It is a profound result that this simple device is all we need to guarantee that every poly-
nomial equation has a solution, and that every polynomial of degree n can be factored into
n polynomials of degree one, as in (B.3).
B.2
Arithmetic of imaginary numbers
The sum of i (or i1) and i is written i2. Sums and differences of imaginary numbers
simplify like real numbers:
i3 + i2 = i5, i3 − i4 = −i.
If iy1 and iy2 are two imaginary numbers, then
iy1 + iy2 = i(y1 + y2),
iy1 − iy2 = i(y1 − y2).
(B.6)
1Here, the operator is ordinary multiplication, not products of sets.
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B.3. COMPLEX NUMBERS
The product of a real number x and an imaginary number iy is
x × iy = iy × x = ixy.
To take the product of two imaginary numbers, we must remember that i2 = −1, and so
for any two imaginary numbers, iy1 and iy2, we have
iy1 × iy2 = −y1 × y2.
(B.7)
The result is a real number. We can use rule (B.7) repeatedly to multiply as many imagi-
nary numbers as we wish. For example,
i × i = −1, i3 = i × i2 = −i, i4 = 1.
The ratio of two imaginary numbers iy1 and iy2 is a real number
iy1
y1
=
.
iy2
y2
B.3
Complex numbers
The sum of a real number x and an imaginary number iy is called a complex number. This
sum does not simplify as do the sums of two reals numbers or two imaginary numbers,
and it is written as x + iy or x + jy.
Examples of complex numbers are
√
2 + i, −3 − i2, −π + i 2.
In general a complex number z is of the form
√
z = x + iy = x +
−1y,
where x, y are real numbers. The real part of z, written Re{z}, is x. The imaginary part
of z, written Im{z}, is y. Notice that, confusingly, the imaginary part is a real number.
The imaginary part times i is an imaginary number. So
z = Re{z} + iIm{z}.
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The set of complex numbers, therefore, is defined by
√
C = {x + iy | x ∈ R, y ∈ R, and i =
−1}.
(B.8)
Every real number x is in C , because x = x + i0; and every imaginary number iy is in C ,
because iy = 0 + iy.
Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal if and only if their real
parts are equal and their imaginary parts are equal, i.e. z1 = z2 if and only if
Re{z1} = Re{z2}, and Im{z1} = Im{z2}.
B.4
Arithmetic of complex numbers
In order to add two complex numbers, we separately add their real and imaginary parts,
(x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2).
The complex conjugate of x + iy is defined to be x − iy. The complex conjugate of a
complex number z is written z∗. Notice that
z + z∗ = 2Re{z}, z − z∗ = 2iIm{z}.
Hence, the real and imaginary parts can be obtained using the complex conjugate,
z + z∗
z − z∗
Re{z} =
, and Im{z} =
.
2
2i
The product of two complex numbers works as expected if you remember that i2 = −1.
So, for example,
(1 + 2i)(2 + 3i) = 2 + 3i + 4i + 6i2 = 2 + 7i − 6 = −4 + 7i,
which seems strange, but follows mechanically from i2 = −1. In general,
(x1 + iy1)(x2 + iy2) = (x1x2 − y1y2) + i(x1y2 + x2y1).
(B.9)
If we multiply z = x + iy by its complex conjugate z∗ we get
zz∗ = (x + iy)(x − iy) = x2 + y2,
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B.5. EXPONENTIALS
which is a positive real number. Its positive square root is called the modulus or magni-
tude of z, and is written |z|,
√
|z| = zz∗ = x2 + y2 .
How to calculate the ratio of two complex numbers is less obvious, but it is equally me-
chanical. We convert the denominator into a real number by multiplying both numerator
and denominator by the complex conjugate of the denominator,
2 + 3i
2 + 3i
=
× 1 − 2i
1 + 2i
1 + 2i
1 − 2i
(2 + 6) + (−4 + 3)i
=
1 + 4
8
=
− 1 i.
5
5
The general formula is
x1 + iy1
x1x2 + y1y2
−x1y2 + x2y1
=
+ i
.
(B.10)
x2 + iy2
x2 + y2
x2 + y2
2
2
2
2
In practice it is easier to calculate the ratio as in the example, rather than memorizing
formula (B.10).
B.5
Exponentials
Certain functions of real numbers, like the exponential function, are defined by an infinite
series. The exponential of a real number x, written ex or exp(x), is
∞
xk
x2
x3
ex = ∑
= 1 + x +
+
+ · · · .
k!
2!
3!
k=0
We also recall the infinite series expansion for cos and sin:
2
4
θ
cos(θ)
=
1 − θ +
− ···
2
4!
3
5
θ
sin(θ)
=
θ − θ +
− ···
3!
5!
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The exponential of a complex number z is written ez or exp(z), and is defined in the same
way as the exponential of a real number,
∞
zk
z2
z3
ez = ∑
= 1 + z +
+
+ · · · .
(B.11)
k!
2!
3!
k=0
Note that e0 = 1, as expected.
The exponential of an imaginary number iθ is very interesting,
(iθ)2
(iθ)3
eiθ
=
1 + (iθ) +
+
+ · · ·
2!
3!
2
4
3
5
θ
θ
=
[1 − θ +
− ···] + i[θ − θ +
− ···]
2
4!
3!
5!
=
cos(θ) + i sin(θ).
This identity is known as Euler’s formula:
eiθ = cos(θ) + i sin(θ).
(B.12)
Euler’s formula is used heavily in this text in the analysis of linear time invariant systems.
It allows sinusoidal functions to be given as sums or differences of exponential functions,
cos(θ) = (eiθ + e−iθ)/2
(B.13)
and
sin(θ) = (eiθ − e−iθ)/(2i).
(B.14)
This proves useful because exponential functions turn out to be simpler mathematically
(despite being complex valued) than sinusoidal functions.
An important property of the exponential function is the product formula:
ez1+z2 = ez1 ez2 .
(B.15)
We can obtain many trigonometric identities by combining (B.12) and (B.15). For example, since
eiθe−iθ = eiθ−iθ = e0 = 1,
and
eiθe−iθ = [cos(θ) + i sin(θ)][cos(θ) − i sin(θ)] = cos2(θ) + sin2(θ),
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B.6. POLAR COORDINATES
so we have the identity
cos2(θ) + sin2(θ) = 1.
Here is another example. Using
ei(α+β) = eiαeiβ,
(B.16)
we get
cos(α + β) + i sin(α + β)
=
[cos(α) + i sin(α)][cos(β) + i sin(β)]
=
[cos(α) cos(β) − sin(α) sin(β)]
+i[sin(α) cos(β) + cos(α) sin(β)].
Since the real part of the left side must equal the real part of the right side, we get the
identity,
cos(α + β) = cos(α) cos(β) − sin(α) sin(β),
Since the imaginary part of the left side must equal the imaginary part of the right side,
we get the identity,
sin(α + β) = sin(α) cos(β) + cos(α) sin(β).
It is much easier to remember (B.16) than to remember these identities.
B.6
Polar coordinates
The representation of a complex number as a sum of a real and an imaginary number,
z = x + iy, is called its Cartesian representation.
Recall from trigonometry that if x, y, r are real numbers and r2 = x2 + y2, then there is a
unique number θ with 0 ≤ θ < 2π such that
x
y
cos(θ) = , sin(θ) = .
r
r
That number is
θ = cos−1(x/r) = sin−1(y/r) = tan−1(y/x).
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We can therefore express any complex number z = x + iy as
x
y
z = |z|( |z| +i|z|) = |z|(cosθ+isinθ) = |z|eiθ,
where θ = tan−1(y/x). The angle or argument θ is measured in radians, and it is written
as arg(z) or ∠z. So we have the polar representation of any complex number z as
z = x + iy = reiθ.
(B.17)
The two representations are related by
r = |z| =
x2 + y2
and
θ = arg(z) = tan−1(y/x).
The values x and y are called the Cartesian coordinates of z, while r and θ are its polar
coordinates. Note that r is real and r ≥ 0.
Figure B.1 depicts the Cartesian and polar representations. Note that for any integer K,
rei(2Kπ+θ) = reiθ.
Basics: From Cartesian to polar coordinates
The polar representation of a complex number z is
z = x + iy = reiθ,
where
r = |z| =
x2 + y2
and
θ = arg(z) = tan−1(y/x).
However, you must be careful in calculating tan−1(y/x). For any angle θ,
tan(θ) = tan(θ + π).
Thus, for any real number y/x, there are two possible values for θ = tan−1(y/x) that lie
within the range [0, 2π). You should select the one of these that yields a non-negative
value for r in
z = x + iy = reiθ = r(cos(θ) + i sin(θ)).
This choice makes it reasonable to interpret r as the magnitude of the complex number.
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B.6. POLAR COORDINATES
Im
y
z = x + iy
r
!
Re
x
Figure B.1: A complex number z is represented in Cartesian coordinates as z =
x + iy and in polar coordinates as z = reiθ. The x-axis is called the real axis, the
y axis is called the imaginary axis. The angle θ in radians is measured counter-
clockwise from the real axis.
This is because
rei(2Kπ+θ) = rei2Kπeiθ
and
ei2Kπ = cos(2Kπ) + i sin(2Kπ) = 1.
Thus, the polar coordinates (r, θ) and (r, θ + 2Kπ) for any integer K represent the same
complex number. Thus, the polar representation is not unique; by convention, a unique
polar representation can be obtained by requiring that the angle given by a value of θ
satisfying 0 ≤ θ < 2π or −π < θ ≤ π. We normally require 0 ≤ θ < 2π.
Example B.1: The polar representation of the number 1 is 1 = 1ei0. Notice that
it is also true that 1 = 1ei2π, because the sine and cosine are periodic with period
2π. The polar representation of the number −1 is −1 = 1eiπ. Again, it is true that
−1 = 1ei3π, or, in fact, −1 = 1eiπ+K2π, for any integer K.
Products of complex numbers represented in polar coordinates are easy to compute. If
zi = |ri|eiθi, then
z1z2 = |r1||r2|ei(θ1+θ2).
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B. COMPLEX NUMBERS
Im
1
Re
Figure B.2: The 5th roots of unity.
Thus the magnitude of a product is a product of the magnitudes, and the angle of a product
is the sum of the angles,
|z1z2| = |z1||z2|, ∠(z1z2) = ∠(z1) + ∠(z2).
Example B.2: We can use the polar representation to find the n distinct roots of
the equation zn = 1. Write z = reiθ, and 1 = 1e2kπ, so
zn = rneinθ = 1ei2kπ,
which gives r = 1 and θ = 2kπ/n, k = 0, 1, · · · , n − 1. These are called the n roots
of unity. Figure B.2 shows the 5 roots of unity.
Whereas it is easy to solve the polynomial equation zn = 1, solving a general polynomial
equation is difficult.
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B.6. POLAR COORDINATES
Theorem The polynomial equation
zn + a1zn−1 + · · · + an−1z + an = 0,
where a1, · · · , an are complex constants, has exactly n factors of the form
(z −αi), where α1, · · · αn are called the n roots. In other words, we can always
find the factorization,
n
zn + a1zn−1 + · · · + an−1z + an = ∏(z − αk) .
k=1
Some of the roots may be identical.
Note that although this theorem ensures the existence of this factorization, it does not
suggest a way to find the roots. Indeed, finding the roots can be difficult. Fortunately,
software for finding roots is readily available, for example using the Matlab roots func-
tion.
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Exercises
Each problem is annotated with the letter E, T, C which stands for exercise, requires some
thought, requires some conceptualization. Problems labeled E are usually mechanical,
those labeled T require a plan of attack, those labeled C usually have more than one
defensible answer.
1. E Simplify the following expressions:
(a)
3 + 4i × 3+6i,
5 − 6i
4 − 5i
(b)
e2+πi.
2. E Express the following in polar coordinates:
2 − 2i, 2 + 2i,
1
,
1
, 2i, −2i.
2 − 2i 2 + 2i
3. E Depict the following numbers graphically as in Figure B.1:
i1, −2, −3 − i, −1 − i.
4. E Find θ so that
Re{(1 + i)eiθ} = −1.
5. E Express the six distinct roots of unity, i.e. the six solutions to
z6 = 1
in Cartesian and polar coordinates.
6. T Express the six roots of −1, i.e. the six solutions to
z6 = −1
in Cartesian and polar coordinates. Depict these roots as in Figure B.2.
7. T Figure out in for all positive and negative integers n. (For a negative integer n,
z−n = 1/zn.)
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EXERCISES
8. T Factor the polynomial z5 + 2 as
5
z5 + 2 = ∏(z − αk),
k=1
expressing the αk in polar coordinates.
√
√
9. C How would you define
1 + i ? More generally, how would you define
z for
any complex number z?
10. T The logarithm of a complex number z is written log z or log(z) . It can be defined
as an infinite series, or as the inverse of the exponential, i.e. define log z = w, if
ew = z. Using the latter definition, find the logarithm of the following complex
numbers:
1, −1, i, −i, 1 + i
More generally, if z = 0 is expressed in polar coordinates, what is log z? For which
complex numbers z is log z not defined?
11. E Use Matlab to answer the following questions. Let z1 = 2 + 3i and z2 = 4 − 2i.
Hint: Consult Matlab help on i, j, exp, real, imag, abs, angle, conj, and
complex. Looking up “complex” in the help desk may also be helpful.
(a) What is z1 + z2? What are the real and imaginary parts of the sum?
(b) Express the sum in polar coordinates.
(c) Draw by hand two rays in the complex plane, one from the origin to z1 and the
other from the origin to z2. Now draw z1 + z2 and z1 − z2 on the same plane.
Explain how you might systematically construct the sum and difference rays.
(d) Draw two rays in the complex plane to z3 = −2 − 3i and z4 = 3 − 3i. Now
draw z3 × z4 and z3/z4.
(e) Consider z5 = 2eiπ/6 and z6 = z∗. Express z
5
6 in polar coordinates. What is
z5z6?
(f) Draw the ray to z0 = 1 + 1i. Now draw rays to zn = z0einπ/4 for n = 1, 2, 3, . . ..
How many distinct zn are there?
(g) Find all the solutions of the equation z7 = 1. Hint: Express z in polar coordi-
nates, z = reiθ and solve for r, θ.
12. E This problem explores how complex signals may be visualized and analyzed.
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(a) Use Matlab to plot the complex exponential function as follows:
plot(exp((-2+10i)*[0:0.01:1]))
The result is a spiraling curve corresponding to the signal f : [0, 1] → C where
∀t ∈ [0,1] f (t) = e(−2+10i)t.
In the plot window, under the Tools menu item, use ‘Axes properties’ to turn
on the grid. Print the plot and on it mark the points for which the function is
purely imaginary. Is it evident what values of t yield purely imaginary f (t)?
(b) Find analytically the values of t that result in purely imaginary and purely real
f (t).
(c) Construct four plots, where the horizontal axis represents t and the vertical
axis represents the real and imaginary parts of f (t), and the magnitude and
angle of f (t). Give these as four subplots.
(d) Give the mathematical expressions for the four functions plotted above in part
(c).
13. T Euler’s formula is: for any real number θ,
eiθ = cos θ + i sin θ,
and the product formula is: for any complex numbers z1, z2,
ez1+z2 = ez1 ez2 .
The following problems show that these two formulas can be combined to obtain
many useful identities.
(a) Express sin(2θ) and cos(2θ) as sums and products of sin θ and cos θ. Hint:
Write ei2θ = eiθeiθ (by the product formula) and then use Euler’s formula.
(b) Express sin(3θ) and cos(3θ) also as sums and products of sin θ and cos θ.
(c) The sum of several sinewaves of the same frequency ω but different phases is
a sinewave of the same frequency, i.e. given Ak, φk, k = 1, . . . , n, we can find
A, φ so that
n
A cos(ωt + φ) = ∑ Ak cos(ωt + φk)
k=1
Express A, φ in terms of {Ak, φk}.
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NOTATION INDEX
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Notation Index
[D → R]
◦
∑
DM
delay (discrete time)
D
delay (continuous time)
τ
[A, b, c, d]
∗
N = {1, 2, · · · }
···
N0 = {0, 1, 2, · · · }
∈
/∈
not a member
A ⊂ B
A ⊆ B
/
0
=
:=
==
℘(X )
2X
703
NOTATION INDEX
|
NewSet = {x ∈ Set | Pred(x)}
prototype for sets
∀
∃
Z
Z+