COMPLEX QUANTITIES 



39 



Thus the first four powers of i give i, - I, i, and 1 ; the 

 next four powers give the same quantities in the same order, as 

 also will the next four powers. It follows, then, that any positive 

 integral power of i will give 1 when the power is even and i 

 when the power is odd. 



Negative Powers : 



-i = i = _L _L _ 



i-2 - l = l 

 i 2 - 1 



1 1 



i 



H 



= i 



i-' = 



= i 



Thus the negative integral powers of i will give 1 when the 

 power is even and i when the power is odd. The complex 

 quantity can be treated algebraically, provided the treatment is 

 combined with a knowledge of the values of the different powers 

 of i. 



25. Multiplication of Complex Quantities. They can be multi- 

 plied algebraically and the value of i 2 put, where it occurs, in the 

 result. 



Thus (5 - 8t)(2 + 5t) = 10 + 9t - 40i 2 



= 50 + 9i since i 2 = - 1 

 Also (5 - 8i)(2 + 5i)(2 - 8i) = (50 + 9i)(2 - 3i) 



= 100 - 132i - 27i 2 

 = 127 - 132i 



When two complex quantities are multiplied together, care 

 should be taken to reduce the product to the form a + bi before 

 multiplying by a third complex quantity. 



26. Division of Complex Quantities. If we consider the com- 

 plex quantities a + bi and a bi, we notice that the product is 

 a 2 b 2 i 2 , which reduces to a 2 + b 2 , and this provides us with a 

 means of removing the imaginary term from a complex quantity. 



Hence if we wish to divide 50 + Qi by 5 8i we can represent 



50 -f- Qi 

 the process by the fraction -. and simplify the fraction. 



O ~ ot> 



By multiplying numerator and denominator of the fraction by 

 5 + 8i, we can make the denominator entirely real without altering 

 the value of the fraction. 



Then 



50 



250 



5 - 8i 5+8i 



25 - 64i 2 

 178 + 445e 

 89 



2+5* 



