Learning Ashram

COMPLEX NUMBERS

 

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.
The complex numbers a + bi and a — bi are called complex conjugates of each other.

 
 
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

 
 


A complex number a + bi can be plotted as a point (a, b) on an xy plane called an Argand diagram or Gaussian plane.
A complex number can also be interpreted as a vector from O to P.

 
 
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 +
iy = r(cos θ + i sin θ)

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, . . .,n-l.