# Surds

We will discuss here about surds and its definition.

First let us recall about rational number and irrational number.

Rational number: A number of the form p/q, where p (may be a positive or negative integer or zero) and q (taken as a positive integer) are integers prime to each other and q not equal to zero is called a rational number or commensurable quantity.

For example, each of the numbers 7, 3/5, 0.73, √25 etc. is a rational number. Evidently, the number 0 (zero) is a rational number.

Irrational number: A number which cannot be expressed in the form p/q where p and q are integers and q ≠ 0, is called an irrational number or incommensurable quantity.

For example, each of the numbers √7, ∛3, $$\sqrt[5]{13}$$ etc. is an irrational number.

Definitions of surd: A root of a positive real quantity is called a surd if its value cannot be exactly determined.

For example, each of the quantities √3, ∛7, ∜19, (16)^2/5 etc. is a surd.

From the definition it is evident that a surd is an incommensurable quantity, although its value can be determined to any degree of accuracy. It should be noted that quantities √9, ∛64, ∜(256/625) etc. expressed in the form of surds  are commensurable quantities and are not surds (since √9 = 3, ∛64 = 4, ∜(256/625) = 4/5 etc.). In fact, any root of an algebraic expression is regarded as a surd.

Thus, each of √m, ∛n, $$\sqrt[5]{x^{2}}$$ etc. may be regarded as a surd when the value of m ( or n or x) is not given. Note that √m = 8 when m = 64; hence, in this case √m does not represent a surd. Thus, √m does not represent surd for all values of m.

Note: All surds are irrationals but all irrational numbers are not surds. Irrational numbers like π and e, which are not the roots of algebraic expressions, are not surds.