Our motive for the introduction of complex numbers is to make every algebraic equation solvable. Let us consider the equation $z^{2}+4=0$. This equation has no solution in the set of real numbers. There is no real no. $\mathrm{x}$ whose square is -4 . In order to remedy this situation, a new kind of numbers were introduced and were given the name complex nos. Eular was the first to introduce the symbol $\mathrm{i}$ for $\sqrt{-1}$ with the property $\mathrm{i}^{2}=-1$. $\mathrm{i}$ is also called the symbol as the imaginary unit. $\mathrm{i}$ is called an imaginary number.

Powers of $i$

Complex numbers

An expression of the form $\mathrm{x}+\mathrm{iy}$, where $\mathrm{x} \& \mathrm{y}$ are real numbers and $\mathrm{i}$ is a symbol, is called a complex number and usually denoted by $z$.

Addition of complex number

Let $z _{1}=a _{1}+i b _{1} \& z _{2}=a _{2}+i b _{2}$ be two complex number Then their sum $z _{1}+z _{2}$ is defined as the complex number $\left(a _{1}+a _{2}\right)+i\left(b _{1}+b _{2}\right)$

Multiplication of Complex Number

Let $z _{1}=a _{1}+i b _{1} \& z _{2}=a _{2}+i b _{2}$ be two complex number. Then the multiplication of $z _{1}$ with $z _{2}$ is denoted by $z _{1} z _{2}$ and is defined as $\left(a _{1} a _{2}-b _{1} b _{2}\right)+i\left(a _{1} a _{2}+b _{1} b _{2}\right)$.

Division of Complex Number

The division of a complex number $z _{1}$ by a non-zero complex $z _{2}$ is defined as the multiplicaiton of $z _{1}$ by the multiplication inverse of $z _{2}$ and is denoted by $\frac{z _{1}}{z _{2}}$

$\frac{z _{1}}{z _{2}}=\frac{\left(a _{1}+i b _{1}\right)}{\left(a _{2}+i b _{2}\right)}=\frac{\left(a _{1}+i b _{1}\right)}{\left(a _{2}+i b _{2}\right)} \frac{\left(a _{2}-i b _{2}\right)}{\left(a _{2}-i b _{2}\right)}$

$=\frac{\left(a _{1} a _{2}+b _{1} b _{2}\right)}{a _{2}^{2}+b _{2}^{2}}+\frac{i\left(a _{2} a _{1}-b _{1} b _{2}\right)}{a _{2}^{2}+b _{2}^{2}}$

Conjugate of $\mathrm{Z}$

Let $z=a+i b$ be a complex number. Then the conjugate of $z$ is denoted by $z$ and is equal to $a-i b$ thus $\mathrm{z}=\mathrm{a}+\mathrm{ib}$

$\Rightarrow \quad \overline{\mathrm{z}}=\mathrm{a}-\mathrm{ib}$

e.g. if $z=3+4 i$ then $\bar{z}=3-4 i$

Modulus of Z

The modulus of a complex number $x+$ iy is denoted by $|x+i y|$ and is denoted as $|x+i y|=$ $\sqrt{x^{2}+y^{2}}=$ non-negative square root of $x^{2}+y^{2}$ e.g. $z=3-4 i$, then $|z|=\sqrt{(3)^{2}+(-4)^{2}}=5$.

Equality of complex number

Two complex nos $\mathrm{z} _{1}=\mathrm{x} _{1}+\mathrm{i} \mathrm{y} _{1} \& \mathrm{z} _{2}=\mathrm{x} _{2}+\mathrm{iy} _{2}$ are said to be equal if and only if $\mathrm{x} _{1}=\mathrm{x} _{2} \& \mathrm{y} _{1}=\mathrm{y} _{2}$ i.e. $Z _{1}=Z _{2} \Rightarrow \operatorname{Re}\left(z _{1}\right)=\operatorname{Re}\left(z _{2}\right) \& \operatorname{Im}\left(z _{1}\right)=\operatorname{Im}\left(z _{2}\right)$.

Examples

1. Evaluate $\left[i^{-53}+\left\{i^{n}+i^{n+1}+i^{n+2}+i^{n+3}+4\right\}^{\frac{1}{2}} i\right]$

2. Show that the polynomial $x^{4 p}+x^{4 q+1}+x^{4 r+2}+x^{4 s+3}$ is divisible by $\mathrm{x}^{3}+\mathrm{x}^{2}+\mathrm{x}+1$, where $\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s} \in \mathrm{N}$.

3. Express $\frac{1}{1-\cos \theta+2 i \sin \theta}$ in the form $\mathrm{x}+\mathrm{iy}$

4. Find the multiplicative inverse of complex no $3+2 \mathrm{i}$

Practice questions

1. If $a<0, b>0$, then $\sqrt{a} \cdot \sqrt{b}$ is equal to

(a). $-\sqrt{|a| b}$

(b). $-\sqrt{|a| \cdot b i}$

(c). $\sqrt{|\mathrm{a}| \mathrm{b}}$

(d). none of these

2. The value of the sum $\sum _{n=1}^{13}\left(i^{n}+i^{n+1}\right)$, where $\mathrm{i}=\sqrt{-1}$, is

(a). $i$

(b). $i-1$

(c). $-\mathrm{i}$

(d). $0$

3. The smallest positive integral value of $\mathrm{n}$ of which $\left(\frac{1-i}{1+i}\right)^{n}$ is purely imaginary with positive imaginary part, is

(a). $1$

(b). $3$

(c). $5$

(d). none of these

4. If $\mathrm{n}$ is an odd integer, $\mathrm{i}=\sqrt{-1}$, then $(1+\mathrm{i})^{6 \mathrm{n}}+(1-\mathrm{i})^{6 \mathrm{n}}$ is equal to

(a). $0$

(b). $2$

(c). $-2$

(d). none of these

5. If $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is a real number and $0<\theta<2 \pi$, then $\theta=$

(a). $\pi$

(b). $\frac{\pi}{2}$

(c). $\frac{\pi}{2}$

(d). $\frac{\pi}{6}$

6. If $\mathrm{b}+\mathrm{c}=(1+\mathrm{a}) \mathrm{z}$ and $\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}=1$, then $\frac{1+i z}{1-i z}$

(a). $\frac{a-i b}{1-c}$

(b). $\frac{a-i b}{1+c}$

(c). $\frac{a+i b}{1-c}$

(d). $\frac{a+i b}{1+c}$

7. If $(x+i y)^{\frac{1}{3}}=a+i b$ then $\frac{x}{a}+\frac{y}{b}=$

(a). $0$

(b). $1$

(c). $-1$

(d). none of these

8. If $(a+i b)^{5}=\alpha+i \beta$, then $(b+a i)^{5}$ is equal to

(a). $\beta+i \alpha$

(b). $\alpha-i \beta$

(c). $\beta-i \alpha$

(d). $-\alpha-i \beta$

9. The set of values of $a \in R$ for which $x^{2}+i(a-1) x+5=0$ will have a pair of conjugate complex roots is

(a). $\mathrm{R}$

(b). $\{1\}$

(c). $\left\{a:\left|\cdot a^{2}-2 a+21\right|>0\right\}$

(d). none of these

10. The relation between the real numbers $a$ and $b$, which satisfy the equation $\frac{1-i x}{1+i x}=a-i b$, for some real value of $x$, is

(a). $(a-b)(a+b)=1$

(b). $\frac{a-b}{a+b}$

(c). $a^{2}+b^{2}=1$

(d). none of these