Complex Numbers
Introduction
Complex numbers are a type of number that extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.
Definition
A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i² = −1. In this expression, a is the real part and b is the imaginary part of the complex number.
Real and Imaginary Parts
A complex number can be visually represented as a pair of numbers (a,b) on a diagram called an Argand diagram, representing the real and imaginary parts on the horizontal and vertical axis respectively. The real part of a complex number Z is denoted as Re(Z) or ℜ(Z) and the imaginary part is denoted as Im(Z) or ℑ(Z).
Complex Conjugate
The complex conjugate of a complex number is the number with the same real part and an imaginary part equal in magnitude but opposite in sign. For example, the complex conjugate of 3 + 4i is 3 - 4i.
Operations with Complex Numbers
Complex numbers can be added, subtracted, multiplied, and divided. The rules for these operations are similar to those for real numbers, but with the additional use of the imaginary unit i, which has the property that i² = −1.
Complex Plane
The complex plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the horizontal axis representing the real part of the complex number and the vertical axis representing the imaginary part.
Polar Form of Complex Numbers
Every nonzero complex number has a polar form, a unique way of writing it that highlights its magnitude and its angle with respect to the real axis. The polar form is written as r(cos θ + i sin θ), where r is the magnitude of the number and θ is the angle.
Applications of Complex Numbers
Complex numbers have many practical applications in a range of fields including physics, engineering, and computer science. They are used in calculations involving the physics of waves and oscillations, electrical circuits, quantum physics and signal processing.