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Section 1.1 The Complex Plane

The complex numbers were originally invented to solve problems in algebra. It was later recognized that the algebra of complex numbers provides an elegant set of tools for geometry in the plane. For an introduction (or for a review) of the basics of the algebra and geometry of the complex numbers, we refer the reader to the section entitled Complex Numbers
in the author’s text "Not Just Calculus" [5]. The remainder of this short section introduces material that will be useful later in this text.
Circles and Lines. Let \(C\) be the circle in the complex plane with radius \(r\gt 0\) and with center \(a\in \C\text{.}\) A point \(z\) lies on \(C\) if and only if the distance from \(z\) to \(a\) equals \(r\text{.}\) In mathematical symbols, \(C\) is the set of complex solutions \(z\) for the following equation.
\begin{equation} |z-a|=r\tag{1.1.1} \end{equation}
The real line \(\R\) in the complex plane is the set of solutions \(z\) of the equation \(\im(z)=0\text{.}\) More generally, let \(L\) be a line that contains the point \(p\in \C\) and makes an angle \(\theta\) with the real axis (set \(\theta=0\) if \(L\) is parallel to the real axis). If \(z\in L\text{,}\) then \(e^{-i\theta}(z-p)\) is real, so \(\im(e^{-i\theta}(z-p))=0\text{.}\) See Figure 1.1.1. Conversely, if \(e^{-i\theta}(z-p)\) is real, then \(z\) lies on \(L\text{.}\) Multiplying by a positive constant \(k\text{,}\) and setting \(a=ke^{-i\theta}\) and \(b=-ke^{-i\theta}p\text{,}\) we conclude that the line \(L\) is the set of solutions to the following equation.
\begin{equation} \im(az+b)=0\tag{1.1.2} \end{equation}
Figure 1.1.1. A line in the complex plane.

Exercises Exercises

1. Solving quadratic equations.

Find all complex solutions of the following equations.
  1. \(\displaystyle \displaystyle z^2 + 3z + 5 = 0\)
  2. \(\displaystyle (z - i)(z + i) = 1\)
  3. \(\displaystyle \displaystyle \frac{2z + i}{-z+3i} = z\)

2. Circles and lines.

  1. For a real variable \(x\) and a real constant \(a\text{,}\) completing the square refers to rewriting the expression \(x^2 - 2ax\) as follows.
    \begin{equation*} x^2-2ax = x^2-2ax +a^2 - a^2 = \left(x-a\right)^2 -a^2. \end{equation*}
    A complex version of completing the square for a complex variable \(z\) and a complex constant \(a\) is the following.
    \begin{equation} |z|^2-2\re(za^\ast)=|z-a|^2 -|a|^2\tag{1.1.3} \end{equation}
    Write a derivation to justify this. Then use completing the square to find the center and radius of the circle given by the equation \(|z|^2 -iz +iz^\ast -5=0\text{.}\)
  2. Write an alternative proof for the general form for the equation of a line (1.1.2), as follows. Let \(a=u+iv\text{,}\) \(b=r+is\text{,}\) \(z=x+iy\text{.}\) Find the equation of the line \(\im(az+b)=0\) in terms of the real variables \(x,y\) and real constants \(u,v,r,s\text{.}\) Explain why it is necessary that \(a\neq 0\text{.}\)

3. Complex numbers as \(2\times 2\) real matrices.

Let \({\mathcal M}_\C\) denote the set of \(2\times 2\) matrices of the form \(\left[\begin{array}{cc}a\amp b\\-b\amp a\end{array}\right]\) with \(a,b\in \R\text{.}\) Given a complex number \(z\) with Cartesian form \(z=a+bi\text{,}\) let \(M(z)\) denote the matrix \(\left[\begin{array}{cc}a\amp b\\-b\amp a\end{array}\right]\) in \({\mathcal M}_\C\text{.}\) Conversely, given a matrix \(M\in {\mathcal M}_\C\) with top left entry \(a\) and top right entry \(b\text{,}\) let \(C(M)\) denote the complex number \(a+bi\text{.}\) It is clear that the mappings \(z\to M(z)\) and \(M\to C(M)\) are inverses to one another, and establish a one-to-one correspondence \(\C\leftrightarrow {\mathcal M}_\C\text{.}\)
  1. Show that \({\mathcal M}_\C\) is closed under addition and multiplication. That is, suppose that \(M,N\) are elements of \({\mathcal M}_\C\text{.}\) Show that \(M+N\) and \(MN\) are also elements of \({\mathcal M}_\C\text{.}\)
  2. Show that complex addition and multiplication are "mirrored" in \({\mathcal M}_\C\text{.}\) That is, show that
    \begin{align} M(z+w)\amp =M(z)+M(w)\tag{1.1.4}\\ M(zw)\amp =M(z)M(w).\tag{1.1.5} \end{align}
Significance of this exercise. Matrix algebra provides a framework for theory and applications in almost every area in mathematics. Using the one-to-one correspondence \(\mathcal{M}_\C\leftrightarrow \C\text{,}\) it is possible to translate all of complex algebra in terms of matrix operations. We will use this same idea to define and prove the basic properties of quaternion algebra in Section 1.2, and we will use correspondences with matrix algebras to prove properties of geometric transformations in Chapter 3.