Calculus introduces complex numbers, which are numbers of the form x + iy, where x and y are real numbers and i = √-1. A complex number has two parts, a real part and an imaginary part. Operations like addition, subtraction, multiplication and division can be performed with complex numbers by treating i as a variable and using the property that i^2 = -1. Complex numbers can be represented graphically on a complex plane with real and imaginary axes.

2. Calculus
Introduction of
Complex Numbers

3. Complex Numbers
A number of the form x + i y, where x and y are
real numbers and ί=√-1 is called a complex
number and is denoted by z.
i.e. z = x + i y OR
can also be denoted by an ordered pair(x; y)
i.e. z = (x; y)

4. ►A complex number consisting of two parts:
a+bi
real imaginary part
►If b is 0, the complex number reduces to ‘a’
which is a pure real number.
►If a is 0, the complex number reduces to ‘bi
’ which is a pure imaginary number.
►In other words all numbers, real and
imaginary, are in the set of complex number.
For example; 3 = 3 + 0i ; - 6i = 0 – 6i

5. Imaginary Unit
You can’t take the square root of a negative
number.
If you use imaginary unit denoted by i which is
given as i = ; then you can!
Imaginary unit is used to write the square root
of a negative number.

6. Index Radicand
If ‘r’ is a positive real number, then
= i
Example:
2i

7. The power of i
If i= −1 then:

8. The Complex plane
►We modify the familiar coordinate system by calling the
horizontal axis the real axis and the vertical axis the imaginary
axis.
►Each complex number a + bi determines a unique position vector
with initial point (0, 0) and terminal point (a, b).
Real Axis
Imaginary Axis

9. Graphing in the complex plane
i34 
.
i52 
.i22 
.
i34 
.

10. ►The distance the complex number is from the origin on the
complex plane.
►If you have a complex number the absolute value
can be found using:
)( bia 
22
ba 
Examples
1. i52 
22
)5()2( 
254 
29
2. i6
22
)6()0( 
360 
36
6

11. Conjugate of Complex number
►The complex conjugate of a complex
number, z = x + jy, denoted by z* , is given
by
z* = x – jy.
By only changing the sign of the imaginary
part of the complex number

12. ►Add or subtract the real parts, then add or subtract the imaginary parts.
►For complex numbers a + bi and c + di ,
►Examples
(10  4i) - (5 - 2i)
= (10 - 5) + [4  (-2)]i
= 5 + 6i
(4 + 6i) + (3 + 7i)
= [4 + (3)] + [6 + 7]i
= 7 + 13i
       idbcadicbia 
       idbcadicbia 

13. ►Treat the i’s like variables, then change any that are not to the first power
►For complex numbers a + bi and c + di,
►The product of two complex numbers is found by multiplying as if the
numbers were binomials and using the fact that i2 = -1.
►Example:-
       ibcadbdacdicbia 
)3( ii 
2
3 ii 
)1(3  i
i31
1. Ex: )26)(32( ii 
2
618412 iii 
)1(62212  i
62212  i
i226 

14. i
i
i
i
Ex
21
21
*
21
113
:




)21)(21(
)21)(113(
ii
ii



2
2
4221
221163
iii
iii



)1(41
)1(2253



i
41
2253



i
5
525 i

5
5
5
25 i



i 5

15. Properties of Complex Numbers
►Two complex numbers a + bi and c + di are equal , if
a = c and b = d
►The following properties of real numbers hold for complex
numbers.
►Associative Properties of Addition and Multiplication
►Commutative Properties of Addition and Multiplication
►Distributive property of Multiplication over Addition