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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 mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P. [1]
A complex number is real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division. Argument φ and modulus r locate a point in the complex plane.
The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an matrix made up of complex numbers. For an explanation of the notation used here, we begin by representing complex numbers e i θ {\displaystyle e^{i\theta }} as the rotation matrix, that is,
In fact, the same proof shows that Euler's formula is even valid for all complex numbers x. A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used ...
In the complex plane, the red points are the fifth roots of unity, and the black points are the sums of a fifth root of unity and its complex conjugate. In the complex plane, the corners of the two squares are the eighth roots of unity. For n = 1, 2, both roots of unity 1 and −1 are integers.
If is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), π (X) is the conjugate of −π(X*) for all X in g. For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.
A split-complex number has two real number components x and y, and is written = +. The conjugate of z is =. Since =, the product of a number z with its conjugate is ():= =, an isotropic quadratic form.