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For example, this notion contains the split-complex numbers, which are elements of the ring [] / (as opposed to [] / (+) for complex numbers). In this ring, the equation a 2 = 1 has four solutions. The field R {\displaystyle \mathbb {R} } is the completion of Q , {\displaystyle \mathbb {Q} ,} the field of rational numbers , with respect to the ...
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.}
The complex numbers are the only 2-dimensional hypercomplex algebra that is a field. Split algebras such as the split-complex numbers that include non-real roots of 1 also contain idempotents and zero divisors (+) =, so such algebras cannot be division algebras.
An imaginary number is the product of a real number and the imaginary unit i, [note 1] which is defined by its property i 2 = −1. [1] [2] The square of an imaginary number bi is −b 2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary. [3]
Transcendental number: Any real or complex number that is not algebraic. Examples include e and π. Trigonometric number: Any number that is the sine or cosine of a rational multiple of π. Quadratic surd: A root of a quadratic equation with rational coefficients. Such a number is algebraic and can be expressed as the sum of a rational number ...
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 split-complex number is an ordered pair of real numbers, written in the form ... For example, when a = 0, then (b,c) is a point on the standard hyperbola.
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 ...