# Real part

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

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

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

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