# Real part

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

In terms of the complex conjugate${\displaystyle {\bar {z}}}$, the real part of ${\displaystyle z}$ is equal to ${\displaystyle z+{\bar {z}} \over 2}$.

For a complex number in polar form, ${\displaystyle z=(r,\theta )}$, or equivalently, ${\displaystyle z=r(cos\theta +i\sin \theta )}$, it follows from Euler's formula that ${\displaystyle z=re^{i\theta }}$, and hence that the real part of ${\displaystyle re^{i\theta }}$ is ${\displaystyle r\cos \theta }$.

Sometimes 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 for example electrical impedance.