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Section 0.6 Complex numbers
Subsection 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 \(\Im z\text{.}\)
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 inverse of \(z\text{.}\)
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*}
Subsection 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*}
Exercises Exercises
1. 2. 3. 4.
