Section 1.1 Definitions and Algebraic Properties
There are many equivalent ways to think about a complex number, each of which is useful in its own right. In this section, we begin with a formal definition of a complex number. We then interpret this formal definition in more useful and easier-to-work-with algebraic language. Later we will see several more ways of thinking about complex numbers.
Definition 1.1.1.
The complex numbers are pairs of real numbers,
\begin{equation*}
\C \ := \ \left\{ (x,y) : \, x,y \in \R \right\} \, \text{,}
\end{equation*}
equipped with the addition
\begin{equation}
(x,y) + (a,b) \ := \ (x+a, \, y+b)\tag{1.2}
\end{equation}
and the multiplication
\begin{equation}
(x,y) \cdot (a,b) \ := \ (xa-yb, \, xb+ya) \,\text{.}\tag{1.3}
\end{equation}
One reason to believe that the definitions of these binary operations are acceptable is that \(\C\) is an extension of \(\R\text{,}\) in the sense that the complex numbers of the form \((x,0)\) behave just like real numbers:
\begin{equation*}
(x,0)+(y,0) \ = \ (x+y,0) \qquad \text{ and } \qquad (x,0) \cdot
(y,0) \ = \ (xy,0) \, \text{.}
\end{equation*}
So we can think of the real numbers being embedded in
\(\C\) as those complex numbers whose second coordinate is zero.
The following result states the algebraic structure that we established with our definitions.
Proposition 1.1.2.
The set of complex numbers endowed with the operations of addition and multiplication
\((\C,+,\cdot)\) is a field. More specifically, this means that for all complex numbers
\((x,y)\text{,}\) \((a,b)\text{,}\) and
\((c,d)\text{,}\)
\begin{align}
\amp (x+y)+(a,b)\in \C\tag{1.4}\\
\amp \bigl( (x,y)+(a,b) \bigr) + (c,d) = (x,y) +
\bigl( (a,b)+(c,d) \bigr)\tag{1.5}\\
\amp (x,y) + (a,b) = (a,b) + (x,y)\tag{1.6}\\
\amp (x,y)+(0,0) = (x,y)\tag{1.7}\\
\amp (x,y)+(-x,-y) = (0,0)\tag{1.8}\\
\amp (x,y) \cdot \bigl( (a,b) + (c,d) \bigr) = (x,y)
\cdot (a,b) + (x,y) \cdot (c,d)\tag{1.9}\\
\amp (x,y) \cdot (a,b) \in \C\tag{1.10}\\
\amp \bigl( (x,y) \cdot (a,b) \bigr) \cdot (c,d) = (x,y)
\cdot \bigl( (a,b) \cdot (c,d) \bigr)\tag{1.11}\\
\amp (x,y)\cdot (a,b) = (a,b) \cdot (x,y)\tag{1.12}\\
\amp (x,y) \cdot (1,0) = (x,y)\tag{1.13}\\
\amp
(x,y) \cdot \left(
\tfrac{ x }{ x^2 + y^2 } , \tfrac{ -y }{ x^2 + y^2 } \right)
= (1,0) \,\text{if}\,
(x,y)\in\C\setminus\{(0,0)\} \, .\tag{1.14}
\end{align}
What we are stating here can be compressed in the language of algebra: equations
(1.4)–
(1.8) say that
\((\C,+)\) is an
Abelian group with
identity \((0,0)\text{;}\) equations
(1.9)–
(1.14) say that
\(\left( \C \setminus \{(0,0)\},\cdot \right)\) is an Abelian group with identity
\((1,0)\text{.}\)
Proof.
By our definition for complex addition and properties of additive inverses in \(\R\text{,}\)
\begin{equation*}
(x,y) + (-x, -y) \ = \ (x+(-x), \, y+(-y)) \ = \ (0,0) \, \text{.}
\end{equation*}
The definition of complex multiplication implies the innocent looking statement
\begin{equation}
(0,1) \cdot (0,1) \ = \ (-1,0) \,\text{.}\tag{1.15}
\end{equation}
This identity together with the fact that
\begin{equation*}
(a,0) \cdot (x,y) \ = \ (ax,ay)
\end{equation*}
allows an alternative notation for complex numbers. The latter implies that we can write
\begin{equation*}
(x,y) \ = \ (x,0) + (0,y) \ = \ (x,0) \cdot (1,0) + (y,0) \cdot (0,1) \,\text{.}
\end{equation*}
If we think—in the spirit of our remark about embedding \(\R\) into \(\C\)—of \((x,0)\) and \((y,0)\) as the real numbers \(x\) and \(y\text{,}\) then this means that we can write any complex number \((x,y)\) as a linear combination of \((1,0)\) and \((0,1)\text{,}\) with the real coefficients \(x\) and \(y\text{.}\) Now \((1,0)\text{,}\) in turn, can be thought of as the real number 1. So if we give \((0,1)\) a special name, say \(i\text{,}\) then the complex number that we used to call \((x,y)\) can be written as \(x \cdot 1 + y \cdot i\) or
\begin{equation*}
x+iy \, \text{.}
\end{equation*}
We invite you to check that the definitions of our binary operations and
Proposition 1.1.2 are coherent with the usual real arithmetic rules if we think of complex numbers as given in the form
\(x+iy\text{.}\)
The open-source mathematics software
SageMath can handle complex numbers and functions; we will provide sample code throughout this book to get you started.
The second complex number in
(1.14) is the (multiplicative) inverse of
\(x+iy\text{.}\)
Definition 1.1.3.
The number
\(x\) is called the
real part and
\(y\) the
imaginary part of the complex number
\(x+iy\) and are often denoted as
\(\Re(x+iy)=x\) and
\(\Im(x+iy)=y\text{.}\)
\begin{equation*}
i^2 = -1 \,\text{.}
\end{equation*}
In fact, much more can now be said with the introduction of the square root of
\(-1\text{.}\) It is not just that
(1.1) has a root, but
every nonconstant polynomial has a root in
\(\C\text{:}\)
Theorem 1.1.4. Fundamental Theorem of Algebra.
Every nonconstant polynomial of degree
\(d\) has
\(d\) roots (counting multiplicity) in
\(\C\text{.}\)
The proof of this theorem requires some (important) machinery, so we defer its proof and an extended discussion of it to
Chapter 5 (see
Theorem 5.3.2).
