Complex Numbers

Complex numbers were created as an extension of algebra to solve equations that have no real solutions. For example, the square root of -1 does not exist in the set of real numbers.

This equation has no real solution, so we introduce the imaginary unit, denoted by the letter i.

Algebraic Form

The general form of a complex number consists of two parts: the sum of the real part and the imaginary part.

Here, the real part is a, the imaginary part is b, and i is the imaginary unit.

Basic Operations

Adding two complex numbers is done by adding the real and imaginary parts separately.

For multiplication, apply the rules for real numbers, taking into account that i² = -1.

Conjugate and Absolute Value

The conjugate of a complex number is obtained by changing the sign of the imaginary part.

The absolute value gives the length of the complex number in the coordinate system and is calculated using the Pythagorean theorem.

Geometric Representation

Complex numbers are often represented on the complex plane, where the horizontal axis shows the real part and the vertical axis shows the imaginary part.

Thus, the complex number z = a + bi corresponds to the point (a, b) in the plane.

Practical Examples

• Electrical Engineering: Complex numbers are used in calculations for alternating current.
• Engineering Sciences: For describing vibrations and waves.
• Mathematics: For solving equations that have no real roots.

Practice Exercise