In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a hypercomplex number

Hypercomplex number

In mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...

 has a real part and an imaginary part associated with it. This is most familiar in the context of complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, but extends to the other hypercomplex algebras such as split-complex number

Split-complex number

In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...

s and quaternion

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s.

A complex number is the sum of a real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 – which is the real part, and an imaginary number

Imaginary number

An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...

 – some real number multiplied by the imaginary unit

Imaginary unit

In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

. By convention, for complex numbers the imaginary part is only a real number – not including the imaginary unit. Similarly, in higher-dimensional algebras such as quaternions, a general element is the sum of a real number and two or more "imaginary units" with real coefficients, where the imaginary part refers to the entire expression excluding the initial real number; in these algebras the imaginary units are retained in the definition of the imaginary part.

In brief: if z is a complex number , where xand y are real, then x is the real part of z, y is the imaginary part – not iy (a pure imaginary number), both the real and imaginary parts are themselves the real numbers which make up the complex number. For a quaternion written as , the real part is a and the imaginary part is .

Definition

The Cartesian representation (x, y) for a complex number z is:

the polar form (r, θ) of z is:

where:

• 
• denotes the imaginary unit
  Imaginary unit
  In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
  
  ,
• x, y, r, θ are real numbers, i.e. elements of the set of real numbers R: .

Both representations can be connected using Euler's formula

Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...

:

Using these equivalent forms ;

• the real part of z is
• the imaginary part of z is not including the imaginary unit i.

The imaginary number constituting z above is which does include the i.

Notation

The real part of a complex number z is denoted by or , and the imaginary part by or , in which and are black-letter or fraktur

Fraktur

Fraktur is a calligraphic hand and any of several blackletter typefaces derived from this hand. The word derives from the past participle fractus of Latin frangere...

 capital R and I, respectively. The notation using "Re" and "Im" is notationally cleaner, asthetic and easier to reproduce, and is more frequently employed than the printed-only fraktur. The notations without parentheses are sometimes used, or and or , whenever there is no danger of ambiguity.

Statement (complex conjugates)

In terms of the complex conjugate

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

 ;

the real part of is:
and its imaginary part:

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

Applications

Real and imaginary parts can be used to simplify some expressions.

In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, and (typically) mechanical

Mechanical engineering

Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...

 and electrical engineering

Electrical engineering

Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

, the use of complex numbers is labour-saving. Oscillations can be represented using complex numbers in the form of both exponentials and circular trigonometric functions, so calculations can be done in whichever form is more convenient. At the end result - the real part is taken as the physical and measurable answer, and the imaginary part can simply be discarded.

Trigonometry

Similarly, trigonometry can often be simplified by directly representing the sinusoidal functions in terms of the real or imaginary part of a complex expression, with manipulations performed on the complex expression. For example:

Because the real part of the first expression equals the real part of the last expression, we can conclude that
which is the angle addition formula for cosines. Similarly, equality of the imaginary parts gives the angle addition formula for sines,

Electrical engineering

Since the voltage in an AC circuit

Circuit

Circuit may mean:In science and technology*Circuit theory, the theory of accomplishing work by routing electrons, gas, fluids, or other matter through a loop*Pneumatic circuit*Hydraulic circuit*Boolean circuit*circuit...

 is oscillating, it can be represented as a phasor

Phasor

Phasor is a phase vector representing a sine wave.Phasor may also be:* Phasor , a stereo music, sound and speech synthesizer for the Apple II computer* Phasor measurement unit, a device that measures phasors on an electricity grid...

:


To obtain the measurable quantity, the real part is taken:


See for example .

Signal transmission engineering

Another example, relevant to the two side bands of amplitude modulation

Amplitude modulation

Amplitude modulation is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. AM works by varying the strength of the transmitted signal in relation to the information being sent...

of AM radio, is: