Chapter 1 Complex Numbers
Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk. (God created the integers, everything else is made by humans.)―Leopold Kronecker (1823–1891)
The real numbers have many useful properties. There are operations such as addition, subtraction, and multiplication, as well as division by any nonzero number. There are useful laws that govern these operations, such as the commutative and distributive laws. We can take limits and do calculus, differentiating and integrating functions. But you cannot take a square root of \(-1\text{;}\) that is, you cannot find a real root of the equation
\begin{equation}
x^2+1=0 \,\text{.}\tag{1.1}
\end{equation}
Most of you have heard that there is a “new” number \(i\) that is a root of (1.1); that is, \(i^2+1=0\) or \(i^2=-1\text{.}\) We will show that when the real numbers are enlarged to a new system called the complex numbers, which includes \(i\text{,}\) not only do we gain numbers with interesting properties, but we do not lose many of the nice properties that we had before.
The complex numbers, like the real numbers, will have the operations of addition, subtraction, multiplication, as well as division by any complex number except zero. These operations will follow all the laws that we are used to, such as the commutative and distributive laws. We will also be able to take limits and do calculus. And, there will be a root of (1.1).
As a brief historical aside, complex numbers did not originate with the search for a square root of \(-1\text{;}\) rather, they were introduced in the context of cubic equations. Scipione del Ferro (1465–1526) and Niccolò Tartaglia (1500–1557) discovered a way to find a root of any cubic polynomial, which was publicized by Gerolamo Cardano (1501–1576) and is often referred to as Cardano’s formula. For the cubic polynomial \(x^3 + px + q\text{,}\) Cardano’s formula involves the quantity \(\sqrt{ \frac{ q^2 }{ 4 } + \frac{ p^3
}{ 27 } }\text{.}\) It is not hard to come up with examples for \(p\) and \(q\) for which the argument of this square root becomes negative and thus not computable within the real numbers. On the other hand (e.g., by arguing through the graph of a cubic polynomial), every cubic polynomial has at least one real root. This seeming contradiction can be solved using complex numbers, as was probably first exemplified by Rafael Bombelli (1526–1572).
In the next section we show exactly how the complex numbers are set up, and in the rest of this chapter we will explore the properties of the complex numbers. These properties will be of both algebraic (such as the commutative and distributive properties mentioned already) and geometric nature. You will see, for example, that multiplication can be described geometrically. In the rest of the book, the calculus of complex numbers will be built on the properties that we develop in this chapter.
