Complex numbers are defined as numbers of the form x+iy, where x and y are real numbers and i = √-1. For example, 3+2i, -2+i√3 are complex numbers. For a complex number z = x+iy, x is called the real part, denoted by Re z and y is called the imaginary part denoted by Im z. For example, if z = 3+2i, Re z = 3 and Im z = 2.

In this section, aspirants will learn about Complex Numbers: their definition, standard form, algebraic operations, conjugate, polar form, Euler’s form, and much more. A Complex Number is a combination of a Real Number and an Imaginary Number.

Table of Contents for Complex Numbers:

What Are Complex Numbers?

Algebraic Operations with Complex Numbers

Conjugate of Complex Number

Properties of Conjugate of Complex Number

Modulus and Argument of a Complex Number

Euler’s Form

Solved Problems on Complex Numbers

Complex numbers are numbers that have both a real and imaginary component. They are usually written in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is equal to the square root of -1.

If $x, y \in \mathbb{R}$, then an ordered pair $(x, y) = x + iy$ is called a complex number. It is denoted by $z$, where $x$ is the real part of $Re(z)$ and $y$ is the imaginary part or $Im(z)$ of the complex number.

(i) If the real part of $z$ is equal to 0, then $z$ is called a purely imaginary number.

(ii) If the imaginary part of a complex number $z$ is equal to $0$, then $z$ is called a purely real number.

Note: The set of all possible ordered pairs, known as the complex number set, is denoted by C.

Powers of Iota Integrals

i² = -1, i³ = -i, i⁴ = 1, where i is an imaginary number defined as √-1.

To find the value of n (where n > 4) first, divide n by 4.

Let q be the quotient and r be the remainder.

n = 4q + r, where 0 < r < 3

in = i4q + r = (i4)q
ir = (i)q
ir = ir

The sum of four consecutive powers of $i$ is equal to zero.

∀n∈Z, n + (n + 1) + (n + 2) + (n + 3) = 0

-1/i = -i

(1 + i)^2 = 2i and (1 - i)^2 = -2i

The equation $\sqrt{a} . \sqrt{b} = \sqrt{ab}$ is valid only when at least one of $a$ and $b$ is non-negative.

If both a and b are negative, then $\sqrt{a} \times \sqrt{b} = -\sqrt{|a|\cdot|b|}$

√-a × √-b = -√(a × b)

Illustration 1: Evaluate i201

Solution: 201 \equiv 1 \pmod 4 \implies i_{201} = i_1 = i

Illustration 2: Evaluate $$1 + (1+i) + (1+i)^2 + (1+i)^3$$

Solution: 1 + (1 + i) + (2i) + (1 + i)² + (1 + i)³ = 1 + (1 + i) + (2i) + (-2 + 2i)

1 + 1 + i + 2i - 2 + 2i = 5i

Illustration 3: [(1 + i)/√2]⁸ⁿ + [(1 – i)/√2]⁸ⁿ = 2⁸ⁿ

Given:

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Solution:

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[{(1 + i)/√2}2]4n + [{(1 – i)/√2}2]4n = [(1 + i)/√2]8n + [(1 – i)/√2]8n