Chapter 1
Complex Numbers
1.1 The âmysteryâ of .
Many years ago a distinguished mathematician wrote the following words, words that may strike some readers as somewhat surprising:
I met a man recently who told me that, so far from believing in the square root of minus one, he did not even believe in minus one. This is at any rate a consistent attitude. There are certainly many people who regard
as something perfectly obvious, but jib at
. This is because they think they can visualize the former as something in physical space, but not the latter. Actually
is a much simpler concept.
1 I say these words are âsomewhat surprisingâ because I spent a fair amount of space in
An Imaginary Tale documenting the
confusion about
that was common among many very intelligent thinkers from past centuries.
It isnât hard to appreciate what bothered the pioneer thinkers on the question of
. In the realm of the ordinary real numbers, every positive number has two real square roots (and zero has one). A
negative real number, however, has
no real square roots. To have a solution for the equation
x2 + 1 = 0, for example, we have to âgo outsideâ the realm of the real numbers and into the expanded realm of the complex numbers. It was the need for this expansion that was the intellectual roadblock, for so long, to understanding what it means to say
i =
âsolvesâ
x2 + 1 = 0. We can completely sidestep this expansion,
2 however, if we approach the problem from an entirely new (indeed, an unobvious) direction.
Figure 1.1.1. A rotated vector
A branch of mathematics called
matrix theory, developed since 1850, formally illustrates (I think) what the above writer may have had in mind. In
figure 1.1.1 we see the
vector of the complex number
x + iy, which makes angle
Îą with the positive real axis,
rotated counterclockwise through the additional angle of
β to give the vector of the complex number
xⲠ+ iyâ˛. Both vectors have the same length
r, of course, and so
. From the figure we can immediately write
x =
r cos(
Îą) and
y =
r sin(
Îą), and so, using the addition formulas for the sine and cosine
Now, focus on the xâ˛, yⲠequations and replace r cos(Îą) and r sin(Îą) with x and y, respectively. Then,
Writing this pair of equations in what is called column vector/matrix notation, we have
where
R(
β) is the so-called
two-dimensional matrix rotation operator (weâll encounter a different sort of operatorâthe
differentiation operatorâin
chapter 3 when we prove the irrationality of
Ď2). That is, the column vector
, when operated on (i.e., when multiplied
3) by
R (
β), is rotated counterclockwise through the angle
β into the column vector
.
Since
β = 90° is the CCW rotation that results from multiplying
x + iy by
i, this would seem to say that
i =
can be associated with the 2 Ă 2 matrix
R (90°)
Does this mean that we might, with merit, call this the imaginary matrix? To see that this actually makes sense, indeed that it makes a lot of sense, recall the 2 Ă 2 identity matrix
which has the property that, if
A is any 2 Ă 2 matrix, then
AI =
IA =
A. That is,
I plays the same role in matrix arithmetic as does 1 in the arithmetic of the realm of the ordinary real numbers. In that realm, of course,
i2 = â 1, and the âmysteryâ of
is that it itself is
not (as mentioned earlier) in the realm of the ordinary real numbers. In the realm of 2 Ă 2 matrices, however, there is no such âmysteryâ because the square of the âimaginary matrixâ (a perfectly respectable 2 Ă 2 matrix) is
That is, unlike the ordinary real numbers, the realm of 2 Ă 2 matrices does have a member whose square is equal to the negative of the 2 Ă 2 matrix that plays the role of unity.
To carry the analogy with the ordinary real numbers just a bit further, the
zero 2 Ă 2 matrix is
, since any 2 Ă 2 matrix multiplied by 0 gives
0. In addition, just as (1/
a) ¡...