Complex Numbers

Complex numbers are an extension of the real numbers, which in turn are an extension of the rational numbers. If we allow for the square root of negative one, denoted as i, then we can generate the set of complex numbers.

Definition

A complex number, commonly denoted as ℂ, can be defined as follows:

ℂ = {a + bi | a, b ∈ ℝ}

In this notation, a + bi represents a complex number, a, b ∈ ℝ means that a and b are both real numbers, and i is the imaginary unit with the property that i² = -1.

Properties of Complex Numbers

Complex numbers have several important properties:

  1. Closure Property: If you add, subtract, multiply, or divide two complex numbers (except for division by zero), the result is always a complex number.

  2. Associative Property: For all complex numbers a, b, and c, (a + b) + c = a + (b + c), and (a * b) * c = a * (b * c).

  3. Commutative Property: For all complex numbers a and b, a + b = b + a and a * b = b * a.

  4. Distributive Property: For all complex numbers a, b, and c, a * (b + c) = a * b + a * c.

  5. Identity Property: The number 0 is the additive identity in the set of complex numbers, and the number 1 is the multiplicative identity.

  6. Additive and Multiplicative Inverses: Every complex number a + bi has an additive inverse -a - bi, such that (a + bi) + (-a - bi) = 0, and a multiplicative inverse 1/(a + bi) (if a + bi ≠ 0), such that (a + bi) * 1/(a + bi) = 1.

Real and Imaginary Parts

Every complex number can be expressed as the sum of a real part a and an imaginary part bi. For example, in the complex number 3 + 4i, 3 is the real part and 4i is the imaginary part.

Applications

Complex numbers are used in many areas of mathematics, including algebra, calculus, geometry, and trigonometry. They're also used in physics and engineering, especially in the fields of electronics, signal processing, control systems, quantum physics, and many others. The concept of the complex plane also allows for a geometric interpretation of complex numbers.