Complex number

A number of the form a + bi, where a and b are real numbers and i² = −1. Its real and imaginary parts form coordinates in a two-dimensional number system.

Real number: A number on the continuous number line, including rational and irrational numbers. The real coefficient a is the real part of every complex number.
Complex addition: The operation that adds real parts and imaginary parts separately. It follows directly from combining like terms in sums of a + bi.
Fundamental theorem of algebra: The theorem that every nonconstant polynomial with complex coefficients has a complex root. Complex numbers provide the setting in which polynomial equations always have roots.
Quaternion: A four-dimensional number system extending the real and complex numbers with noncommutative multiplication. Quaternions extend complex arithmetic to represent three-dimensional rotations.
Imaginary unit: The number i defined by i² = −1. Multiplying the real coefficient b by i creates the imaginary part.
Complex multiplication: The operation defined by distributivity and i² = −1 for complex numbers. Its rule turns multiplication into a combination of scaling and rotation.
Complex analysis: The study of functions of a complex variable and their differentiability, integration, and singularities. It develops a rich theory from functions whose inputs and outputs are complex numbers.
Dual number: A number of the form a + bε, where ε² = 0. Dual numbers add a nilpotent unit instead of the square root of −1.
Complex conjugate: The complex number formed by changing the sign of a number’s imaginary part. Conjugation pairs a + bi with a − bi and helps simplify division.
Polar form of a complex number: A representation of a complex number using its modulus and angle from the positive real axis. Polar form makes multiplication’s effects on magnitude and angle explicit.

Linked from 85 pages