What is a surd in mathematics?

A surd is an irrational number that can't be simplified to remove a square root (or cube root, etc.).

In mathematics, surds are expressions that include roots which cannot be simplified into a rational number. For example, √2 is a surd because it cannot be expressed as a fraction or a terminating or repeating decimal. Surds are important because they allow us to work with exact values rather than approximations, which is crucial in many areas of mathematics.

When dealing with surds, you might encounter expressions like √3, √5, or even more complex forms like 3√2 + 2√3. These expressions remain in their root form because simplifying them would result in an irrational number. For instance, √2 is approximately 1.41421356..., a non-repeating, non-terminating decimal, which means it cannot be written as a simple fraction.

In GCSE Maths, you'll often be asked to simplify surds, which means expressing them in their simplest form. For example, √50 can be simplified to 5√2 because 50 is 25 × 2, and √25 is 5. You'll also learn how to rationalise the denominator, which involves removing surds from the bottom of a fraction. For instance, to rationalise 1/√2, you multiply the numerator and the denominator by √2 to get √2/2.

Understanding surds is essential for solving various mathematical problems, including those involving geometry, algebra, and trigonometry. They help maintain precision in calculations and are a fundamental concept in higher-level mathematics.