What is Complex Number?
Complex numbers are defined as expressions of the form x = iy
where x, y ∈ R, and i = √-1 . It is denoted by z i.e. z= x + iy
The numbers x and y are called respectively real and imaginary parts of complex number z.
I.e. x = Re (z) and y = Im (z)
Purely Real and Purely Imaginary Complex Number
A complex number z is a purely real, if its imaginary part is 0.
i.e. Im (z) = 0. And purely imaginary, if its real part is 0 i.e. Re (z) = 0.
- The Set R of real numbers is a proper subset of the Complex Numbers. Hence the Complex Number system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C.
- Zero is both purely real as well as purely imaginary but not imaginary.
- √x √y = √xy Only if atleast one of either x or y is non negative.
- i = √-1 is called the imaginary unit. Also i2 = -1; i3 = -i; i4 = 1 etc.
Equality of Complex Numbers
Two complex numbers Z1 = a1 + ib1 and z2 = ac2 + ib2 are equal, iff a1 = a2 and b1 = b2 i.e. Re (Z1) = Re (z2) and Im (Z1) = Im (z2).
Addition and subtraction of complex numbers
Addition of complex numbers is defined by separately adding real
and imaginary parts; so if
z = a + bi, w = c + di
z + w = (a + c) + (b + d)i .
Similarly for subtraction.
Example complex numbers
QUESTION 1. Express each of the following in the form
x + yi.
(a) 3 + 5i + =2 − 3i
(b)3 + 5i + 6
(c) 7i − = 4 + 5i
(a) 3 + 5i + ( ) 2 − 3i = 3 + 2 + ( ) 5 − 3 i = 5 + 2i
(b)3 + 5i + 6 = 9 + 5i
(c)7i − ( ) 4 + 5i = 7i − 4 − 5i = −4 + 2i
Multiplication of complex numbers
Multiplication is straightforward provided you remember that
i2 = −1.
Simplify in the form x + yi :
(a) 3( 2 + 4i)
(b) (5 + 3i)i
(c) (2 − 7i ) (3 + 4i)
(a) 3 (2+ 4i)
= 3 ( 2) + 3 (4i)
= 6 +12i
(b) (5 + 3i) i
= (5 i +3 (i) i
= 5i + 3 (i2)
= 5i + −(1 ) 3
= − 3 + 5i
(c) (2 − 7i) (3 + 4i )
= ( 2) (3 ) − (7i ) ( 3) + ( 2) (4i ) − ( 7i) ( 4i)
= 6 − 21i + 8i − −( 28)
= 6 − 21i + 8i + 28
= 34 −13i
In general, if
z = a + bi ,
w = c + di,
z w = (a + bi)(c + di)
= a c − b d + (a d + b c)i
Division of complex numbers
The complex conjugate of a complex number is obtained by
changing the sign of the imaginary part. So if
z = a + bi, its
complex conjugate, z , is defined by
z = a − bi
Any complex number
a + bi has a complex conjugate
a − bi
and from Activity 5 it can be seen that
( a + bi ) (a − bi ) is a real
number. This fact is used in simplifying expressions where the
denominator of a quotient is complex.
Simplify the expressions:
To simplify these expressions you multiply the numerator and
denominator of the quotient by the complex conjugate of the
(a) The complex conjugate of i is −i, therefore