Real part

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An illustration of the complex plane. The real part of a complex number

z = x + i y

x

In mathematics, the real part of a complex number $z$ , is the first element of the ordered pair of real numbers representing $z$ , i.e. if $z = (x, y)$ , or equivalently, $z = x + i y$ , then the real part of $z$ is $x$ . It is denoted by Re{z} or $\Re$ {z}, where $\Re$ is a capital R in the Fraktur typeface. The complex function which maps $z$ to the real part of $z$ is not holomorphic.

In terms of the complex conjugate $\bar{z}$ , the real part of $z$ is equal to $z+\bar z\over2$ .

For a complex number in polar form, $z = (r,θ)$ , the Cartesian (rectangular) coordinates are $z = (r cosθ, r sinθ)$ , or equivalently, $z = r (cosθ + i sinθ).$ It follows from Euler's formula that $z = r e i θ$ , and hence that the real part of $r e i θ$ is $r cosθ$ .

Computations with real periodic functions such as alternating currents and electromagnetic fields are simplified by writing them as the real parts of complex functions. (see Phasor (sine waves))

Similarly, trigonometry can often be simplified by representing the sinusoids in terms of the real part of a complex expression, and perform the manipulations on the complex expression. For example:

$\begin{align} \cos(n\theta)+\cos[(n-2)\theta] & = \operatorname{Re}\left\{e^{in\theta} + e^{i(n - 2)\theta}\right\} \\ & = \operatorname{Re}\left\{(e^{i\theta} + e^{-i\theta})\cdot e^{i(n - 1)\theta}\right\} \\ & = \operatorname{Re}\left\{2\cos(\theta) \cdot e^{i(n - 1)\theta}\right\} \\ & = 2\cos(\theta) \cdot \operatorname{Re}\left\{e^{i(n - 1)\theta}\right\} \\ & = 2 \cos(\theta)\cdot \cos[(n - 1)\theta]. \end{align}$

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