Download - Complex numbers org.ppt
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PRESENTATION BYOSAMA TAHIR
09-EE-88
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COMPLEX NUMBERS
&COMPLEX PLANE
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A complex number is a number consisting of a Real and Imaginary part. It can be written in the form
COMPLEX NUMBERS
1i
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COMPLEX NUMBERS Why complex numbers are
introduced??? Equations like x2=-1 do not have a solution within the real numbers
12 x
1x
1i
12 i
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COMPLEX CONJUGATE
The COMPLEX CONJUGATE of a complex number
z = x + iy, denoted by z* , is given by
z* = x – iy The Modulus or absolute value
is defined by
22 yxz
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Complex Numbers
Real Numbers Imaginary Numbers
Real numbers and imaginary numbers are subsets of the set of complex numbers.
COMPLEX NUMBERS
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COMPLEX NUMBERS
Equal complex numbers
Two complex numbers are equal if theirreal parts are equal and their imaginaryparts are equal.
If a + bi = c + di, then a = c and b = d
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idbcadicbia )()()()(
ADDITION OF COMPLEX NUMBERS
i
ii
)53()12(
)51()32(
i83
EXAMPLE
Real Axis
Imaginary Axis
1z
2z
2z
sumz
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SUBTRACTION OF COMPLEX NUMBERS
idbcadicbia )()()()(
i
i
ii
21
)53()12(
)51()32(
Example
Real Axis
Imaginary Axis
1z
2z
2z
diffz
2z
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MULTIPLICATION OF COMPLEX NUMBERS
ibcadbdacdicbia )()())((
i
i
ii
1313
)310()152(
)51)(32(
Example
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DIVISION OF A COMPLEX NUMBERS
dic
bia
dic
dic
dic
bia
22
2
dc
bdibciadiac
22 dc
iadbcbdac
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EXAMPLE
i
i
21
76
i
i
i
i
21
21
21
76
22
2
21
147126
iii
41
5146
i
5
520 i
5
5
5
20 i i4
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Slide 14
COMPLEX PLANE
A complex number can be plotted on a plane with two perpendicular coordinate axes The horizontal x-axis, called the real axis The vertical y-axis, called the imaginary axis
P
z = x + iy
x
y
O
The complex plane
x-y plane is known as the complex plane.
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Pz = x + iy
x
y
O
Im
Re
θ
Geometrically, |z| is the distance of the point z from the origin while θ is the directed angle from the positive x-axis to OP in the above figure.
x
y1tan
θ is called the argument of z and is denoted by arg z. Thus,
0tanarg 1
z
x
yz
COMPLEX PLANE
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So any complex number, x + iy, can be written inpolar form:
Expressing Complex Number in Polar Form
sinry cosrx
irryix sincos
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Real axis
Imaginary axis
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De Moivre’s TheoremDe Moivre's Theorem is the theorem which shows us how to take complex numbers to any power easily.
Let r(cos F+isin F) be a complex number and n be any real number. Then
[r(cos F+isin F]n = rn(cosnF+isin nF)
[r(cos F+isin F]n = rn(cosnF+isin nF)
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Euler Formula
jre
jyxjrz
)sin(cos
yjye
eeee
jyxz
x
jyxjyxz
sincos
This leads to the complex exponential function :
The polar form of a complex number can be rewritten as
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A complex number, z = 1 - j has a magnitude
2)11(|| 22 z
Example
rad24
21
1tan 1
nnzand argument :
Hence its principal argument is :
rad
Hence in polar form :
4
zArg
4sin
4cos22 4
jezj
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APPLICATIONS
Complex numbers has a wide range of applications in Science, Engineering, Statistics etc. Applied mathematics Solving diff eqs with function of complex roots
Cauchy's integral formula
Calculus of residues
In Electric circuits to solve electric circuits
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Examples of the application of complex numbers:
1) Electric field and magnetic field.2) Application in ohms law.
3) In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes
4) A complex number could be used to represent the position of an object in a two dimensional plane,
How complex numbers can be applied to “The Real World”???
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REFERENCES..
Wikipedia.com Howstuffworks.com Advanced
Engineering Mathematics
Complex Analysis
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THANK YOUFOR YOUR ATTENTION..!