Post on 18-Jan-2016
Review of Complex numbers
1
Exponential Form:
π§=π₯+ ππ¦ π§=π πππ=|π§|ππ πRectangular Form:
Real
Imag
x
y
f
r=|z|
π§
π π (π§ )=π₯
πΌπ (π§ )=π¦
The real and imaginary parts of a complex number in rectangular form are real numbers:
Real
Imag
x=Re(z)
y=Im(z)
π§
π₯+ ππ¦=π π (π§ )+ππΌπ(π§)
Therefore, rectangular form can be equivalently written as:
Real & Imaginary Parts of Rectangular Form
Real
Imag
x
f
r=|z|
π§
The real and imaginary components of exponential form can be found using trigonometry:
cosπ=πππhπ¦π
=π₯π
Geometry Relating the Forms
y
Real
Imag
f
r=|z|
π§
sin π=πππhπ¦π
=π¦π
π₯=π cosπ
π¦=π sin π
β
β
πΌπ (π§ )=π¦=π sin π=|π§|sin π
Geometry Relating the Forms: Real & Imaginary Parts
Real
Imag
π
r=|z|
π§The real and imaginary parts of a complex number can be expressed as follows:
Geometry Relating the Forms: Quadrants
In exponential form, the positive angle, , is always defined from the positive real axis. If the complex number is not in the first quadrant, then the βtriangleβ has lengths which are negative numbers.
cosπ=πππhπ¦π
=ΒΏ π₯β¨ ΒΏπ
ΒΏ
Real
Imag
x
y
f
r=|z|
π§
π
π₯=βπ cosπ=π cosπReal
Imag
π₯>0π¦>0
π₯>0π¦<0
π₯<0π¦<0
π₯<0π¦>0
Real
Imag
x
y
r=|z|
π§
π2=|π§|2=π₯2+π¦ 2
Use Pythagorean Theorem
π=|π§|=βπ₯2+π¦2
to find in terms of and :
Geometry Relating the Forms: in terms of and
Geometry Relating the Forms: in terms of and
tanπ= π sin ππ cosπ
=πΌπ(π§)π π(π§)
= π¦π₯
Real
Imag
x
y
f
r=|z|
π§
tanπ=ππππππ
adj
opp
π
hypUse trigonometry
to find in terms of and
π=tanβ1( π¦π₯ )
Summary of Algebraic Relationships between Forms
Real
Imag
x
y
f
r=|z|
π§ π₯=π cosπ
π¦=π sin π
π=|π§|=βπ₯2+π¦2
π=tanβ1( π¦π₯ )
πππ=cosπ+ π sinπ
Eulerβs Formula
π§=π₯+ ππ¦
ΒΏπ cosπ+ππ sin π=|π§|cosπ+π|π§|sin π
)
ΒΏπ πππ=ΒΏ π§β¨πππ
Rectangular Form:
Exponential Form:
Consistency argument
If these represent the same thing, then the assumed Euler relationship says:
π§=π₯+ ππ¦ π§=π πππ=ΒΏ π§β¨πππ
11
πππ=exp (ππ )=cosπ+π sinπ
Eulerβs Formula
πππ 0π‘=exp (ππ0π‘)=cosπ0 π‘+π sinπ0 π‘
Can be used with functions:
Addition and subtraction of complexnumbers is easy in rectangular form
12
Addition & Subtraction of Complex Numbers
π§1=π+ππ π§ 2=π+ππ
π§=π§1+π§ 2=π+ππ +π+ππΒΏ (π+π)+π (π+π)
Addition and subtraction are analogous to vector addition and subtraction
Real
Imag
ab
d
c
π§ 2
π§1
οΏ½βοΏ½1=π οΏ½ΜοΏ½+π οΏ½ΜοΏ½ οΏ½βοΏ½ 2=π οΏ½ΜοΏ½+π οΏ½ΜοΏ½
οΏ½βοΏ½= οΏ½βοΏ½1+π§ 2=(π+π ) οΏ½ΜοΏ½+(π+π) οΏ½ΜοΏ½
x
y
ab
d
c
οΏ½βοΏ½1
οΏ½βοΏ½1
οΏ½βοΏ½ 2
οΏ½βοΏ½
π§
Multiplication of Complex Numbers
13
Multiplication of complex numbers is easy in exponential form
π§1=π1ππ π π§ 2=π2π
ππ
π§=π§1 π§2=π1ππ ππ 2π
π π
ΒΏπ1π2ππ(π+π )
ΒΏΒΏ π§ 1β¨ΒΏ π§ 2β¨ππ(π+π )
Multiplication by a complex number, , can be thought of as scaling by and rotation by
Real
Imag
π§
π§π ππ π
πMagnitude scaled by
Angle rotated counterclockwise by
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Division of Complex NumbersDivision of complex numbers is
easy in exponential form
π§1=π1ππ π π§ 2=π2π
ππ
π§=π§1
π§2
=π1π
π π
π2ππ π
ΒΏπ1
π2
ππ(πβπ )
ΒΏΒΏ π§ 1β¨ΒΏ
ΒΏ π§2β¨ΒΏππ(πβπ )ΒΏΒΏ
Division of complex numbers is sometimes easy in rectangular form
π§=π+πππ+ππ
ΒΏπ+πππ+ ππ
πβ πππβ ππ
ΒΏππ+ππ+π (ππβππ)
π2+π2
ΒΏ ππ+πππ2+π2 +π
(ππβππ)π2βπ2
ΒΏπ π(π§ )+ππΌπ(π§ )ΒΏβ¨π§β¨ππ(πβπ )
Multiply by 1 using the complex conjugate of the denominator
Complex Conjugate
Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c.: change i -i
π§=π₯+ ππ¦π§β=π₯βπ π¦
π§=π πππ
π§β=π πβ ππ
Real
Imag
x
y
f
r=|z|
π§
π§β
The complex conjugate is a reflection about the real axis
The product of a complex number and its complex conjugate is REAL.
Common Operations with the Complex Conjugate
Addition of the complex number and its complex conjugate results in a real number
π§+π§β=π₯+ ππ¦+π₯βππ¦ΒΏ2 π₯
π§ π§β=π ππ ππ πβπ π
ΒΏπ 2ππ(πβπ )
ΒΏπ 2
ΒΏΒΏ π§β¨ΒΏ2 ΒΏ
Real
Imag
x
y
f
r=|z|
π§
π§β
x