Complex numbers, a captivating extension of real numbers, introduce us to a realm where real and imaginary elements seamlessly intertwine. In this exploration, we will delve into the definition, formulas, properties, and examples of complex numbers, unraveling the intricacies that make them a fundamental concept in mathematics.
Table of Contents
Definition of Complex Numbers
A complex number takes the form a + bi, where aa and bb are real numbers, and i represents the imaginary unit \((i = \sqrt{1}\). The real part (aa) and the imaginary part (bi) combine to form a unique number in the complex plane.
Formulas for Complex Numbers:
 Addition and Subtraction:

 (a + bi) + (c + di) = (a + c) + (b + d)i
 (a + bi) – (c + di) = (a – c) + (b – d)i

 Multiplication:

 a+bi)×(c+di)=(ac−bd)+(ad+bc)i

 Conjugate:

 The conjugate of a + bia+bi is a – bia−bi.

 Division:


 \(\frac{a + bi}{c + di} = \frac{(a + bi) \times (c – di)}{c^2 + d^2}\)

5. Modulus (Magnitude):


 \(a + bi = \sqrt{a^2 + b^2}\)

Properties of a Complex Number
Complex numbers possess several key properties that define their behavior and relationships. Here are the fundamental properties of a complex number:
Real and Imaginary Components:
A complex number a + bia+bi consists of a real part a and an imaginary part bi.
Imaginary Unit (ii):
The imaginary unit is denoted by ii and is defined as \(i = \sqrt{1}\).
Complex Conjugate:
The complex conjugate of a complex number \( a + bi is a – bi\) . The conjugate is obtained by changing the sign of the imaginary part.
Addition and Subtraction:
Complex numbers are added or subtracted by combining their real and imaginary parts separately. For \( (a + bi) + (c + di), the result is \(a + c) + (b + d)i\) .
Multiplication:
The product of two complex numbers \( (a + bi) \times (c + di)\) is calculated using the distributive property and results in \( (ac – bd) + (ad + bc)i\) .
Modulus (Magnitude):
The modulus or magnitude of a complex number a + bi is given by a + bi =\( \sqrt{a^2 + b^2}\)
. It represents the distance of the complex number from the origin in the complex plane.
Division:
Complex numbers are divided by multiplying the numerator and denominator by the conjugate of the denominator. The result is \( \frac{a + bi}{c + di} \)=\( \frac{(a + bi) \times (c – di)}{c^2 + d^2}\).