Bielefeld, June 1997

Complex Numbers.

1. Genesis. Mathematical evolution has repeatedly expanded the notion of number in order to overcome arithmetic restrictions. For instance, since the system N of natural numbers 1, 2, 3, 4 … does not allow subtraction a − b unless b < a, it is augmented by 0 and the negatives to form the integers Z. To permit unrestricted division (except by 0), the latter in turn is enlarged to the rationals Q, which encompass all fractions or “ratios” of integers.

Q is fine for elementary arithmetic but cannot solve simple equations like x^{2} = 2, let alone 10^{x} = 3. Hence it is extended to the reals R, which include such “irrationals” as \sqrt{2} or log 3. Real numbers can be depicted as points on an infinite “number line”, and are often presented as “decimal expansions that go on and on indefinitely.” While this is not exactly a crisp definition, it underlines the fact that every real number can be approximated by rationals to any desired degree of precision, so that any problem solvable in R is approximately solvable in Q.

R is the work-horse of undergraduate science. However, large parts of mathematics and physics would be seriously hobbled without recourse to the system C of complex numbers, which parametrizes a plane instead of a line, and has the virtue of providing square roots (and more) for all its inhabitants. The passage from R to C is technically far simpler than the one from Q to R — but may at first seem more mysterious, because something like \sqrt{−5} does not even make approximate sense in R.

The journey from N to R took thousands of years and included long periods of confusion, which left us with such giddy terms as “irrational”, “transcendental”, “imaginary”, etc. Every time a concept is expanded, its original meaning must be replaced by a more subtle one — which invariably raises the question of whether it all makes sense. If you are accustomed to counting sheep or days, the idea of a “negative number” may be puzzling indeed. To make room for such a notion, the original counting numbers from N have to be reinterpreted as increments rather than absolute quantities. Then they can interact with decrements in a new happy family Z.

To prepare them for insertion into C, the familiar reals will likewise be reinterpreted, this time as matrices. Instead of seeing a real number u as a magnitude, think of it as the scalar matrix uI. This makes no difference at all to the customary arithmetical or analytical operations with these objects, but it does allow them to interact with the so-called “imaginary numbers”— which are matrices of the form vJ. Since (vJ)^{2} equals −v^{2}I, imaginary numbers are square roots of negative real numbers (taken as scalar matrices). A complex number is just the sum of a real and an imaginary number, in other words, a matrix of the form uI + vJ.

Unfortunately, traditional notation is not as straightforward as that. Just as we write 5 instead of +5, 5/1, or 5.000 ⋯, — whether that number appears in the context of N, Z, Q, or R — we wish to write 5 intead of 5I in the context of C as well, even if this condemns the imaginary part to sit there as an inexplicable appendage. Thus, the complex number 5I + 3J is traditionally written as

5 + 3\sqrt{−1},  5 + 3i,  or 5 + 3j

depending on the mathematical subculture in which it appears. We would prefer the last of these options (already used by electrical engineers) but shall regretfully choose the second one, because it is the most popular.

In this terse notation, complex numbers appear as expressions u + vi, where u, v are actual real numbers but +, i are just punctuation marks with i behaving like \sqrt{−1}. The best way to handle complex numbers is to treat the symbol i exactly like the symbol x in computations with polynomials, and to simplify all results by substituting −1 for i^{2} wherever possible. If anyone should challenge you to explain what it all means, just take him aside and tell him about the matrices uI + vJ, of which the mysterious u + vi is but an abbreviation.

2. Properties. Addition of such “numbers” w = u + vi and w′ = u′ + v′i goes as expected:

(u + vi) + (u′ + v′i) = [u + u′] + [v + v′] i ,

but multiplication follows the more intricate pattern

(∗)  (u + vi) · (u′ + v′i) = [uu′ − vv′] + [uv′ + u′v] i

because of the rule i^{2} = −1. It is evident that w · w′ = w′ · w, i.e., complex multiplication is commutative. Moreover, since det(uI + vJ) = u^{2} + v^{2} is positive unless u = v = 0, every complex number w ≠ 0 is invertible, and commutativity allows us to write w′/w instead of w^{−1}w′ = w′w^{−1}. In C, we therefore have unrestricted addition, subtraction, multiplication, and division (by non-zero elements) with all the familiar rules of arithmetic. What more could we ask from a number system? Oh yes, there is no order, no dichotomy between positive and negative.

But there is a notion of size: the absolute value |w| of a complex number w = uI + vJ is defined to be \sqrt{u^{2} + v^{2}}. Since this is just the square root of det(uI + vJ), it is compatible with multiplication and division, i.e.,

|w · w′| = |w| · |w′| and |w^{−1}| = |w|^{−1} .

With respect to addition, however, it satisfies the more delicate relation

(∗∗)  |w + w′| ≤ |w| + |w′| ,

known as the “triangle inequality”. To prove it, we multiply both sides by the positive real number |w|^{−1} (the case w = 0 being trivial) and obtain |1 + c| ≤ 1 + |c|, with c = w′/w, as the thing to check. An inequality between non-negative reals is neither created nor destroyed by squaring both sides, so the question is whether |1 + c|^{2} ≤ 1 + 2|c| + |c|^{2} or not. For c = a + bi, the left hand side of this is (1 + a)^{2} + b^{2} = 1 + 2a + a^{2} + b^{2}, while the right comes to 1 + 2|c| + a^{2} + b^{2}. Therefore it all boils down to a ≤ |c|, which is certainly true — and is a strict inequality unless c = a is a non-negative real number.

In some sense, a complex number w = u + vi is just a pair (u, v) of real numbers, and as such can be depicted as a point or a vector in the plane. Addition is just the usual composition of vectors. To see the geometric manifestation of multiplication, however, it is good to remember that w actually stands for the matrix uI + vJ =

(†)  [ \begin{matrix}u & −v \\ v & u \end{matrix} ] = ρ [ \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ]

= ρR(θ), where ρ, θ are the polar coordinates of the point (u, v). In these terms, it is easy to see how multiplication by w affects a complex number z = x+yi. After all, the vector (x, y) is just the first column of xI +yJ, and analogously the coordinates of w·z form the first column of the matrix (uI +vJ)·(xI +yJ)

— which is the same as (x, y) transformed by ρR(θ).

Conclusion: complex multiplication by w = u + vi means matrix multiplication by uI + vJ = ρR(θ), in other words, rotating by the angle θ and scaling by the positive number ρ.

The scaling factor ρ = \sqrt{u^{2} + v^{2}} is what we had previously called the absolute value |w|. By the Theorem of Pythagoras, it corresponds to the length of the vector (u, v), which explains the term “Triangle Inequality” used above: if you picture the sum of w and w′ as one arrow following another, |w + w′| is the length of the third side in the triangle so obtained.

In C it is easy to find any n-th root, i.e. solve the equation z^{n} = w for any given w ∈ C and n ∈ N. Because of (†), you just have to choose z = σ(cos ϕ + i sin ϕ) where σ^{n} = ρ and n · ϕ = θ, so that the underlying matrix σR(ϕ) has it n-th power equal to ρR(θ). Since angles obey the special rule 360^{∘} = 0^{∘}, we actually get n different solutions by setting ϕ_{k} = (θ + k · 360^{∘})/n for k = 0, 1, …, (n − 1).

We have tacitly assumed that angles were measured in degrees. If you measure them in radians — i.e., by the parametrization obtained in “Reihenpower” — you can rewrite (†) using Euler’s formula

cos(t) + i sin(t) = e^{it} .

3. Polynomials. Over the complex numbers any polynomial equation f(z) = 0 turns out to have a solution. Though often taken on faith, this ”Fundamental Theorem of Algebra” is not exactly plausible. Why should \sqrt{−1} open the gate for all polynomials? Our answer will focus on the continuous real-valued function |f(z)| and use the fact that, restricted to any closed planar disc, such a function must reach a minimum. For the rest of this dicussion, we consider a polynomial

f(z) = c_{0} + c_{l}z^{l} + ⋯ + c_{h}z^{h},

with non-zero complex coefficients c_{0}, c_{l}, …, c_{h} and the powers of z arranged in ascending order. Our argument will be based on the following two properties of its absolute value |f(z)|:

(i) |f(z)| > |f(0)| for all sufficiently large values of z, and (ii) |f(z)| < |f(0)| for certain small values of z.

These have nothing to do with fancy concepts like continuity. They result from some straightforward digging in C, which will be performed at the end of this section. For the moment, let us see how they would bring about the proof of the theorem. Note that (ii) depends vitally on the assumption that f(0) = c_{0} ≠ 0 (without which there would be nothing to prove, anyway).

1. The absolute value |f(z)| attains a minimum. Like every continuous function, |f(z)| takes on a minimum value m(r) ≤ |f(z)|, for z ranging over a closed disc of radius r, i.e. |z| ≤ r. In particular, m(r) ≤ |f(0)|. Hence, if r is chosen large enough, (i) guarantees that m(r) < |f(z)| for z outside that disc, i.e. |z| > r, as well. In other words, for sufficiently large r, the minimal value m = m(r) is a global minimum. (This is unusual for complex functions; even such a decent one as e^{z} does not achieve it.)

2. This minimum cannot be positive. Let a ∈ C be the place where the minimum is attained, i.e. |f(a)| = m. Now form the polynomial g(z) = f(a − z), whose range of values is obviously the same as that of f(z). In particular, the minimum of |g(z)| is also equal to m = |f(a)| = |g(0)|. However, if m were not zero, (ii) could be applied to g(z) and would say that |g(0)| is not minimal. Since this cannot be, we conclude that g(0) = f(a) = 0, as was to be shown.

It remains to establish the famous properties (i) and (ii) used in the proof. Consider f(z) written as

f(z) = φ(z) + c_{h}z^{h}  or  f(z) = c_{0} + c_{l}z^{l} + ψ(z),

and note that φ(z) and ψ(z) are negligible for very large z and very small z, respectively. In fact, if K is a real number greater than the sum of the absolute values |c_{0}| + |c_{l}| + ⋯ + |c_{h}|, you can easily check that |φ(z)| < K|z|^{h−1} for |z| ≥ 1, and |ψ(z)| < K|z|^{l+1} for |z| ≤ 1 (see?).

The reason behind (i) is that, for very large values of z, the order of magnitude of f(z) is dictated entirely by that of the term c_{h}z^{h}, which overpowers φ(z).

On the other hand, (ii) is due to the fact that, for very small values of z, the behaviour of f(z) resembles that of c_{0} + c_{l}z^{l}, because ψ(z) counts for so little. To make this more pecise, let us first look at the special case f(0) = c_{0} = 1, and try to find a particular z = s such that |f(s)| < 1.

Start with a complex number w such that c_{l}w^{l} = −1, and scale it down by a real factor 0 < ε < 1, putting s = εw. By adjusting ε we can make s so small that ψ(s) is less than half the size of c_{l}s^{l} = −ε^{l} in absolute value. Then f(s) = 1 − ε^{l} + ψ(s) with |ψ(s)| < \frac{1}{2}ε^{l}, and hence

|f(s)| ≤ |1 − ε^{l}| + |ψ(s)| < |1 − ε^{l}| + \frac{ε^{l}}{2} < 1.

If f(0) = c_{0} ≠ 0 is not equal to 1, we simply apply this argument to the polynomial c_{0}^{−1}f(z). It then produces an s such that |c_{0}^{−1}f(s)| < 1, whence |f(s)| < |c_{0}| as required.