CHAPTER III 



22. The Complex Quantity. The quantity V 9 can be taken 

 as V9 x 1, which reduces to 3V 1. Treating V 8 in the 

 same way, it becomes 2-828V 1, the number being correct to 

 four significant figures. Thus the square root of any negative 

 quantity can be reduced to the form bV 1 or bi where b is a 

 real number which can be exact, or given correct to as many 

 significant figures as desired. The result of multiplying a real 

 quantity by i or V 1 is to make the product imaginary. 



Many quadratic equations are spoken of as having imaginary 

 roots, but it is only in a few special cases for which the roots 

 are wholly imaginary. 



Taking the quadratic equation x 2 16x + 100 = 

 Then x* - IGx + 64 = - 100 + 64 = - 36 



x - 8 = 6i 

 and x = 8 + 6i or 8 6i 



We can thus have quantities consisting of two distinct parts, 

 a real part and an imaginary part. Such quantities are spoken 

 of as being complex. 



A complex quantity can be expressed generally in the form 

 a + bi, where a and b are numbers which can be exact or given 

 correct to so many significant figures. 



23. Two complex quantities can only be equal providing the 

 real parts are equal and the imaginary parts are equal. 



Thus if a + bi = c + di, then a = c and b = d. 

 For, if not, suppose a > c, then a = c + x, where x is the differ- 

 ence between two real quantities and must therefore be real. 



Then c+x+bi = c+di 



x = di bi 



This makes x imaginary, because it is equal to the difference 

 of two imaginary quantities, but x must be real, therefore a cannot 

 be greater than c. In the same way it can be shown that a 

 cannot be less than c. Hence a must be equal to c. 

 If a = c, then b must be equal to d. 



24. The Powers ofi. Positive Powers : 



= ^ 



i 6 = i 2 x i* = i 2 = - 1 



38 



i 9 = i x i 8 = i = I 



"12 *4 *8 *4 "I 



