Complex numbers

Section 0.6 Complex numbers

In this section we begin with a brief treatment of the complex numbers \(\C\text{.}\) As mentioned below, you should think of the complex numbers as an additional link in our chain of number systems (0.1.1). One major advantage of working with \(\C\) is that any polynomial factors as a product of linear terms over \(\C\) (Theorem 0.7.7). This fact, along with a review of basic definitions and properties of polynomials, will be taken up in Section 0.7. This section is meant mainly as a reference, and as such all theorems will be stated without proof.

Subsection 0.6.1 Definition of \(\C\)

The complex numbers constitute a number system built by taking the set of all pairs of real numbers and defining operations on these pairs that we call complex addition and complex multiplication.

Definition 0.6.1. Complex numbers.

The set \(\C\) of complex numbers is defined as the set of all pairs of real numbers: i.e.,

\begin{equation*} \C=\{(a, b)\colon a, b\in \R\}\text{.} \end{equation*}

Given a complex number \(z=(a,b)\text{,}\) its first entry \(a\) is called the real part of \(z\text{,}\) denoted \(\Re z\text{,}\) and its second entry \(b\) is called the imaginary part of \(z\text{,}\) denoted \(\Im z\text{.}\)

Notation 0.6.2.

Henceforth we will primarily denote a complex number \(z=(a,b)\) as

\begin{equation} z=a+bi\text{.}\tag{0.6.1} \end{equation}

Since \(a=\Re z\) and \(b=\Im z\text{,}\) you can think of the notation (0.6.1) as a decomposition of \(z\) as a sum of its real and imaginary components. In what follows we give actual arithmetic meaning to the symbols ‘\(+\)’ and ‘\(i\)’ used in this notation, but first and foremost (0.6.2) should simply be thought of as an alternative manner of denoting the pair \((a,b)\text{.}\)

Additionally we will adopt the following notational conventions: \(a-bi=a+(-b)i\text{,}\) \(a=a+0i\text{,}\) \(a+i=a+1i\text{,}\) \(bi=0+bi\text{,}\) and \(i=0+1i\text{.}\)

Remark 0.6.3. Complex equality.

When moving to the notation (0.6.1) do not lose sight of the essential nature of complex numbers as pairs of real numbers. This is important, for example, for understanding what it means for complex numbers \(z=(a,b)=a+bi\) and \(w=(c,d)=c+di\) to be equal. According to the general definition of equality for tuples (0.3.4), we have

\begin{equation} z=w\iff a=c \text{ and } b=d \iff \Re z=\Re w \text{ and } \Im z=\Im w\text{.}\tag{0.6.2} \end{equation}

Definition 0.6.4. Complex addition and multiplication.

We define addition and multiplication operations on \(\C\) as follows. Let \(z=a+bi\) and \(w=c+di\text{,}\) where \(a,b,c,d\in \R\text{.}\)

Complex addition.

The sum \(z+w\) is the complex number defined as

\begin{equation*} z+w=(a+c)+(b+d)i\text{.} \end{equation*}
Complex multiplication.

The product \(zw\) is the complex number defined as

\begin{equation*} z\, w=(ac-bd)+(ad+bc)i\text{.} \end{equation*}

Example 0.6.5.

Let \(z=2+3i\) and \(w=-1+2i\text{.}\) We have

\begin{align*} z+w \amp =(2-1)+(3+2)i=1+5i\\ zw \amp=(-2-6)+(4-3)i=-8+i \end{align*}

Our first theorem indicates that complex addition and multiplication behave in much the same way as real addition and multiplication. A number system that satisfies the properties of Theorem 0.6.6 is called a field. You should think of the field properties as guaranteeing that we can perform arithmetic in the complex numbers (or any fied) essentially as we do with real numbers.

Theorem 0.6.6. Basic properties of complex arithmetic.

Commutativity.

For all \(z, w\in \C\) we have

\begin{align*} z+ w \amp = w+z\\ z w \amp= wz \text{.} \end{align*}
Associativity.

For all \(z, w, u \in \C\) we have

\begin{align*} z+( w+ u) \amp = (z+ w)+ u\\ z( w u) \amp=(z w) u \text{.} \end{align*}
Distributivity.

For all \(z, w, u \in \C\) we have

\begin{equation*} z( w+ u)=z w+z u\text{.} \end{equation*}
Additive identity.

The complex number \(0=0+0i\in \C\) satisfies

\begin{equation*} 0+z=z \end{equation*}

for all \(z\in \C\text{.}\) We call \(0\) the additive identity of \(\C\text{.}\)
Additive inverse.

For all \(z=(a,b)=a+bi\in \C\) the element \(-z=(-a,-b)=-a+(-b)i\in \C\) satisfies

\begin{equation*} z+ -z=0\text{.} \end{equation*}

We call \(-z\) the additive inverseof \(z\text{.}\)
Multiplicative identity.

The complex number \(1=1+0i\) satisfies

\begin{equation*} 1z=z \end{equation*}

for all \(z\in \C\text{.}\) We call \(1\) the multiplicative identity of \(\C\text{.}\)
Multiplicative inverse.

For all nonzero complex numbers \(z=(a,b)=a+bi\in \C\) (i.e., \(a\ne 0\) of \(b\ne 0\)), the complex number \(z^{-1}=\frac{a}{a^2+b^2}+\frac{-b}{a^2+b^2}i \in \C\) satisfies

\begin{equation*} z\,z^{-1}=1\text{.} \end{equation*}

The complex number \(z^{-1}\text{,}\) also denoted \(1/z\text{,}\) is called the multiplicative inverse of \(z\text{.}\)

Example 0.6.7.

Let \(z=-2+i\text{.}\) According to Theorem 0.6.6, the multiplicative inverse of \(z\) is

\begin{equation*} z^{-1}=\frac{-2}{4+1}+\frac{-1}{4+1}i=\frac{-2}{5}+{-1}{5}i\text{.} \end{equation*}

Let’s check that \(zz^{-1}=1\text{:}\)

\begin{align*} zz^{-1} \amp =(-2+i)(\frac{-2}{5}+{-1}{5}i)\\ \amp = (\frac{4}{5}+\frac{1}{5})+(\frac{2}{5}+\frac{-2}{5})i \\ \amp = 1+0i=1\text{.} \end{align*}

Remark 0.6.8. Complex numbers as extension of the reals.

From now on we identify the real numbers as the set of complex numbers of the form \(a=a+0i\text{,}\) where \(a\in \C\text{:}\) equivalently, the set of complex numbers \(z\) satisfying \(\Im z=0\text{.}\) Under this identification \(\R\) can be thought of as a subset of \(\C\text{,}\) and we may add one more link to the chain of subsets given in (0.1.1):

\begin{equation} \Z\subseteq\Q\subseteq\R\subseteq\C\text{.}\tag{0.6.3} \end{equation}

Furthermore, it is easy to verify that the various operations on \(\C\) agree with their real counterparts when restricting to \(\R\text{:}\) e.g.,

\begin{align*} (a+0i)+(b+0i) \amp = (a+b)+0i\\ (a+0i)(b+0i)\amp=ab+0i \\ -(a+0i) \amp =-a+0i\\ \frac{1}{a+0i} \amp =\frac{1}{a}+0i\text{.} \end{align*}

This allows us to think of the complex numbers as a larger number system containing the reals, whose arithmetic operations are extensions of real number operations.

Subsection 0.6.2 Absolute value and complex conjugation

We end our introduction to the complex numbers with two further operations: the absolute value and complex conjugation. Theorem 0.6.10 is an indication of their usefulness, and articulates how they interact with the other operations on \(\C\text{.}\)

Definition 0.6.9. Absolute value and complex conjugation.

Let \(z=(a,b)=a+bi\) be a complex number. The absolute value (or modulus) of \(z\text{,}\) denoted \(\abs{z}\text{,}\) is defined as

\begin{equation*} \abs{z}=\sqrt{a^2+b^2}\text{.} \end{equation*}

The complex conjugate of \(z\text{,}\) denoted \(\overline{z}\text{,}\) is defined as

\begin{equation*} \overline{z}=a-bi\text{.} \end{equation*}

Theorem 0.6.10. Properties of conjugation and modulus.

The following properties hold for all \(\boldz, \boldw\in \C\text{.}\)

\(\overline{z+w} =\overline{z}+\overline{w}\text{.}\)
\(\overline{zw}=\overline{z}\overline{w}\text{.}\)
\(\abs{zw}=\abs{z}\abs{w}\text{.}\)
\(z\overline{z}=\abs{z}^2\text{.}\)
\(z^{-1}=\frac{z}{\abs{z}}\) (if \(z\ne 0\)).
\(z\in \R\iff \overline{z}=z\text{.}\)

Example 0.6.11.

Let \(z=3-i\text{.}\) Observe that

\begin{align*} z\overline{z} \amp=(3-i)(3+i) \\ \amp=(9+1)+0i \\ \amp =\abs{z}^2 \end{align*}

and

\begin{align*} z^{-1} \amp =\frac{3}{10}+\frac{1}{10}i\\ \amp=\frac{3+i}{10} \\ \amp = \frac{\overline{z}}{\abs{z}^2}\text{,} \end{align*}

as claimed in Theorem 0.6.10.

Exercises 0.6.3 Exercises

1.

Rewrite the following expression into the form of a+b\(i\text{:}\)

\(\displaystyle{ \frac{{-6+2i}}{{1-i}} = }\)

Answer.

\(-4-2i\)

Solution.

By definition, \(i=\sqrt{-1}\text{,}\) so we have \(i^{2}=-1\text{.}\)

To get rid of \(i\) in the denominator, we use the difference of squares formula:

\(\displaystyle{ \begin{aligned} \amp \phantom{{}=}(1+{-i})(1-{-i}) \\ \amp = (1)^{2}-({-i})^{2} \\ \amp = (1)-(-1) \\ \amp = 2 \end{aligned} }\)

The full solution is:

\(\displaystyle{ \begin{aligned} \amp \phantom{{}=} \frac{{-6+2i}}{{1-i}} \\ \amp = \frac{({-6+2i})(1-{-i})}{({1-i})(1-{-i})} \\ \amp = \frac{(-6)(1)+(-6)(-{-i})+({2i})(1)+({2i})(-{-i})}{2} \\ \amp = \frac{(-6)+(-6i)+(2i)+(2)(i^{2})}{2} \\ \amp = \frac{(-6)+(-4i)+(2)(-1)}{2} \\ \amp = \frac{(-6)+(-4i)+(-2)}{2} \\ \amp = \frac{(-8)+(-4i)}{2} \\ \amp = \frac{-8}{2}+\frac{-4i}{2} \\ \amp = {-4-2i} \end{aligned} }\)