written as √d with d a positive integer that is not a perfect square (for example, √2, √3, √5). If the radicand has a square factor, the radical can be simplified: √18 = 3√2. After simplification, expressions with square-free radicands (the canonical surd form) are usually regarded as the surd itself. Higher roots, such as ∛7, are also surds when the radicand is not a perfect power of the appropriate degree.

Surds are manipulated according to standard algebraic rules: like surds can be added or subtracted only if they have the same radicand, surds multiply and divide with simplification, and surds in denominators can be rationalized by multiplying numerator and denominator by a suitable factor. Rationalization often aims to remove radicals from denominators.

Important distinctions exist: every surd is an irrational number, but not every irrational number is a surd. For example, π is irrational but not a surd, since it is not a radical expression. Conversely, √2 is an irrational number that is also a surd in its simplest radical form.