COMPLEX NUMBERS 

DEFINITIONS INVOLVING COMPLEX NUMBERS


A complex number is generally written as a + bi where o and 6 are real numbers and i, called the imaginary unit, has the property that i2 = —1. The real numbers a and 6 are called the real and imaginary parts of a + bi respectively. 

EQUALITY OF COMPLEX NUMBERS


a + bi = c + di if and only if a = c and 6 = d  
ADDITION OF COMPLEX NUMBERS


(a + 6i) + (c + di) = (a + c) + (6 + d)i  
SUBTRACTION OF COMPLEX NUMBERS


(a + bi)  (c + eft) = (a  c) + (6  d)i  
MULTIPLICATION OF COMPLEX NUMBERS


(a + bi)(c + di) = (ac — bd) + (ad + bc)i  
DIVISION OF COMPLEX NUMBERS


Note that the above operations are obtained by using the ordinary rules of algebra and replacing t2 by 1 wherever it occurs. 

GRAPH OF A COMPLEX NUMBER 



POLAR FORM OF A COMPLEX NUMBER


Point P with coordinates (x, y) represents the complex number x + iy. Point P can also be represented by polar coordinates (r, θ). Since x= r cos θ, y = r sin θ we have z + called the polar form of the complex number. We often call r = \/x² + y² the modulus and θ the amplitude of x + iy. 

MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS IN POLAR FORM


DE MOIVRE'S THEOREM


If p is any real number, De Moivre's theorem states that [r(cos θ + i sin θ)]p = rp(cos pθ + i sin pθ) 

ROOTS OF COMPLEX NUMBERS 

If p =1/n where n is any positive integer, where k is any integer. From this the n nth roots of a complex number can be obtained by putting k = 0,1,2, . . .,nl. 
