Appendix

Fundamental theorem of algebra

We mentioned previously that, when we extend to complex numbers, all polynomial equations have a solution. This fact, the fundamental theorem of algebra, is proved in university-level mathematics courses, but we can say something about it here.

There are many different types of numbers. Perhaps the simplest is the set of natural numbers \(\mathbb{N} = \{1, 2, 3,…\}\). (For some mathematicians \(\mathbb{N}\) also includes zero.) The natural numbers are good for counting, but not so good to solve an equation like \(x+2 = 0\). To solve that equation, we have to introduce a new exotic type of number called a negative number. When we add in these new numbers, we obtain a number system called the integers \(\mathbb{Z} = \{…, -3, -2, -1, 0, 1, 2, 3, …\}\).

However, if we ever come across an equation like \(2x-1=0\), the integers will not be sufficient to solve it. We will need to invent another exotic species of number called a fraction or rational number to solve it. The rational numbers are

\[ \mathbb{Q} = \Big\{\, \frac{p}{q} \ : \ p, q \text{ are integers with } q \ne 0 \,\Big\}. \]

This extension of the number system, however, is still not sufficient to solve an equation like \(x^2 - 2 = 0\); as it turns out, \(\sqrt{2}\) is not a rational number. The number system is thus extended to include irrational numbers, and we end up with an enormous number system called the real numbers.

Still, the real numbers are not enough to solve such an innocent quadratic equation as \(x^2+1=0\). In a similar vein, we might add in a new number \(i\) such that \(i^2 = -1\); so '\(i\) is the square root of \(-1\)'. The numbers so obtained are called the complex numbers,

\[ \mathbb{C} = \big\{\, a+bi \ : \ a,b \text{ are real numbers} \,\big\}. \]

The amazing fact about the complex numbers is that we do not need to invent any new numbers to solve polynomial equations. Until now we found that very simple polynomial equations, such as \(2x-1=0\) and \(x^2 - 2 = 0\), did not have solutions; now they do. Amazingly, now any polynomial equation has a complex number solution: for any polynomial

\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \]

where the coefficients \(a_0, \dots, a_n\) are complex numbers, we can find complex number roots \(\alpha_1, \alpha_2, \dots, \alpha_n\) and factorise the polynomial completely as

\[ f(x) = a_n ( x - \alpha_1 ) (x-\alpha_2) \dotsm (x-\alpha_n). \]

We don't need to go any further than complex numbers to solve polynomial equations. This property is called algebraic closure, and it's another way to state the fundamental theorem of algebra.

Theorem (Fundamental theorem of algebra)

The complex numbers \(\mathbb{C}\) are algebraically closed.

That's not to say that we can't make up more numbers as we need! An important, even larger number system is called the quaternions; these involve three different square roots of \(-1\) called \(i\), \(j\) and \(k\). The quaternions are very useful in 3-dimensional geometry and are important in higher mathematics and physics.

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