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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 imaginary 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 second element of the ordered pair of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s representing i.e. if , or equivalently, , then the imaginary part of is . It is denoted by or , where is a capital I in the Fraktur typeface. The complex function which maps to the imaginary 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 imaginary part of z is equal to .

For a complex number in polar form, , 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 imaginary part of is .

In electric power
Electric power

Electric power is defined as the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt .When electric current flows in a circuit, it can transfer energy to do mechanical work or work ....
, 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 in the power wires can be represented as a complex number (electrical engineers use to indicate the imaginary unit rather than , which also represents current
Electric current

Electric current is the flow of electric charge. The electric charge may be either electrons or ions.The International System of Units unit of electric current intensity is the ampere....
).






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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 imaginary 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 second element of the ordered pair of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s representing i.e. if , or equivalently, , then the imaginary part of is . It is denoted by or , where is a capital I in the Fraktur typeface. The complex function which maps to the imaginary 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 imaginary part of z is equal to .

For a complex number in polar form, , 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 imaginary part of is .

In electric power
Electric power

Electric power is defined as the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt .When electric current flows in a circuit, it can transfer energy to do mechanical work or work ....
, 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 in the power wires can be represented as a complex number (electrical engineers use to indicate the imaginary unit rather than , which also represents current
Electric current

Electric current is the flow of electric charge. The electric charge may be either electrons or ions.The International System of Units unit of electric current intensity is the ampere....
). The "real current" 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" 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.

See also

  • Real part
    Real part

    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 ....
  • 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....