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Relatively Prime ( Co-Prime ) : Two positive integers are said to be relatively prime to each other if their highest common factor ( H.C.F. ) is 1.

(a, b) = H.C.F. of a and b & [a, b] = L.C.M. of a and b.

e.g. 18 and 25 are co-primes. Because (18, 25) = 1.

Rationalization : Consider a fraction of type $\frac{1}{\sqrt{3}} or \frac{1}{\sqrt{5}+\sqrt{3}}$ . Here, the denominator is an irrational number. Its very difficult to perform any mathematical operation using such numbers. Hence we convert the denominator of these numbers to a rational number. This process is called as rationalization.

e.g. (i) $\frac{1}{\sqrt{3}}=\frac{1}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

(ii) $\frac{1}{\sqrt{5}+\sqrt{3}}=\frac{1}{\sqrt{5}+\sqrt{3}}\times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}} = \frac{\sqrt{5}-\sqrt{3}}{5-3}=\frac{\sqrt{5}-\sqrt{3}}{2}$

(iii) $\frac{1}{\sqrt{7}-\sqrt{2}}=\frac{1}{\sqrt{7}-\sqrt{2}}\times \frac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}} = \frac{\sqrt{7}+\sqrt{2}}{7-2}=\frac{\sqrt{7}+\sqrt{2}}{5}$

Conjugates of a Complex Numbers : If (a + ib) is any given complex number, then the conjugate of ( a + ib ) is ( a – ib ) and is denoted by $\overline{a+ib}$ .