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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the real part of a complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 , is the first element of the ordered pair
Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate ....
 of real numbers representing , i.e. if , or equivalently, , then the real part of is . It is denoted by Re or , where is a capital R in the Fraktur typeface. The complex function which maps to the real part of is not holomorphic.

In terms of the complex conjugate
Complex conjugate

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

For a complex number in polar form, , the Cartesian (rectangular) coordinates are , or equivalently, It follows from Euler's formula
Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematics formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function....
 that , and hence that the real part of is .

Computations with real periodic functions such as alternating currents and electromagnetic fields are simplified by writing them as the real parts of complex functions.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the real part of a complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 , is the first element of the ordered pair
Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate ....
 of real numbers representing , i.e. if , or equivalently, , then the real part of is . It is denoted by Re or , where is a capital R in the Fraktur typeface. The complex function which maps to the real part of is not holomorphic.

In terms of the complex conjugate
Complex conjugate

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

For a complex number in polar form, , the Cartesian (rectangular) coordinates are , or equivalently, It follows from Euler's formula
Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematics formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function....
 that , and hence that the real part of is .

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:



See also

  • Imaginary part
    Imaginary part

    In mathematics, the imaginary part of a complex number , is the second element of the ordered pair of real numbers representing i.e. if , or equivalently, , then the imaginary part of is ....
  • Imaginary number
    Imaginary number

    In mathematics, an imaginary number is a complex number whose square value is a real number not greater than zero. The imaginary unit, denoted by i or j, is an example of an imaginary number....