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 + iy, then the real part of z is x. It is denoted by Re{z} or , where is a capital R in the Fraktur typeface. The complex function which maps z to the real part of z is not holomorphic. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ... Please refer to Real vs. ... The German word Fraktur (pronounced in the International Phonetic Alphabet (IPA)) refers to a specific sub-group of blackletter typefaces. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

In terms of the complex conjugate , the real part of z is equal to . In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

For a complex number in polar form, z = (r,θ), or equivalently, z = r(cosθ + isinθ), it follows from Euler's formula that z = re^iθ, and hence that the real part of re^iθ is rcosθ. This article describes some of the common coordinate systems that appear in elementary mathematics. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

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. Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ...

See also

• Complex number
• Imaginary part
• Imaginary number
• Real number