Before getting to the next round of puzzles, we require some math background, which we will try to build up now.
The complex numbers, \(\mathbb{C}\) , can be represented as points in the plane. In polar coordinates, we may write any \(z \in \mathbb{C}\) as \(z= re^{i \theta}=r (\cos(\theta )+i \sin(\theta ))\) where \(r\) is the distance to the origin and \(\theta\) is the polar angle. The complex conjugate \(z^*=r e^{-i\theta}\) . Given two complex numbers \(z_1 = r_1 e^{i \theta_1}\) and \(z_2 = r_2 e^{i \theta_2}\) , their product is \[z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}.\] Note that \(zz^*=r^2\) . We sometimes denote \(|z|=r\) .
We next prove a theorem about complex numbers. The reason we present this theorem here is not only that it illustrates the power of topological arguments in mathematics, but in addition this example mirrors what we will find later in this chapter in physical examples.
Fundamental Theorem of Algebra
Let \(f(z) = z^n +a_{n-1} z^{n-1} + \ldots + a_0 = 0\) . If \(n \geq 1\) , then then \(f(z)\) has a solution in \(\mathbb{C}\) .
A simple consequence of this is that a polynomial of degree \(n\) with coefficients in \(\mathbb{C}\) has exactly \(n\) solutions, counted with multiplicity. The fundamental theorem lies at the root of the ubiquity of complex numbers. So how do we go about proving this theorem? Suppose \(f(z)\) had no solutions. We now show that this assertion leads to a contradiction. If \(f(z)\) had no zeroes, the function \[g(z) = \frac{f(z)}{|f(z)|}\] exists where \(|f(z)|=\sqrt{f(z)f(z)^*}\) and \(f(z)^*\) is the complex conjugate of \(f(z)\) . Note that this division is allowed precisely because we have assumed that \(f(z)\) is never \(0\) .
The function \(g(z)\) was constructed to have the property that \(|g(z)| = 1\) for all \(z \in \mathbb{C}\) . In other words, \(g(z)\) lies on the unit circle. Therefore, as we vary the value of \(z \in \mathbb{C}\) , \(g(z)\) maps the whole complex plane to the unit circle.
Consider the image under \(g\) of an extremely large circle, whose radius is much, much bigger than all of the coefficients of the polynomial. Then, to a very good approximation, \(f(z) \approx z^n\) for \(|z| \gg 0\) because the other powers are much smaller by comparison. Similarly, \(g(z) \simeq \frac{z^n}{|z|^n} = e^{i n \theta}\) for \(|z|\gg 0\) . The conclusion is that for really big circles, \(g\) wraps the big circle on the complex plane around the unit circle \(n\) times (Fig. 39 ), because as \(\theta\) varies from \(0\) to \(2\pi\) , \(g\) varies from \(1\) to \({\rm exp}(2\pi i n)\) .
