Introduction – Surds

Rational Numbers:

Some numbers can be expressed as ratios of two integers, e.g. \( \frac {a}{b} \) where a and b are integers and b≠0, such numbers are called rational numbers.

Examples of rational numbers: \( \scriptsize 3, \; \normalsize \frac {2}{5}, \; \scriptsize 2.7 \; e.t.c \)

All integers are rational numbers since each integer n can be written in the form \( \frac{n}{1} \). For example \( \frac{5}{1} \)

A repeating decimal is always a rational number. Repeating decimals are those in which one number or a pattern of numbers repeats indefinitely. For example, the numbers 0.2222, 0.66666, 0.17171717, 1.333… and 8.1414… are repeating decimals.

Irrational Numbers:

Other numbers cannot be expressed as ratios of integers. Such numbers are called irrational numbers. They are irrational because their exact values are not determinable, only the approximate values can be known e.g.

⇒ \( \scriptsize \pi = 3.14159265359 \)

⇒ \( \scriptsize \sqrt{3} = 1.73205080757\)

⇒ \( \scriptsize \sqrt{7} = 2.64575131106 \)

Non-terminating and non-repeating decimal numbers are irrational numbers.

What are Surds?

Surds are irrational numbers. They are the roots of rational numbers whose values cannot be expressed as exact fractions.

Examples of irrational numbers: \(\scriptsize \sqrt {3}, \; \sqrt {12}, \; \sqrt {18} \; e.t.c \)

Rules of Surds:

i. \( \scriptsize \sqrt {ab} = \sqrt{a} \: \times \: \sqrt{b} \)

ii. \(\sqrt { \frac{a}{b}} = \frac {\sqrt {a}}{\sqrt {b}} \)

iii. \( \scriptsize \sqrt {a + b} \neq \sqrt{a} \: + \: \sqrt {b} \)

iv. \( \scriptsize\sqrt {a \: – \: b} \neq \sqrt{a}\: – \: \sqrt {b} \)

where: \( \scriptsize \neq \: means\: “not\: equal \: to” \)

Note, all values of a, b > 0.