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Complex numbers
The need for imaginary and complex numbers arises when finding the
two roots of a quadratic equation.
The two roots are given by the quadratic formula
There are no problems as long as (β/2α)
2 ≥ γ/α – there are two real roots
and everything is clean. But if (β/2α)
2 < γ/α, then we are faced with
having to take the square-root of a negative number.
In “ancient” times, such situations were deemed impossible and simply
ignored. And yet, physical systems described by the “impossible”
parameters continued to function, generally with very interesting
results. Clearly, ignoring the problem is not helpful.
αx
2
x =
β
2 α
β
2 α
2
γ
α
So what to do when faced with such situations?
z = a +
b
2
It took a couple of hundred years, but the people working on the
problem realized that the square-root term had useful physical
information and could not be ignored. However, square-root term was
different from the real number represented by the first term. The second
term had to be treated in a special way, and a new algebra had to be
developed to handle these special numbers. (Actually, the new algebra
is an extension of the old real number algebra.
The special nature of the square-root term is signified by introducing a
new symbol.
b
2
=
b
2
= jb
where and b is conventional real number.
(Note: In almost all other fields, it is conventional to use.
However, in EE/CprE, we use i for current, and so it has become
normal practice in our business to use j .)
j = − 1
i = − 1
How to work with this new type of number? Clearly, an imaginary
number is somehow different from a familiar real number. In thinking
about how real numbers relate to each other and when visualizing
functions of real numbers, we often start with a real number line. All
real numbers are represented by a point on the line. Similarly,
imaginary numbers can be represented by points on an imaginary
number line.
real
imaginary
Now we have two number lines – how are they related? In working this
out, the early mathematicians came to the conclusion that the
imaginary axis is perpendicular to the real axis, so that the two axes
form what is essentially an x-y set of coordinates. The real and
imaginary parts of a complex number give the coordinates of a point in
the complex plane.
Complex number plane
1 + j 1
2 – j 1
! + j 2
6 + j
2
0 + j 2.
–1.5 + j 0
–1 – j 2
Complex math – multiplication
Multiplication is also straight-forward. It is essentially the same as
multiplying polynomials — just make sure that every term is multiplied
by every other term. The result will be a mixing of the reals and
imaginaries from the two factors, and these will need to be sorted out
for the final result.
z 1 · z 2 = ( a + jb ) · ( c + jd ) = ac + jad + jbc + ( j )
2 bd
Note that the two imaginary terms multiply together to give a real, since
j
2 = –1. Collect the real and imaginary parts to write the complex
number in standard form.
z 1 · z 2 = ( a + jb ) · ( c + jd ) = ( ac – bd ) + j ( ad + bc )
Complex math – complex conjugates
The two roots that are the solutions to a quadratic equation may be
complex. In that case, the roots come as set:
z 1 = a + jb and z 2 = a – jb
The same real part and the imaginary parts have
opposite signs.
Numbers having this relationship are known as
complex conjugates. Every complex number, z, has a
conjugate, denoted as z*. From above
z 1
* = a – jb and z 2
* = a + jb
Again, the two roots are complex conjugates of each other.
Conjugates in the
complex plane.
z + z
= ( a + jb ) + ( a jb ) = 2 a
purely real!
purely real
z z purely imaginary
= ( a + jb ) ( a jb ) = j ( 2 b )
z · z
= ( a + jb ) · ( a jb ) = a
2
jab + jab + b
2
= a
2
2
re
im
z
1
z
2
z
3
z
2
z
3
z
1
Polar representation
re
im
z
1
a
b
M
θ
Specifying a complex number in the complex
plane using the real and imaginary parts is
quite simple — it the same as specifying
points with rectangular coordinates.
However, we also know that a point can be specified using polar
coordinates. In our case, we would locate the complex number in the
plane by specifying an angle (or heading) and distance from the origin
to the point along that heading. To describe the complex number in
polar form, we use the magnitude ( M ) and the angle ( θ ). A commonly
used notation for specifying a complex number in polar form is to list
the magnitude followed by the angle inside a “angle bracket” simple.
z 1
= M (^) θ
You might see this notation in many circuits texts. We will not use it in
EE 201, because there is a better notation that is more descriptive.
re
im
z
1
a
b
M
θ
rectangular to polar (and back)
From the plot in the complex plane, we see
that the conversion from rectangular form
( a + jb ) to polar form ( M θ ) is a simple
application of trigonometry.
M = a
2
2
θ = arctan
b
a
b = M sin θ
It is equally easy to convert from polar to rectangular.
a = M cos θ
pseudo-proof
exp ( θ ) = 1 +
θ
θ
2
θ
3
θ
4
θ
5
θ
6
θ
7
= 1 + j
θ
θ
2
− j
θ
3
θ
4
θ
5
θ
6
− j
θ
7
[
θ
2
θ
4
θ
6
]
[
θ
θ
3
θ
5
θ
7
]
cos θ = 1 −
θ
2
θ
4
θ
6
θ
θ
3
θ
5
θ
7
cos θ + j sin θ =
[
θ
2
θ
4
θ
6
]
[
θ
θ
3
θ
5
θ
7
]
cos θ + j sin θ = exp (
jθ )
The expression exp( jθ ) is a complex number pointing at an angle of θ and
with a magnitude of 1. (M = 1). We can use this notation to express other
complex numbers with M ≠ 1 by multiplying by the magnitude.
M exp (
jθ )
This is just another way of expressing a complex number in polar form.
M θ
same as
z = M exp (
jθ )
Using Euler’s formula:
z = M exp (
jθ )
= M cos θ + jM sin θ = a + jb
z 1
= 4 + j 3
z 2
= 2 + j 2
z 1
= 5 exp (
j 36.
∘
z 2
= 2.83 exp (
j 45
∘
z
p
= z 1
⋅ z 2
4 + j 3 ) (
2 + j 2 )
= ( 4 ⋅ 2 − 3 ⋅ 2 ) + j ( 4 ⋅ 2 + 3 ⋅ 2 ) = 2 + j 14
z p
= 2 + j (^14) z
p
= 14.1 exp (
j 81.
∘
z p
= z 1
⋅ z 2
[
5 exp (
j 36.
∘
] [
2.83 exp (
j 45
∘
]
= 14.1 exp (
j 81.
∘
z q
z 1
z 2
4 + j 3
2 + j 2
( 4 ⋅ 2 + 3 ⋅ 2 ) + j (− 4 ⋅ 2 + 3 ⋅ 2 )
2
2
= 1.75 − j 0.
z q
= 1.75 − j 0.25 z q
= 1.77 exp (
− j 8.
∘
z q
z 1
z 2
5 exp (
j 36.
∘
2.83 exp (
j 45
∘
= 1.77 exp (
− j 81.
∘
The complex conjugate in polar form is also quite easy.
z = a + jb
z * = a − jb
z = M exp (
jθ )
z * = M exp (
− jθ )
M * = a
2
2
= M
θ * = arctan
− b
a
= − arctan
b
a
= − θ
M = a
2
2 θ = arctan
b
a
z ⋅ z * =
[
M exp (
jθ ) ] [
M exp (
− jθ ) ]
= M
2
= a
2
2
As expected.
re
im
z
a
b
M
z*
θ
- θ
–b
M
