Complex numbers are an extension of the real numbers, defined by introducing a new special number called \(i\) whose square is defined to be \(-1\text{.}\)
Definition7.1.1.Complex Numbers.
A complex number is a pair \((x,y)\) of real numbers. We think of the pair \((x,y)\) as \(x+iy\text{,}\) where \(i\) is a new number in which \(i^2=-1\text{.}\)
The set of all complex numbers is denoted \(\CC\text{.}\)
The real and imaginary parts of \(z \in \CC\) are \(\mathrm{Re}(z)=x\) and \(\mathrm{Im}(z)=y\text{,}\) respectively.
Just like vectors, we represent complex numbers on an \(xy\)-plane, where the \(x\)-coordinate is the real part and the \(y\) coordinate is the imaginary part. Using the plane to represent complex numbers in this way is sometimes called an Argand diagram.
Multiplication, addition and subtraction of complex numbers maintains how addition works for real numbers, and treating \(i\) as a formal symbol with \(i^2=-1\text{.}\) Thus for two complex numbers \(z=a+i b\) and \(w=c+i d\text{,}\)
where we have re-grouped terms with the \(i\) symbol and terms without the \(i\) symbol separately. These are the real and imaginary parts of the product \(zw\) respectively.
For an integer \(n\text{,}\) we write \(z^{n}=z\cdot z\cdots z\) as we did with real numbers.
Complex addition can be interpreted as vector addition in the complex plane. Consider any two complex numbers \(z=a+i b\) and \(w=c+i d\text{.}\)
\begin{equation*}
\begin{pmatrix}a \\ b \end{pmatrix} + \begin{pmatrix}c \\ d \end{pmatrix} = \begin{pmatrix}a+c \\ b+d \end{pmatrix}
\end{equation*}
We will examine the geometry of complex multiplication fully in a short while.
Given \(z\in\CC\) not equal to zero, we let \(z^{-1}=\frac{1}{z}\) denote the complex number such that \(z\cdot z^{-1}=1\text{.}\) Such a number (i) exists and (ii) is unique (a theorem we omit to save space). In particular, we can divide complex numbers by each other, but before we explain how to do this, there are a couple of important quantities related to a complex number that will make this task easier.
Definition7.1.2.Complex Conjugate.
Given a complex number \(z=x+iy\text{,}\) its complex conjugate is defined to be
Geometrically, the modulus of \(z\) is its distance from the origin, which is the length of the hypotenuse of a triangle of base \(x\) and height \(y\text{,}\) computed using the Pythagorean theorem.
This extends the definition of the absolute value to complex numbers, and in fact, if \(x\) is real, the modulus of \(x\) is equal to the absolute value. This definition should remind you of the vector norm: for a vector \((x,y)\text{,}\) its norm was \(||(x,y)|| = \sqrt{x^2+y^2}\) as well.
We can use these concepts to show how to divide two complex numbers. For example, let's look at \(\frac{3+i}{1+i}\text{:}\) we can make the denominator real by multiplying and dividing by the conjugate of \(1+i\) (which is \(\overline{1+i}=1-i\)):