5.1 Introduction

A Complex Number is a combination of a Real Number and an Imaginary Number and is expressed in the form of \(a+i b\) where, \(a, b\) are real numbers and ‘ \(i\) ‘ is an imaginary number called “iota”. The value of \(i=\sqrt{-1}\). Because when we square \(i\) we get \(-1\), \(i^2=-1\).

For example, in the complex number \(3+i 2\), the real part is 3 and the imaginary part is 2.

Definition of Complex Numbers

A number of the form \(x+i y\), where \(x, y \in R\) and \(i=\sqrt{-1}\) is called a complex number and ‘ \(i\) ‘ is called iota. A complex number is usually denoted by \(z\) and the set of complex numbers is denoted by \(C\).
\(
C=\{x+i y ; x \in R, y \in R, i=\sqrt{-1}\}
\)
For example, \(5+3 i,-1+i, 0+4 i, 4+0 i\), etc. are complex numbers.
Here ‘ \(x\) ‘ is called the real part of \(z\) and ‘ \(y\) ‘ is known as the imaginary part of \(z\). The real part of \(z\) is denoted by \(\operatorname{Re}(z)\) and the imaginary part by \(\operatorname{Im}(z)\). If \(z=3-4 i\), then \(\operatorname{Re}(z)=3\) and \(\operatorname{Im}(z)=-4\).
Note that the sign ‘ + ‘ does not indicate addition as normally understood, nor does the symbol \(i\) denote a number. These things are parts of the scheme used to express numbers of a new class and they signify the pair of real numbers \((x, y)\) to form a single complex number.

A complex number \(z\) is purely real if its imaginary part is zero, i.e. \(\operatorname{Im}(z)=0\) and purely imaginary if its real part is zero, i.e. \(\operatorname{Re}(z)=0\).

Remarks:

• For any positive real number \(a\), we have \(\sqrt{-a}=\sqrt{-1 \times a}\)
  \(
  =\sqrt{-1} \sqrt{a}=i \sqrt{a} .
  \)
• The property \(\sqrt{a} \sqrt{b}=\sqrt{a b}\) is valid only if at least one of \(a\) and \(b\) is non-negative. If \(a\) and \(b\) are both negative, then \(\sqrt{a} \sqrt{b}=-\sqrt{|a||b|}\).
• Inequality in complex numbers are never talked. If \(a+i b >c+\) id has to be meaningful then \(b=d=0\). Equalities
  however in complex numbers are meaningful. Two complex numbers \(z_1=a_1+i b_1\) and \(z_2=a_2+i b_2\) are equal if \(a_1 =a_2\) and \(b_1=b_2\), i.e. \(\operatorname{Re}\left(z_1\right)=\operatorname{Re}\left(z_2\right)\) and \(\operatorname{Im}\left(z_1\right)=\operatorname{Im}\left(z_2\right)\).
• In real number system \(a^2+b^2=0 \Rightarrow a=0=b\). But if \(z_{\text {, }}\) and \(z_2\) are complex numbers then \(z_1^2+z_2^2=0\) does not imply \(z_1=z_2=0\), e.g. \(z_1=1+i\) and \(z_2=1-i\). However, if the product of two complex numbers is zero then at least one of them must be zero, same as in case of real numbers.

Integral Power of lota (\(i\))

Since \(i=\sqrt{-1}\), we have \(i^2=-1, i^3=-i\) and \(i^4=1\).
To find the value of \(i^n(n>4)\), first divide \(n\) by 4.
Let \(q\) be the quotient and \(r\) be the remainder, i.e. \(n=4 q+r\) where \(0 \leq \mathrm{r} \leq 3\)
\(
i^n=i^{4 q+r}=\left(i^{4}\right)^q(i)^r=(1)^q(i)^r=i^r
\)
In general, we have the following results: \(i^{4 n}=1, i^{4 n+1}=i\), \(i^{4 n+2}=-1, i^{4 n+3}=-i\), where \(n\) is any integer.

Algebraic Operations with Complex Numbers

Let two complex number be \(z_1=a+i b\) and \(z_2=c+i d\)
Addition \(\left(z_1+z_2\right)\) :
\(
(a+i b)+(c+i d)=(a+c)+i(b+d)
\)
Subtraction \(\left(z_1-z_2\right)\) :
\(
(a+i b)-(c+i d)=(a-c)+i(b-d)
\)
Multiplication \(\left(z_1 z_2\right)\) :
\(
(a+i b)(c+i d)=(a c-b d)+i(a d+b c)
\)
Division \(\left(z_1 / z_2\right)\) :
\(
\frac{a+i b}{c+i d}=\frac{(a+i b)}{(c+i d)} \frac{(c-i d)}{(c-i d)} \text { (Rationalization) }
\)
(where at least one of \(c\) and \(d\) is non-zero)
\(
\Rightarrow \quad \frac{a+i b}{c+i d}=\frac{(a c+b d)}{c^2+d^2}+\frac{i(b c-a d)}{c^2+d^2}
\)

Properties of Algebraic Operations on Complex Numbers

Let \(z_1, z_2\) and \(z_3\) are any three complex numbers. Then their algebraic operations satisfy the following properties:
1. Addition of complex numbers satisfies the commutative and associative properties, i.e.
\(
z_1+z_2=z_2+z_1 \text { and }\left(z_1+z_2\right)+z_3=z_1+\left(z_2+z_3\right)
\)
2. Multiplication of complex numbers satisfies the commutative and associative properties, i.e.
\(
z_1 z_2=z_2 z_1 \text { and }\left(z_1 z_2\right) z_3=z_1\left(z_2 z_3\right)
\)
3. Multiplication of complex numbers is distributive over addition, i.e.
\(
z_1\left(z_2+z_3\right)=z_1 z_2+z_1 z_3 \text { and }\left(z_2+z_3\right) z_1=z_2 z_1+z_3 z_1
\)

Example 1: Evaluate:
a. \(i^{135}\)
b. \(\quad(-\sqrt{-1})^{4 n+3}, n \in N\)
c. \(\sqrt{-25}+3 \sqrt{-4}+2 \sqrt{-9}\)

Solution: (a) 135 leaves remainder as 3 when it is divided by 4
\(
\therefore \quad i^{135}=i^3=-i
\)
(b)
\(
\begin{aligned}
(-\sqrt{-1})^{4 n+3} & =(-i)^{4 n+3} \\
& =(-i)^{4 n}(-i)^3 \\
& =\left\{(-i)^4\right\}^n(-i)^3 \\
& =1 \times(-i)^3=i
\end{aligned}
\)
(c) \(\sqrt{-25}+3 \sqrt{-4}+2 \sqrt{-9}=5 i+6 i+6 i=17 i\)