SURDS

CONTENT

1. Meaning of rational and irrational numbers leading to the definition of surds.
2. The rules guiding the basic operation with surd i.e \(\sqrt{a} + \sqrt{b} ≠ \sqrt{a + b}\); \(\sqrt{a} -\sqrt{b} ≠ \sqrt{a -b}\); \(\sqrt{a} × \sqrt{b} = \sqrt{a × b}\); \(\sqrt{a} ÷ \sqrt{b} = \sqrt{\frac{a}{b}}\)
3. Conjugates of a binomial surd using the idea of the difference of two squares
4. Application to solving triangles involving trigonometric ratios of special angles \(30^o, 60^o,\) and \(45^o\) .
5. Evaluation of expressions involving surds.

Meaning of Rational and Irrational Numbers Leading to the Definition of Surds

Rational Numbers (Fractions)

Rational numbers are any number that can be expressed as a ratio of two integers (i.e can be expressed as a fraction in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers and where \(b ≠ 0\). Any integer can be expressed as \(\frac{a}{1}\), hence integers are rational numbers such as \(\frac{1}{3},\frac{5}{17},\frac{7}{10},\frac{-4}{7}, \frac{9}{1},\) etc. are rational numbers. Therefore Natural numbers are subsets of Integers while Integers are subset of Rational numbers N⊂ Z ⊂ Q

Examples are:

(i) Proper and improper fractions: \(\frac{3}{4}, \frac{2}{3}\) and \(\frac{14}{9}, \frac{17}{10}\)

(ii) Mixed numbers: \(2\frac{3}{4}, 5\frac{3}{7}\)

(iii) Integers i.e counting numbers : \(0 = \frac{0}{1}, 6\frac{6}{1}, -9 = \frac{-9}{1}\)

(iv) Terminating decimals, e.g.