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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.
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.
The set of the p-adic numbers contains the rational numbers, but is not contained in the complex numbers. The elements of an algebraic function field over a finite field and algebraic numbers have many similar properties (see Function field analogy). Therefore, they are often regarded as numbers by number theorists.
Complex conjugate: If z is a complex number, then ¯ is its complex conjugate. For example, + ¯ =. 2. Topological closure: If ...
A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real. In other words, real numbers are the only fixed points of conjugation. Conjugation does not change the modulus of a complex number: | ¯ | = | |. Conjugation is an involution, that is, the conjugate of the conjugate of a complex ...
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 ...
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]).
"A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system." [1]: 38 The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. [1]