Introduction to Surds

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A Surd is an expression containing a root with an irrational solution that cannot be expressed exactly. Expressions such as √4, √25 have exact numerical values, viz. √4 = 2, √25 = 5. These expressions 4, 25 are called roots. But expressions such as √2, √3, √5 cannot be written as numerically exact quantities. For example, we might say that √2 is equal to 1.414 (correct to 3 decimal places), but we can never find an exact quantity equal to √2. Such numbers are called irrational and they are then called surds. An algebraic sum involving an irrational number along with one or more rational numbers is termed a surd.

Conjugate surds

Conjugate surds are two surds of the form a√b + c√d and a√b - c√d, where a and c are rational and √b and √d are irrational. A surd when multiplied by its conjugate results in a rational number. Hence the conjugate is often, but not always, the rationalizing factor of a surd.

If two surds a + √b and c + √d are equal, then a = c and b = d.


Consider the fraction  where the denominator is a surd. To rationalize the denominator, both numerator and denominator are multiplied by the conjugate of the denominator a-√b. The table below gives typical surds and their rationalizing factors.