Conjugate of Complex Number

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History of Complex Numbers

A fact that is surprising to many is that complex

numbers arose from the need to solve cubic

equations, and not (as it is commonly believed)

quadratic equations.

These notes track the development of complex

numbers in history, and give evidence that

supports the above statement.

Al-Khwarizmi (780-850) in his Algebra has

solution to quadratic equations of various types.

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In early ages the problem arise for the solution

of problems in which negative numbers comes

in under root.

To solve this problem, Rafael Bombelli authored

lAlgebra (1572, and 1579), a set of three books.

Bombelli introduces a notation for , and

calls it piu di meno.1

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The term Conjugate

Inversely or oppositely related with respect to

one of a group of otherwise identical

properties, especially designating either or

both of a pair of complex numbers differingonly in the sign of the imaginary term.

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In Mathematics In mathematics, the complex conjugate of acomplex number is given by changing the signof the imaginary part. Thus, the conjugate of the

complex number

z=a+ib

(where a and b are `numbers) is

z*=a-ib

The complex conjugate is also very commonly

denoted by z * . 7


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For example,

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Complex numbers are often depicted

as points in a plane with a Cartesian

coordinate . The x-axis contains the

real numbers and the y-axis contains

the multiples of i. In this view, complexconjugation corresponds to reflection

at the x-axis.

In polar form, however, the conjugate

of rei is given by re i. This can

easily be verified by using Euler's

formula.

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Properties

These properties apply for all complex

numbers zand w, unless stated otherwise.

ifwis non-zero

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Use as a variable

Once a complex number or

is given, its conjugate is sufficient to reproduce

the parts of the z-variable:

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Graphical Explanation

The complex number z* is just

like a mirrored image of z.

The graph shows us that the

function containing a complex

number and its complex conjugate

will be a discontinuous function so

creating a non analyticity property in

the function.

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f/z* =0

If f is an continuous function in Domain D

then f will be analytic iff/z* =0

Proof:Since, f = u + iv

z

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0!x

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This shows that the complex function whendifferentiated with its conjugate reruns zero inresult.

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Triangular Inequality

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Applications of Complex

Conjugates

When a real positive definite quantity is needed from a

real function, the square of the function can be used.

In the case of a complex function, the complex conjugate

is used to accomplish that purpose. The product of a complex number and its complex

conjugate is the complex number analog to squaring a

real function.

The complex conjugate is used in the rationalization of

complex numbers and for finding the amplitude of the

polar form of a complex number.

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One application of the complex conjugate in

physics is in finding the probability in quantum

mechanics.

Since the wave function which defines theprobability amplitude may be a complex

function, the probability is defined in terms of the

complex conjugate to obtain a real value.

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Complex Numbers in Real Life

In electronics, the state of a circuit element is described bytwo real numbers (the voltage Vacross it and the current Iflowing through it). A circuit element also may possess acapacitance Cand an inductance L that (in simplistic terms)describe its tendency to resist changes in voltage and current

respectively. These are much better described by complex numbers.

Rather than the circuit element's state having to be describedby two different real numbers Vand I, it can be described by asingle complex numberz= V+ iI. Similarly, inductance andcapacitance can be thought of as the real and imaginary parts

of another single complex numberw= C+ iL. The laws ofelectricity can be expressed using complex addition andmultiplication.

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Another example is electromagnetism. Rather than trying

to describe an electromagnetic field by two real

quantities (electric field strength and magnetic field

strength), it is best described as a single complex

number, of which the electric and magnetic components

are simply the real and imaginary parts.

A sinusoidal voltage of frequency can be thought of as

the real-valued part of a complex-valued exponential

function

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When such a voltage is passed through a circuit

of resistance R, capacitance C, and inductance

L, the circuit impedes the signal. The amount by

which it impedes the signal is called theimpedance and this is an example of the first

kind of application of complex numbers I

described above: a quantity with direct physical

relevance that is described by a complexnumber. It is given by

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Conjugate of Complex Number

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