Complex number - Practice Questions & MCQ

A number of the form $\mathrm{a}+\mathrm{ib}$ is called a complex number (where a and b are real numbers and i is iota). We usually denote a complex number by the letter $z, z_1, z_2$, etc

For example, $z=5+2 i$ is a complex number.
5 here is called the real part and is denoted by $\operatorname{Re}(z)$, and 2 is called the imaginary part and is denoted by $\operatorname{Im}(z)$

Note: 2 i is not the imaginary part, only 2 is called the imaginary part.
For $z=-2-i, \operatorname{Re}(z)=-2, \operatorname{Im}(z)=-1$
We denote the set of all complex numbers by C .
Purely Real and Purely Imaginary Complex Number
A complex number is said to be purely real if its imaginary part is zero, $\operatorname{Im}(z)=0$
i.e. $z=4+0 i, z=4$.

A complex number is said to be purely imaginary if its real part is zero, $\operatorname{Re}(z)=0$
i.e. $z=0+3 i, z=3 i$

All real numbers are also complex numbers (with $b=0$ ). Eg 4 can be written as $4+0 \mathrm{i}$
So, $R$ is a proper subset of $C$.
Equality of Complex Numbers
Two complex numbers are said to be equal if and only if their real parts are equal and their imaginary parts are equal.

$
a+i b=c+i d
$

$\Rightarrow \mathrm{a}=\mathrm{c}$ and $\mathrm{b}=\mathrm{d}$
$a, b, c, d \in R$ and $i=\sqrt{ }-1$
Note: In complex numbers, inequalities do not exist. $z_1>z_2$ does not make any sense in complex numbers

We can only compare two complex numbers if their imaginary parts are $0 . \mathrm{Eg}, 5+0 \mathrm{i}>4+0 \mathrm{i}$ is correct.
Argand Plane
A complex Number can be represented on a rectangular coordinate system called Argand Plane.
In this $\mathrm{z}=\mathrm{a}+\mathrm{ib}$ is represented by a point whose coordinates are $(\mathrm{a}, \mathrm{b})$
So, the $x$-coordinate of the point is the Real part of $z$ and $y$ coordinate is the imaginary part of $z$
$\operatorname{Eg} z=-2+3 i$ is represented by the point $(-2,3)$ and it lies in the second quadrant.