This document provides information about complex numbers. It begins by introducing complex numbers and defining them as numbers of the form a + bi, where a and b are real numbers and i = √-1. It then discusses various representations of complex numbers including Cartesian (a + bi), polar (r(cosθ + i sinθ)), and Euler (re^iθ) forms. It also covers operations on complex numbers such as addition, subtraction, multiplication, division, and powers. De Moivre's theorem relating powers of complex numbers to trigonometric functions is presented. The document concludes by stating some basic algebraic laws for complex numbers.

1. Complex number
Course- Diploma
Semester-II
Subject- Advanced Mathematics
Unit- II
RAI UNIVERSITY, AHMEDABAD

2. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
Introduction— Indian mathematician Mahavira (850 A.D.) was first to mention in his work
'Ganitasara Sangraha'; 'As in nature of things a negative (quantity) is not a square (quantity), it
has, therefore, no square root'. Hence there is no real number which satisfies the polynomial
equation + 1 = 0.
A symbol√−1, denoted by letter “i” was introduced by Swiss Mathematician, Leonhard Euler
(1707-1783) in 1748 to provide solutions of equation + 1 = 0. “i” was regarded as a
fictitious or imaginary number which could be manipulated algebraically like an ordinary real
number, except that its square was – 1. The letter “i” was used to denote√−1 , possibly
because “i” is the first letter of the Latin word 'imaginarius'.
2.1Definition— A number of the form = + , [ , ∈ & = √−1 ], is called a complex
number.
Figure 2.01 represents the complex plane. It consists of a Real axis and an Imaginary axis.
2. 2 Real & Imaginary part of a complex number—
Let = + is a complex number. Then it’s real part & imaginary part is given by—
( ) = ( ) =
2. 3 Representation of a complex number = + –
A complex number = + can be represented on the co-ordinate as given below—
Fig - 2.01
This system of representing complex number is
also called as Argand diagram.

3. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
2. 4 Modulus of a complex number— Let = + is a complex number,
then its modulus is given by—
| |= + .
Example— Find the modulus of the complex number = + .
Solution— Given complex number is z = 3 + 4i.
Since, modulus of z = x + iy is given by | |= +
Hence, required modulus of z is | |=√3 + 4 =√9 + 16 = √25
∴ | | = 5
2. 5 Argument of a complex number—
Let = + is a complex number,
then its argument is given by—
arg( ) = = tan .
Note— is measured in radians and positive in the counterclockwise sense.
For = , ( ) is not defined.
The value of that lies in the interval − ≤ ≤ is called as Principal value of the argument
of (≠ 0).
Hence, − ≤ ( ) ≤
Example— Find the argument of the complex number = + .
Solution— given complex number is z = 1 + i.
Since, argument of z = x + iy is given by arg( ) = tan and the point (1,1) lies in first
quadrant.
Hence, arg( ) = tan = tan (1) =
Example— Find the argument of the complex number = − + .
Solution— given complex number is z = −1 + i.
Since, argument of z = x + iy is given by arg( ) = tan and the point (−1,1) lies in second
quadrant.
Hence, arg( ) = tan = tan (−1) = − =
Example— Find the argument of the complex number = − − .
Solution— given complex number is z = −1 − i.
Since, argument of z = x + iy is given by arg( ) = tan and the point (−1, −1) lies in third
quadrant.
Hence, arg( ) = tan = tan (1) = − + =
Example— Find the argument of the complex number = − .
Solution— given complex number is z = 1 − i.
Fig - 2.02

4. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
Since, argument of z = x + iy is given by arg( ) = tan and the point (1, −1) lies in fourth
quadrant.
Hence, arg( ) = tan = tan (−1) = −
2.6 Cartesian representation of a complex number—
A complex number of the form = + is known as Cartesian representation of complex
number. This can also be written in paired form as ( , ). Cartesian representation of complex
number = + is shown in Fig- 1.02.
2.7 Polar representation of a complex number—
A complex number of the form = + can be represented in polar form as = ( +
). Where and is given by—
= + and
= tan
This is shown in Fig- 2.03.
2.8 Euler representation of a complex number—
A complex number of the form = + or = ( + ) can be represented in
“Euler form” form as---
=
Where and is given by—
= + , = tan
This is shown in Fig- 2.03.
2.9 Conversion of a complex number given in cartesian system to polar system—
If = + is a complex number given in Cartesian system then it can be written in polar
form by writing—
= ( + )
Where = + and = tan .
Example— Convert the following complex numbers in polar form—
( ) + ( ) + ( ) − + ( ) + √
Solution—
( ) Given complex number is = 3 + 4 ,
Hence, = √3 + 4 = √9 + 16 = √25 = 5 ∴ = 5 and
= tan = tan
Fig - 2.03

5. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
Thus, = 3 + 4 can be written in polar form as = 5( + ), where = tan
( ) Given complex number is = 1 + ,
Hence, = √1 + 1 = √1 + 1 = √2 ∴ = √2 and
= tan = tan =
Thus, = 1 + can be written in polar form as = √2( + )
( ) Given complex number is = −1 + ,
Hence, = (−1) + 1 = √1 + 1 = √2 ∴ = √2 and
= tan = tan =
Thus, = −1 + can be written in polar form as = √2( + )
( ) Given complex number is = 1 + √3 ,
Hence, = (1) + (√3) = √1 + 3 = √4 = 2 ∴ = 2 and
= tan = tan
√
=
Thus, = 1 + √3 can be written in polar form as = 2( + )
2.10 Conversion of a complex number given in polar system to cartesian system —
Example— Convert the followings complex numbers in Cartesian form—
( ) ( ) √2 ( )√3 ( ) √2
Solution—
( ) Given complex number is .
Here, = 1 = , let = + is Cartesian representation.
Then, = = 1. = 0 and
= = 1.
2
= 1
Fig - 2.04
If = (cos + ) is a complex
number given in Polar system then it
can be written in Cartesian system as—
= +
Where = and = .

6. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
Thus, (1, ) can be written in Cartesian form as = 0 +
( ) Given complex number is √2
Here, = √2 = , let = + is Cartesian representation.
Then, = = √2. = 1 and = = √2. = 1
Thus, √2 can be written in Cartesian form as = 1 +
( ) Given complex number is √3
Here, = √3 = , let = + is Cartesian representation.
Then, = = √3. =
√
and
= = √3. =
Thus, √3 can be written in Cartesian form as =
√
+
( ) Given complex number is √2
Here, = √2 = , let = + is Cartesian representation.
Then, = = √2. = −1 and
= = √2.
3
4
= 1
Thus, √2 can be written in Cartesian form as = −1 +
2.10 Conjugate of a complex number—
Example— Find the conjugate of the complex number = + .
Solution— Given complex number is z = 2 + 3i.
Since, complex conjugate of z = x + iy is given by = − .
Hence, required complex conjugate is = − .
2.11 Addition of two complex numbers—
Fig - 1.05
Let a complex number is given by = + , then the
conjugate of complex number is denoted by ̅ and it is given
by = − .
A complex number = + and its conjugate = − is
represented in Argand plane as shown in Fig- 2.05.
Hence,
⇒ ( ) = = ( + ) & ( ) = = ( − )

7. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
Let = + and = + are two complex numbers, then addition of these two
complex numbers and is given by—
= ( + ) + ( + )
Example— Find the addition of the complex numbers = + and = + .
Solution— given complex numbers are = 2 + 3 and = 4 + 2 .
Hence, + =(2 + 4) + (3 + 2)
∴ = 6 + 5
2.12 Subtraction of two complex numbers—
Let = + and = + are two complex numbers, then subtraction of complex
numbers from is given by—
= ( − ) + ( − )
Example— Subtract the complex numbers = + from = + .
Solution— given complex numbers are = 2 + 3 and = 4 + 2 .
Hence, − =(4 − 2) + (2 − 3)
∴ = 2 −
2.13 Multiplication of two complex numbers—
Let = + and = + are two complex numbers, then multiplication of complex
numbers and is given by—
= ( − ) + ( + )
Example—Multiply the complex numbers = + and = + .
Solution— Given complex numbers are = 3 + 4 and = 4 + 3 .
Hence, ( ) =(3 + 4 )(4 + 3 ) = (12 − 12) + (9 + 16)
∴ = 25
2.14 Division of two complex numbers—
Let = + and = + are two complex numbers, then division of complex
numbers by is given by—
= =
+
+
Now, multiplying and devide by = − .
Hence,
= = (
+
+
)(
−
−
)
Now, multiplying numerator and denominator,
= =
( + ) + ( − )
( + )
∴ = =
( + )
( + )
+
( − )
( + )

8. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
Example—Divide the complex numbers = + by = + .
Solution— given complex numbers are = 1 + and = 3 + 4 .
Hence,
= =
3 + 4
1 +
Now, multiplying and devide by = 1 −
Hence, = = =
( ) ( )
=
∴ = = +
2.15 De Moivre’s Theorem—
In mathematics, De Moivre's formula or De Moivre's identity states that for any complex
number = and integer " " it holds that—
( + ) = cos( ) + ( )
Example— Solve by using De Moivre’s Theorem—
(a) ( + ) ( ) ( + ) ( ) +
√
( )
Solution—
(a) Given that = ( + ) .
From De Moivre’s Theorem we know that—
( + ) = cos( ) + ( )
Hence, ( + ) = 2 + 2 .
(b) Let = (1 + )
Now, converting to polar form—
| | = √1 + 1 = √2 and = tan =
Hence, = √2( + )
Now, there is given (1 + )
So, (1 + ) = √2 +
From De Moivre’s Theorem we know that—
( + ) = cos( ) + ( )
Hence, (1 + ) = 2 + = 2 cos + + +
= 2 −cos − = 2 −
√
−
√
= −4(1 + )
(c)Let = +
√
= +
From De Moivre’s Theorem we know that—
( + ) = cos( ) + ( )

9. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
Hence, +
√
= + = +
= cos 3 + + 3 + = − −
= − −
√
( ) Let =
= ∗ =
( )
= ( + )
From De Moivre’s Theorem we know that—
( + ) = cos( ) + ( )
Hence, = cos (24 ) + (24 )
2.16 Powers to " ”—
Let = , here, = 1 & = tan ( ) = tan ∞ =
Hence, = can be written in Cartesian form as—
=
2
+
2
Thus, = +
From De Moivre’s Theorem we know that—
( + ) = cos( ) + ( )
Therefore, = +
Note—
= , = −1, = − , = 1
Hence, = , = −1, = − , = 1
Where, is an integer.
Example—Solve the following complex numbers—
(a) (b) (c) (d)
Solution—
(a) Given complex number is =
It can be written as = ( ∗ )
Hence, = −
(b) Given complex number is =
It can be written as = ( ∗ )
Hence, = −1

10. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
(c) Given complex number is =
It can be written as = ( ∗ )
Hence, = −1
(d) Given complex number is =
It can be written as = ( ∗ )
Hence, =
2.17 Laws of algebra of complex numbers—
The closure law— The sum of two complex numbers is a complex number, i.e., + is a
complex number for all complex numbers and .
The commutative law— For any two complex numbers and ,
+ = + .
The associative law— For any three complex numbers , , ,
( + ) + = + ( + ).
The distributive law— For any three complex numbers , , ,
(a) ( + ) = +
(b) ( + ) = +
The existence of additive identity— There exists the complex number 0 + 0 (denoted as 0),
called the additive identity or the zero complex number, such that, for every complex number
, + 0 = .
2.18 Properties of complex numbers—
1. =
2. If = + , then = +
3. = | |
4. ± = ±
5. =
6. = , ≠

11. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
EXERCISE-I
(a) Find the modulus of the following complex numbers—
1. 2 + 3
2. −1 + 2
3. 3 +
4. 3 − 4
5. 4 + 3
(b) Find the argument of the following complex numbers—
1. −1 +
2. 1 −
3. −1 −
4. √3 +
5. 1 + √3
6. 1 −
7. + (1 − )
8. (1 − ) +
9. −
10.1 + +
(c) Find the complex conjugate of the following complex numbers—
1. 12 + 3
2. √3 + 12
3. 4 + √3
4. 1 −
5. + (1 − )

12. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
6. (1 − ) +
7. −
8. 1 + +
EXERCISE-II
(a) If = 2 + 3 , = −3 + & = 1 + . Evaluate the followings—
1. + +
2. + 2 −
3.
4.
5.
(b) If = , = −1 + & = 1 + √3 . Evaluate the argument of the followings—
1. + +
2. + 2 −
3.
4.
5.
EXERCISE-III
(a) Convert the following complex numbers in polar form—
1. √3 +
2. 3 + 3
3.
4. −1 + √3
5. 2
(b) Convert the following complex numbers in Euler’s form—
1. √3 −
2. −3 + √3
3.
4. −1 + √3
5. 1 +
(c) Convert the following complex numbers in Euler’s form—
1.
2.
√
+
√
3. − −
√
4. −1 + √3
5. 1 +

13. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
EXERCISE-IV
(a) Solve the following complex numbers—
1.
√
+
√
2. − +
√
3.
4.
5. (( + )( − ))
(b) Solve the following Complex numbers—
1.
2.
3.
4.
5.
Crackers Attack
LEVEL-I
(a) Find the modulus of the following complex numbers—
1. 1 −
2. + (1 − )
3. (1 − ) +
4. −
5. 1 + +
(b) Find the argument of the following complex numbers—
1. 1 −
2. + (1 − )
3. (1 − ) +
4. −
5. 1 + +
LEVEL-II
(a) Find the square root of the following complex numbers—
1. 8 − 6
2. −15 + 8
3. (1 − ) +
4. −
5. 1 + +
Advanced Problems

14. Unit-II COMPLEX NUMBER
RAI UNIVERSITY, AHMEDABAD
LEVEL-I
1. Evaluate .
2. Express and in terms of Euler’s expressions.
3. Prove that sin + cos = 1, by using complex numbers.
4. If = + , evaluate ln z in terms of " " and " ".
LEVEL-II
1. Evaluate (i.e., find all possible values of ) 1 .
2. Evaluate (1 + )√ .
3. Evaluate √ .
Reference—
1. en.wikipedia.org/wiki/Complex_number
2. https://www.khanacademy.org
3. www.stewartcalculus.com
4. www.britannica.com
5. mathworld.wolfram.com
6. www.mathsisfun.com
7. www.purplemath.com
8. www.mathwarehouse.com
9. www.clarku.edu
10. home.scarlet.be