1 Numbers and Equations
Numbers have often been invented to solve equations .
For example, by introducing negative numbers the natural numbers N = {0, 1, 2, .
. .} can be
extended to the integers Z = {. . . ,-2,-1, 0, 1, 2, . . .} so that we may solve
simple equations
such as x+2 = 0. Likewise, by introducing integer quotients, the integers can to
be extended
to the rational numbers Q = {a/b : a, b ∈ Z, b ≠ 0} so that we may solve simple
equations
such as 2x = 1.
The rationals seem like a very nice set in which to do arithmetic. There is a
well-defined addition
operation and a well-defined multiplication operation, and Q is closed with
respect to
these operations. Addition and multiplication are associative (i.e., for all x,
y, z ∈ Q, x(yz) =
(xy)z and likewise for addition) and commutative (i.e., for all x, y ∈ Q, x + y
= y + x and
likewise for multiplication). Every element x ∈ Q has an additive inverse -x
(from which
we may define a subtraction operation ) and every nonzero element x ∈ Q, x ≠ 0,
has a multiplicative
inverse 1/x (from which we may define a division operation ). Furthermore,
multiplication
and addition satisfy the distributive law , i.e., for all x, y, z ∈ Q, x(y+z) =
xy+xz.
Mathematically speaking , the rational numbers form a field. Who could ask for
anything
more?
The trouble is that certain simple equations such as
have no solutions in Q , i.e., no rational number x
satisfies (1). However, following the
progression from N to Z to Q, we might try to get around this problem by
extending Q, i.e.,
by adjoining an element—let's call it θ for now—that satisfies θ2
- 2 = 0, or,
equivalently,
θ2
= 2. We will demand that θ be combinable with ordinary rational numbers (and
with
itself) via addition and multiplication, while satisfying all of the formal
arithmetic properties
(such as closure with respect to addition and multiplication, associativity,
commutativity,
the distributive law, etc.) that we have grown to expect.
If we denote this extended set by Q(θ ), then certainly Q(θ ) must
contain all numbers of
the form a + bθ , where a, b ∈ Q. Numbers involving higher powers of θ
do not arise, since
any such higher power can be reduced to a multiple of a lower power, i.e., θ2
= 2, θ3 = 2θ ,
θ4 = 4, etc. Indeed, the sum, difference, product and quotient
of any two elements of this
form is another element of this form (provided that we don't attempt to divide
by zero). To
see this, observe that
and since a±c and b±d are rational when a, b, c and d are rational, we have
another number
of the same form. Likewise
Again, since ac+2bd and ad+bc are rational when a, b, c and d are rational, we
have another
number of the same form. Finally, to show that we can form quotients, it is
enough to show
that we can form reciprocals. Note that if a and b are not both zero then
Now, a2 - 2b2 ≠ 0 (why?), so a/(a2 - 2b2) and -b/(a2 -
2b2) are rational when a, b, c and
d are rational; thus once again we have another number of the same form. Indeed,
we have
shown that Q( θ ) is a field, and unlike Q, in this field (1) does indeed have a
solution.
Since every element of Q(θ ) can be written as a+bθ , with a, b ∈ Q, we could
associate such
an element with the ordered pair (a, b). We might refer to the first component,
a, as the
"rational part" of a + bθ and the second component, b, as the "irrational part"
of a + bθ .
(Note that the "irrational part" is in fact itself a rational number!) The
arithmetic in Q(θ )
could be defined via operations that occur on such ordered pairs, e.g.,
From this viewpoint there is no mention of the
"irrational" number θ; indeed, from this
viewpoint one can simply regard θ as a "tag", pointing out which of the two
rational numbers
in question is the de signated "irrational part."
By the way, a more usual notation for θ is
2 Complex Numbers and the Complex Plane
So far we have "extended" Q by adjoining , thus taking one small step toward the construction
of the real numbers R. Let us now proceed to the point where we have constructed
all of R. Certain real numbers (e.g., 
) appear as the zeros of a polynomial with rational
coefficients and can be adjoined to Q in the same manner that we used to adjoin
; others
(e.g., π) appear as the limit points of certain sequences.
Unfortunately, even in R, many polynomials have no zeros, e.g., there is no real
number x
that satisfies
Emboldened by our experience in extending Q, let us extend R by introducing a
new element,
a purely "imaginary" number, j, that satisfies j2 + 1 = 0. Once again, we will
demand that
j satisfy all of the formal arithmetic properties (such as closure with respect
to addition and
multiplication, associativity, commutativity, the distributive law, etc.) that
we have grown
to expect in R.
Let us denote the extended set by C. Certainly, as we compute all possible sums
and products
involving real numbers and j, we find that C must contain all numbers of the
form a + b j,
where a, b ∈ R. Numbers involving higher powers of j do not arise, since any
such higher
power can be reduced to a multiple of a lower power, i.e., j2 = -1, j3 = -j, j4
= 1, etc. We
have, for all a, b, c and d in R,
and, provided a and b are not both zero,
This shows that the sum, difference, product and quotient of any two elements of
this form
is another element of this form (provided that we don't attempt to divide by
zero), and so C
is a field. The elements of C are called complex numbers and C is referred to as
the complex
field.
We usually denote individual complex numbers with a single
symbol , such as z. Now, since
every such element can be written in the form z = a + b j, with a, b ∈ R, we can
associate
such an element with the ordered pair (a, b) of real numbers. We refer to the
first component,
a, as the "real part" of z (written Re(z)) and the second component, b, as the "imaginary
part" of z (written Im(z)). Thus for all z ∈ C, we have
z = Re(z) + Im(z) j.
Note that the "imaginary part" is in fact itself a real number! If Im(z) = 0,
then z is referred
to as a (purely) real number; likewise if Re(z) = 0, then z is referred to as a
purely imaginary
number. Thus, in particular, j itself is purely imaginary.
Two complex numbers 
and
are equal if and only if their real and imaginary
parts agree,
i.e.,
if and only if
.
Thus every complex equality can always be regarded as a pair of real equalities.
The complex conjugate z* of a complex number z = Re(z) + Im(z) j is the complex
number
z* = Re(z) - Im(z) j.
Note that z* = z if and only if Im(z) = 0, i.e., if and only if z is real.
The complex conjugate obeys the following properties for all w , z ∈ C:
Furthermore, for all z ∈ C we have
It is possible to show (see the problems) that if
is a
polynomial with real coefficients (i.e., 
) and f(z) = 0
for some complex
value z, then we must also have f(z* ) = 0. In other words, if z is a zero of a
polynomial
with real coefficients then so is its complex conjugate z* .
Note that, unlike the real numbers, complex numbers are not in general ordered,
i.e., it
makes no sense to ask which is larger: 2 + 3 j or 3 + 2 j. However, we can
always compare
the magnitudes of two complex numbers. The magnitude (or absolute value or
modulus) |z|
of a complex number z is defined as
Figure 1: (a) Several points in the complex plane. (b) The
polar form of a complex number.
Clearly |z| is a non-negative real number, and |z| = 0 if and only if z = 0.
Note that
|z* | = |z|, i.e., a complex number and its complex conjugate have the same
magnitude.
It is very convenient to visualize C as a two-dimensional vector space over R,
i.e., as a plane.
Naturally, this plane is referred to as the complex plane. The number a + b j
corresponds to
the point with coordinates (a, b) in the complex plane, as shown in Fig. 1(a).
In Figure 1, the two axes have been labelled x and y. Since each point on the
x-axis represents
a real number, this axis is called the real axis. Similarly, the y-axis is
called the imaginary
axis. These two axes intersect at 0, which is the only complex number that is
simultaneously
purely real and purely imaginary.
The magnitude |z| of a complex number z has a geometric inter pretation in the
complex
plane: |z| measures the (Euclidean) distance between the origin 0 and z. Or if z
is regarded
as a vector in the complex plane, then |z| is the length of this vector.
Similarly, z* represents
the geometric reflection of z in the real axis.
