Complex number

A plus bi

Complex numbers intheplane

Complex number illustration

Vector Addition

Complex conjugate picture

Complex numbers are a fundamental concept in mathematics, extending the idea of one-dimensional real numbers to two-dimensional numbers, thereby providing a more comprehensive system for solving a wider range of mathematical problems. A complex number is expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i² = -1. The real part of the complex number is a, and the imaginary part is b. Complex numbers are used in various fields, including engineering, physics, and applied mathematics, to solve equations that have no real solutions.

Definition and Basics

A complex number is defined as an ordered pair of real numbers (a, b), where a represents the real part and b represents the imaginary part. The set of all complex numbers is denoted by **C** or **ℂ**. The basic form of a complex number is written as a + bi, where i is the square root of -1. This imaginary unit i is the cornerstone that allows for the extension of the real number system to include solutions to equations like x² + 1 = 0.

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided, following specific rules:

- **Addition and Subtraction**: These operations are performed by adding or subtracting the corresponding real parts and the imaginary parts of the complex numbers. For example, if z₁ = a + bi and z₂ = c + di, then z₁ + z₂ = (a + c) + (b + d)i.

- **Multiplication**: To multiply two complex numbers, apply the distributive property and use the fact that i² = -1. For example, (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i.

- **Division**: To divide one complex number by another, multiply the numerator and denominator by the conjugate of the denominator and simplify. The conjugate of a complex number a + bi is a - bi.

Geometric Representation

Complex numbers can be represented geometrically on the complex plane, also known as the Argand plane, where the x-axis represents the real part and the y-axis represents the imaginary part. This representation allows for a geometric interpretation of complex number operations. The magnitude (or modulus) of a complex number z = a + bi is given by |z| = sqrt(a² + b²), and represents the distance of the point from the origin in the complex plane.

Applications

Complex numbers have wide-ranging applications in various scientific fields:

- In electrical engineering, they are used to analyze alternating current (AC) circuits. - In quantum mechanics, the wave functions of particles are described using complex numbers. - In applied mathematics and physics, complex numbers are essential for solving differential equations and in Fourier analysis.

Further Topics

- **Complex Conjugate**: The complex conjugate of a + bi is a - bi. It has the same real part but an opposite imaginary part. - **Euler's Formula**: This formula establishes a deep relationship between trigonometric functions and complex exponentials: e^(iθ) = cos(θ) + i sin(θ). - **Roots of Unity**: These are complex numbers that satisfy the equation z^n = 1 for a given positive integer n.