Complex number

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