Complex numbers are an essential concept in mathematics, especially in algebra, engineering, and physics. They extend our understanding of numbers beyond the real number line and allow us to solve equations that have no real solutions.

What Are Complex Numbers?

A complex number is a number of the form:

z = a + bi

Where:
– a is called the real part of the complex number.
– b is called the imaginary part of the complex number.
– i is the imaginary unit, defined by the property i^2 = -1.

Example:
3 + 4i where 3 is the real part and 4 is the imaginary part.

Types of Complex Numbers

1.  Purely Real Number: If b = 0, the complex number is real.
Example: 5 + 0i = 5

2.Purely Imaginary Number: If a = 0, the complex number is purely imaginary.
Example: 0 + 3i = 3i

3.General Complex Number: Both a and b are non-zero.
Example: 2 + 5i

Operations on Complex Numbers

1. Addition

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

Example:
(3 + 2i) + (1 + 4i) = 4 + 6i

2. Subtraction

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

Example:
(5 + 6i) – (2 + 3i) = 3 + 3i

3. Multiplication

(a + bi)(c + di) = (ac – bd) + (ad + bc)i

Example:
(2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i^2 = 2 + 11i – 12 = -10 + 11i

4. Division

Divide by multiplying numerator and denominator by the conjugate of the denominator:
(a+bi)/(c+di) = ((a+bi)(c-di)) / (c^2+d^2)

Example:
(3 + 2i)/(1 + i) = ((3+2i)(1-i))/(1^2 + 1^2) = (3 – 3i + 2i – 2i^2)/2 = (5 – i)/2 = 2.5 – 0.5i

Conjugate of a Complex Number

The conjugate of a + bi is a – bi. Conjugates are useful in division and in finding magnitudes.

Example:
Conjugate of 4 + 5i is 4 – 5i

Magnitude (Modulus) of a Complex Number

The magnitude of z = a + bi is given by:
|z| = √(a^2 + b^2)

Example:
For 3 + 4i, |3 + 4i| = √(3^2 + 4^2) = √(9 + 16) = 5

Why Complex Numbers Are Important

Solve equations like x^2 + 1 = 0, which have no real solution.
Used in electrical engineering for AC circuits.
Applied in physics, quantum mechanics, and signal processing.
Helpful in computer graphics and control systems.