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An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.

In mathematics, the real and imaginary parts of a complex number z are respectively the first and the second elements of the ordered pair of real numbers representing z. If z = x + iy (with i denoting the imaginary unit), then x and y are respectively the real part and the imaginary part of z.

The real part of a complex number z is denoted by Re(z) or (z) and the imaginary part by Im(z) or (z), where and are black-letter capital R and I (Unicode U+211C and U+2111, HTML entities ℜ and ℑ, TeX \Re and \Im), respectively. The notations without parentheses are also used, Re z or z and Im z or z, whenever there is no danger of ambiguity.

For a complex number in polar form, z = (r,θ), the Cartesian (rectangular) coordinates are z = (rcosθ,rsinθ), or equivalently, z = r(cosθ + isinθ). It follows from Euler's formula that z = reiθ, and hence that the real part of reiθ is rcosθ and its imaginary part is rsinθ.

In terms of the complex conjugate \bar{z}, the real part of z is equal to z+\bar z\over2, and its imaginary part to \frac{z-\bar{z}}{2i}.

The complex function which maps z to the real part of z is not holomorphic, and neither is the function mapping it to its imaginary part.

Applications

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)).

In electric power, when a sine wave voltage drives a "linear" load (in other words, a load that makes the current also be a sine wave), the current I in the power wires can be represented as a complex number I = x + jy (electrical engineers use j to indicate the imaginary unit rather than i, which also represents current). The "real current" x is related to the current when the voltage is maximum. The real current times the voltage gives the actual power consumed by the load (often all that power is dissipated as heat). The "imaginary current" y is related to the current when the voltage is zero. A load with purely imaginary current (such as a capacitor or inductor) dissipates no power; it merely accepts power temporarily then later pushes that power back on the power lines.

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:

 \begin{align} \cos(n\theta)+\cos\left((n-2)\theta\right) & = \operatorname{Re}\left(e^{in\theta} + e^{i(n - 2)\theta}\right) \ & = \operatorname{Re}\left((e^{i\theta} + e^{-i\theta})\cdot e^{i(n - 1)\theta}\right) \ & = \operatorname{Re}\left(2\cos(\theta) \cdot e^{i(n - 1)\theta}\right) \ & = 2\cos(\theta) \cdot \operatorname{Re}\left(e^{i(n - 1)\theta}\right) \ & = 2 \cos(\theta)\cdot \cos\left((n - 1)\theta\right). \end{align}

References

  • Conway, John B., Functions of One Complex Variable I (Graduate Texts in Mathematics), Springer; 2 edition (September 12, 2005). ISBN 0387903283.

R symbol]]

. The real part of a complex number z = x+iy is x.]] 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 \Re{z}, where \Re is a capital R in the Fraktur typeface. The complex function which maps z to the real part of z is not holomorphic.

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

For a complex number in polar form, z = (r, \theta ), the Cartesian (rectangular) coordinates are z = (r \cos\theta, r \sin\theta), or equivalently, z = r (\cos \theta + i \sin \theta). It follows from Euler's formula that z = r e^{i\theta}, and hence that the real part of r e^{i\theta} is r \cos\theta.

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:

\begin{align} \cos(n\theta)+\cos[(n-2)\theta] & = \operatorname{Re}\left\{e^{in\theta} + e^{i(n - 2)\theta}\right\} \\ & = \operatorname{Re}\left\{(e^{i\theta} + e^{-i\theta})\cdot e^{i(n - 1)\theta}\right\} \\ & = \operatorname{Re}\left\{2\cos(\theta) \cdot e^{i(n - 1)\theta}\right\} \\ & = 2\cos(\theta) \cdot \operatorname{Re}\left\{e^{i(n - 1)\theta}\right\} \\ & = 2 \cos(\theta)\cdot \cos[(n - 1)\theta]. \end{align}

See also








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