40 PRACTICAL MATHEMATICS 



18 + I3i 



Example. Simplify 



(2+ 3i)(5- 6i) 

 18 + 13* 18 + 13* 



(2+8*)(5-6t) 10 +3* -18* 2 



18 + I3i 

 = 28 + 3i 



IS + 13 ^ 28 



28 + 3i 28 - 3i 

 504 + 310* - 39* 2 



784 - 9i 2 

 _ 543 + 31 Oi 



793 

 = 0-6847 + 0-3909* 



27. Extraction of the Square Root of Complex Quantities. This 

 can be done by a purely algebraic process. 



For if V bi = x yi 



Then a bi = x 2 2xyi + y 2 i 2 = x z y 2 2xyi 



Equating the real and imaginary parts, we get 



x 2 - y 2 = a 

 2xy = b 



a pair of simultaneous equations to be solved for x and y. 

 To find the square root of 21 161 



Then \/21 - I6i = x- yi 



and 21 - IQi = x z - y 2 - 2xyi 



Hence x 2 - y 2 = 2l\ 



2xy = 16) 



x* - 2x 2 y 2 + y* = 441 ^ 

 4a; 2 ?/ 2 = 256 j 

 x 4 + 2x 2 y 2 +y* = 697 



x 2 + y 2 = 26-40| 

 x 2 - y 2 = 21 J 



a? 2 = 23-70 x = 4-868 

 ?/ 2 = 2-70 y = 1-645 



C/ i/ 



Then \/21 - 16i = 4-868 - 1-645* 



If, in the complex quantity of which we wish to extract the 

 square root, the real part is negative, it is better to make it 

 positive in the following manner : 



V - a bi = V - I(a T bi) = iVa =p bi 



and we can extract the square root of a ^ bi and afterwards 

 multiply the result by i. 



