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As (+) = and (+) + =, the sum and the product of conjugate expressions do not involve the square root anymore. This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation).
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.
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 other words, the scalar multiplication of ¯ satisfies = ¯ where is the scalar multiplication of ¯ and is the scalar multiplication of . The letter v {\displaystyle v} stands for a vector in V , {\displaystyle V,} α {\displaystyle \alpha } is a complex number, and α ¯ {\displaystyle {\overline {\alpha }}} denotes the complex conjugate ...
The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if p and q are quaternions, then (pq) ∗ = q ∗ p ∗, not p ∗ q ∗. The conjugation of a quaternion, in stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:
The self dot product of a complex vector =, involving the conjugate transpose of a row vector, is also known as the norm squared, = ‖ ‖, after the Euclidean norm; it is a vector generalization of the absolute square of a complex scalar (see also: Squared Euclidean distance).
The division process is analogous to complex division in that the denominator is multiplied by its conjugate in order to cancel the non-real parts. Therefore, to evaluate an expression of the form + + we multiply the numerator and denominator by the conjugate of the denominator:
In it, geometrical shapes can be made, as well as expressions from the normal graphing calculator, with extra features. [8] In September 2023, Desmos released a beta for a 3D calculator, which added features on top of the 2D calculator, including cross products, partial derivatives and double-variable parametric equations. [9]