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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 i 2 = −1.
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]).
The complex numbers contain a number i, the imaginary unit, with i 2 = −1, i.e., i is a square root of −1. Every complex number can be represented in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number respectively.
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
A split-complex number is an ordered pair of real numbers, written in the form z = x + j y {\displaystyle z=x+jy} where x and y are real numbers and the hyperbolic unit [ 1 ] j satisfies
Irrational numbers (): Real numbers that are not rational. Imaginary numbers: Numbers that equal the product of a real number and the imaginary unit , where =. The number 0 is both real and imaginary. Complex numbers (): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers.
Alternatively, the same system of complex numbers may be defined as the complex numbers whose real and imaginary parts are both constructible real numbers. [13] For instance, the complex number i {\displaystyle i} has the formulas − 1 {\displaystyle {\sqrt {-1}}} or 0 − 1 {\displaystyle {\sqrt {0-1}}} , and its real and imaginary parts are ...
The imaginary unit or unit imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x 2 + 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 ...
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