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A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. It is common to write a + 0i = a, 0 + bi = bi, and a + (−b)i = a − bi; for example, 3 + (−4)i = 3 − 4i.
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.}
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying =, where . A split-complex number has two real number components x and y , and is written z = x + y j . {\displaystyle z=x+yj.}
The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra ), it follows that every polynomial with real coefficients can be factored into ...
Figure 1. This Argand diagram represents the complex number lying on a plane.For each point on the plane, arg is the function which returns the angle . In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in ...
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.
In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. [1] Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts ...
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.