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An imaginary number is the product of a real number and the imaginary unit i, [ note 1] which is defined by its property i2 = −1. [ 1][ 2] The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary. [ 3]
Complex number. A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i2 = −1. In mathematics, a complex number is an element of a number system ...
The number 0 is both real and imaginary. Complex numbers (): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers. Hypercomplex numbers include various number-system extensions: quaternions (), octonions (), and other less common variants. [4]
The imaginary unit or unit imaginary number ( i) is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal x -axis, called the real axis, is formed by the real numbers, and the vertical y -axis, called the imaginary axis, is formed by the imaginary numbers . The complex plane allows for a geometric interpretation ...
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955 [1] [2]) or complex number (proposed by S. Khmelnik in 1964 [3] and Walter F. Penney in 1965 [4] [5] [6] ).
Hypercomplex number. In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.
In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples include imaginary quadratic fields, cyclotomic fields, and, more generally, CM fields . Any number field that is Galois over the rationals must be either totally real or totally imaginary.