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A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are generally assumed to have a domain that contains a nonempty open subset of the complex plane.
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 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 of ...
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.}
All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation
More precisely, let f be a function from a complex curve M to the complex numbers. This function is holomorphic (resp. meromorphic) in a neighbourhood of a point z of M if there is a chart ϕ {\displaystyle \phi } such that f ∘ ϕ − 1 {\displaystyle f\circ \phi ^{-1}} is holomorphic (resp. meromorphic) in a neighbourhood of ϕ ( z ...
For each nonzero complex number , the principal value is the logarithm whose imaginary part lies in the interval (,]. [2] The expression Log 0 {\displaystyle \operatorname {Log} 0} is left undefined since there is no complex number w {\displaystyle w} satisfying e w = 0 {\displaystyle e^{w}=0} .
The complex numbers with integer part all zeroes in the base i – 1 system Of particular interest are the quater-imaginary base (base 2 i ) and the base −1 ± i systems discussed below, both of which can be used to finitely represent the Gaussian integers without sign.
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