Complex number

Generally, two types of numbers exist in mathematics first one is real numbers and the second one is complex numbers. Here we focus on complex numbers. A number of the form a + 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 $\mathrm{z}, \mathrm{z}_1, \mathrm{z}_2$, etc

In this article, we will cover the concept of complex numbers. This concept falls under the broader category of complex numbers. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of eleven questions have been asked on this concept, including two in 2016, five in 2019, one in 2020, and three in 2023.

Complex Numbers

The number which has no real meaning then these numbers are represented in complex forms. The general form of complex numbers are $a+i b$ where i is iota or $a+i b$.

For example, $
z=5+2 i
$ is a complex number.

5 here is called the real part and is denoted by Re(z), and 2 is called the imaginary part and is denoted by Im(z)

What are Real Numbers?

Any number that is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers. It is represented as Re(). For example: $
1,-4,0,1 / 3,2.83, \sqrt{5} \text {, etc., }
$, are all real numbers.

Real numbers can be integers, whole numbers, natural naturals, fractions, or decimals. Real numbers can be positive, negative, or zero.

Thus, real numbers broadly include all rational and irrational numbers. They are represented by the symbol ∞ and have all numbers from negative infinity, denoted -∞, to positive infinity, denoted ∞, written in interval notation as $
-\infty, \infty) \text {. }
$

What are Imaginary Numbers?

The numbers that are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. It is represented as Im(). Example: $
\sqrt{ }-3, \sqrt{ }-5, \sqrt{ }-1
$ are all imaginary numbers.

Types of a complex number

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
  $

$
\text { i.e. } z=4+0 i, z=4 \text {. }
$

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

$
\text { i.e. } z=0+3 i, z=3
$

All real numbers are also complex numbers (with b=0). Eg 4 can be written as $
4+0 i
$

So, R is a proper subset of C.

Equality of Complex Numbers

The equality of complex numbers is similar to the equality of real 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 $i=\sqrt{ }-1$

Also, the two complex numbers in the polar form are equal, if and only if they have the same magnitude and their argument (angle) differs by an integral multiple of 2π.

Argand Plane

A complex Number can be represented on a rectangular coordinate system called Argand Plane.

In this z = a + ib is represented by a point whose coordinates are (a,b)

So, the x-coordinate of the point is the Real part of z, and the y-coordinate is the imaginary part of z

Eg $
z=-2+3 i
$ is represented by the point $
(-2,3)
$ and it lies in the second quadrant.

Summary

Complex numbers are the powerful tools of mathematics that help in solving different types of complex problems. It is usually used in cases where real numbers are unable to represent the whole numbers then those numbers are represented by using real as well as complex numbers.

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Solved Examples Based On Complex Numbers

Example 1: Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re} z=1$, then it is necessary that

Solution:

As we learned in

Definition of Complex Number -

$z=x+i y, x, y \in R \quad \& i^2=-1

- where Real part of z = Re (z) = x & Imaginary part of z = Im (z) = y

$
z=x+i y, x, y \in R_{\& i^2=-1}
$
- where Real part of $z=\operatorname{Re}(z)=x$ \& Imaginary part of $z=\operatorname{Im}(z)=y$
$
z^2+\alpha z+\beta=0
$

Let $z=1+$ iy
$
\begin{aligned}
& \text { So that }(1+i y)^2+\alpha(1+i y)+\beta=0 \\
& \Rightarrow \quad 1-y^2+i 2 y+\alpha+i \alpha y+\beta=0 \\
& \therefore \quad\left(1-y^2+\alpha+\beta\right)+i(2+\alpha) y=0 \\
& \therefore \quad \alpha=-2 \text { and } 1-y^2-2+\beta=0 \\
& \Rightarrow \quad y^2=\beta-1>0 \\
& \therefore \quad \beta-1>0
\end{aligned}
$

$\beta \in(1, \infty)$

Hence, the answer is $\beta \in(1, \infty)$.

Example 2: The complex number having the sum of the real part and imaginary part 6 and the real part is also double of an imaginary part will be:

1) $8-21$
2) $4+21$
3) $3+32$
4) $5+7$

Solution

Let the complex number $x+i y$

where x will be the real part and y will be the imaginary part.

According to the given conditions

$
x+y=6
$
$\qquad$
$
x=2 y
$
$\qquad$
Solving (1) and (2) we get $\mathrm{x}=4$ and $\mathrm{y}=2$

So, the complex number will be $4+2 \mathrm{i}$ Hence, the answer is option (2).

Example 3: A complex number z is such that the sum of its real and imaginary parts is zero, if 5 is added to the imaginary part, then the imaginary part becomes 8, then z equals:

Solution:

As we learned in

Definition of Complex Number -

$
z=x+i y, x, y \in R_{\&} \mathrm{i}^2=-1
$
- wherein

Real part of $z=\operatorname{Re}(z)=x$ \& Imaginary part of $z=\operatorname{Im}(z)=y$
Let $z=x+i y$ then $\operatorname{Re}(z)=x$ and $\operatorname{Im}(z)=y$
According to the given conditions $\rightarrow$
$
x+y=0
$
$
y+5=8
$

From (1) and (2): $x=-3, y=3$
$
\therefore z=-3+3 i
$

Hence, the answer is $-3+3 i$.

Example 4: The complex number z is such that its real part is thrice the imaginary part, if both real and imaginary parts are increased by 1 then the real part becomes double of the imaginary part. Then z equals:

Solution:

As we learned in

Definition of Complex Number -

$
\begin{aligned}
& z=x+i y, x, y \epsilon R \& \mathrm{i}^2=-1 \\
& \text { - wherein }
\end{aligned}
$
- wherein

Real part of $z=\operatorname{Re}(z)=x$ \& Imaginary part of $z=\operatorname{Im}(z)=y$
Let $z=x+i y$ then according to the given conditions $\rightarrow$
$
x=3 y
$
$\qquad$
$
x+1=2(y+1) \Rightarrow x=2 y+1
$
$\qquad$
From (1) and (2) we get
$x=3$ and $y=1$
$
\therefore z=3+i
$

Hence, the answer is $3+\mathrm{i}$.

Example 5: A value of $\theta$ for which $\frac{2+3 i \sin \theta}{1-2 i \sin \theta}$ is purely imaginary is :

Solution:

As we learned in

Purely Real Complex Number -
$
z=x+i y, x \in R, \boldsymbol{y}=\mathbf{0}
$
& $i^2=-1$
wherein
Real part of $z=\operatorname{Re}(z)=x$ \& Imaginary part of $z=\operatorname{Im}(z)=y$
Let $Z=\frac{2+3 i \sin \theta}{1-2 i \sin \theta}$
$Z=\frac{2+3 i \sin \theta}{1-2 i \sin \theta} \times \frac{1+2 i \sin \theta}{1+2 i \sin \theta}$
$=\frac{(2+3 i \sin \theta)(1+2 i \sin \theta)}{1+4 \sin ^2 \theta}$
$=2-6 \sin ^2 \theta=0$ for purely imaginary, a real part must be zero.
$\sin ^2 \theta=\frac{2}{6}=\frac{1}{3}$
$\sin ^2 \theta=\frac{1}{\sqrt{3}}$
$\theta=\sin ^{-1} \frac{1}{\sqrt{3}}$

Hence, the answer is $
\theta=\sin ^{-1} \frac{1}{\sqrt{3}}
$