Download Complex numbers

First year:
(I)
Complex numbers
Definition, addition and multiplication
A complex number is a number of the form a  ib , where a and b are real
numbers and where i is an imaginary number such that
i  1 .
Note that since i  1 , we have i 2  1 .
Let z  a  ib .
-
The real part of z is a . We write Re( z )  a .
The imaginary part of z is b . We write Im( z )  b .
(1) Adding complex numbers:
Say z1  a  ib and z2  c  id . Then z1  z2  a  c  i(b  d ).
Example: z1  1  i3,
z2  2  i5  z1  z2  (1  2)  i(3  5)  1  i2.
(2) Multiplying complex numbers:
Say z1  a  ib and z2  c  id . Then z1 z2  ac  bd  i(bc  ad ) .
Example: z1  1  i2,
z2  2  i3  z1 z2  (1  i2)(2  i3),
 z1 z 2  2  i3  i 2  2  (i 2)  (i3),

z1 z 2  2  i3  i 4  i 2  2  3,

z1 z 2  2  6  i (4  3)  4  i 7.
(3) Exercises
(i)
Find z1  z2 and z1 z2 , where
z1  1  i3, z2  2  i4 ,
z1  1  i3, z2  2  i 4,
z1  1  i, z2  1  i 2.
(ii)
Find the real and imaginary part of z1 ,
where z1  (1  i3)(2  i7)  i.
(II)
Modulus and complex conjugate
The modulus of a complex number z  a  ib is z  a 2  b 2 .
For example if z  1  i3 , then z  12  32  10 .
The complex conjugate of z  a  ib is z  a  ib .
For example if z  1  i3 , then z  1  i3 .
Note that
z  z z,
2
2
for, if z  a  ib , then z z  (a  ib )( a  ib )  a 2  iab  iba  (ib )( ib )  a 2  b 2  z .
Complex fractions:
You will often be asked to find the real and imaginary parts of complex
numbers of the form
z
1
.
1  i2
To do so, you need to write z in the form a  ib :
-
Multiply the numerator and the denominator by the complex conjugate of
the denominator
z
-
Make use of the equality z  z z :
2
z
-
(1  i 2)
(1  i 2)(1  i 2)
1  i2 1  i2

5
11  2 2
Finally split the fraction into real and imaginary parts:
1 2
z  i
5
5
and conclude: Re( z ) 
1
2
, Im( z ) 
.
5
5
(III)
Polar form and De Moivre’s formula
z  a  ib is called the Cartesian form of the complex number z .
If the real numbers a are represented on a horizontal axis and the complex
numbers ib are represented on a vertical axis, then you can locate the
a
complex number z  a  ib the way you would locate a vector   in R 2 .
b
Another way of representing a complex number is to give its modulus z
together with the angle  between the line [0, z ) and the positive semi-axis
[0,) .
Note: the angle  is called the argument of the complex number z and is
measured anti-clockwise.
Once you have the modulus and the argument of a complex number
z  a  ib , you can write it in polar form:
z  re i ,
with
r  z  a 2  b 2 ,   tan 1 (b / a) and
ei  cos( )  i sin(  ).
Example:
Say z  4  i3 . Then r  42  32  5 and   tan 1 (3 / 5)  0.54 rad  30.96o , so
that
z  5e i 0.54 .
The angle  is usually expressed in radian.
De Moivre’s formula
z  r cos( )  i sin(  ) 
z n  r n cos(n )  i sin( n )   r n ein .
Example:
z  3cos( / 3)  i sin(  / 3)  z 3  27cos( )  i sin(  )   27.