Surd definition
A surd is an exact root that cannot be simplified into a whole number or a terminating decimal. In GCSE maths, surds are most often square roots such as the square root of 2, the square root of 3 or the square root of 5. These values are irrational, which means their decimal forms go on forever without repeating in a regular pattern. Writing them as surds keeps the answer exact.
Not every square root is a surd. The square root of 9 is 3, so it is not a surd because it simplifies to a whole number. The square root of 16 is 4, so it is also not a surd. However, the square root of 8 is a surd because it does not simplify to a whole number, although it can be simplified into a neater exact form.
Surds are used in number, algebra, geometry, trigonometry, Pythagoras’ theorem, exact values, indices and problem solving. At GCSE level, students simplify surds, multiply surds, expand brackets involving surds and rationalise denominators. At A-Level, surds continue to appear in algebraic manipulation, exact trigonometric values, coordinate geometry, calculus and proof-style questions.
Simplifying a surd means writing it in the simplest exact form. The key method is to look for a square factor. For example, the square root of 18 can be written as the square root of 9 times the square root of 2. Since the square root of 9 is 3, the simplified form is 3 times the square root of 2. This is exact and simpler than the original form.
Surds can be multiplied when they have the same type of root. For example, the square root of 2 multiplied by the square root of 8 equals the square root of 16, which equals 4. Another useful idea is that a surd multiplied by itself can become rational. The square root of 5 multiplied by the square root of 5 equals 5. This idea is used when simplifying expressions and rationalising denominators.
Rationalising a denominator means removing a surd from the bottom of a fraction. For example, if a fraction has the square root of 3 in the denominator, multiplying the top and bottom by the square root of 3 creates a denominator of 3. The value of the fraction has not changed because the numerator and denominator were multiplied by the same value. The result is usually easier to work with exactly.
Surds are important because they allow exact answers. A calculator decimal for the square root of 2 is only an approximation, even if it shows many digits. In geometry, exact surd answers are often better than rounded decimals because rounding too early can make later calculations inaccurate. This is especially important in Pythagoras’ theorem, trigonometry and coordinate geometry.
Related ideas include square roots, irrational numbers, indices, powers, exact values, simplifying, rationalising denominators and Pythagoras’ theorem. A clear definition of surd helps students understand why some roots are left in root form rather than changed into decimals. In exams, look for square factors, keep answers exact when requested and avoid rounding unless the question asks for a decimal approximation.