In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i.e. if , or equivalently, , then the real part of is . It is denoted by or . The complex function which maps to the real part of is not holomorphic.
In terms of the complex conjugate, the real part of is equal to .
For a complex number in polar form, , or equivalently, , it follows from Euler's formula that , and hence that the real part of is .
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.