Introduction to Complex Numbers

Complex Numbers

Complex numbers are an important concept in mathematics. They extend the idea of numbers beyond the real number line. A complex number has a real part and an imaginary part. The imaginary part is based on the imaginary unit, denoted as i, where i is defined as the square root of -1.

Definition of Complex Numbers

A complex number can be expressed in the form:

z = a + bi

In this expression:

• z is the complex number.
• a is the real part.
• b is the imaginary part.

For example, in the complex number 3 + 4i, 3 is the real part and 4 is the imaginary part.

Imaginary Numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit i. For instance, 5i is an imaginary number. The square of an imaginary number is negative:

(5i)² = 25i² = 25(-1) = -25

Real Numbers as Complex Numbers

Every real number can be considered a complex number. A real number can be expressed as a complex number with an imaginary part equal to zero. For example, the real number 7 can be written as:

7 = 7 + 0i

Examples of Complex Numbers

Here are some examples of complex numbers:

• 2 + 3i (real part is 2, imaginary part is 3)
• -1 - 2i (real part is -1, imaginary part is -2)
• 0 + 5i (real part is 0, imaginary part is 5)
• 4 + 0i (real part is 4, imaginary part is 0)

Equality of Complex Numbers

Two complex numbers are equal if their real parts are equal and their imaginary parts are equal. For example, the complex numbers 2 + 3i and 2 + 3i are equal because both their real parts (2) and imaginary parts (3) are the same.

Complex Conjugate

The complex conjugate of a complex number z = a + bi is denoted as z* and is defined as:

z* = a - bi

The complex conjugate has the same real part but the opposite imaginary part. For example, the complex conjugate of 3 + 4i is 3 - 4i.

Properties of Complex Conjugates

• The product of a complex number and its conjugate is a non-negative real number:

z × z* = (a + bi)(a - bi) = a² + b²

The result is always a non-negative real number, since both a² and b² are non-negative.

Geometric Representation

Complex numbers can be represented geometrically on the complex plane. The horizontal axis represents the real part, while the vertical axis represents the imaginary part. For example, the complex number 3 + 4i is represented by the point (3, 4) in the complex plane.

Magnitude of a Complex Number

The magnitude (or modulus) of a complex number z = a + bi is given by:

|z| = √(a² + b²)

This formula calculates the distance from the origin (0, 0) to the point (a, b) in the complex plane. For example, for the complex number 3 + 4i:

|3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5

Applications of Complex Numbers

Complex numbers are widely used in various fields such as engineering, physics, and applied mathematics. They are especially useful in solving problems involving oscillations, waves, and electrical circuits.

Summary

• A complex number has a real part and an imaginary part.
• The imaginary unit i is defined as the square root of -1.
• Complex numbers can be equal if both their real and imaginary parts are equal.
• The complex conjugate has the same real part and the opposite imaginary part.
• The magnitude of a complex number is calculated using the formula |z| = √(a² + b²).

Check your understanding

1. What is the definition of a complex number?
2. How do you find the complex conjugate of a complex number?
3. What is the magnitude of the complex number 1 + 2i?
4. Explain why the complex numbers 2 + 3i and 3 + 2i are not equal.