Analysis of a Complex Kind - Part 01

Brief History of Complex Numbers

Consider a quadratic equation
\begin{align*}
{x^2 = mx + b}
\end{align*}

Solutions are
\begin{align*}
x = \frac{m}{2} \pm \sqrt{ \frac{m^2}{4} + b}
\end{align*}

and represent the intersection of
\begin{align*}
y = x^2 \mbox{ and } y = mx + b.
\end{align*}

What if
\begin{align*}
\frac{m^2}{4} + b < 0 ?
\end{align*}

  • In particular, $x^2 = -1$ has no real solutions.
  • It is often argued that this led to $i = \sqrt{-1}.$
  • But... Historically, not interest in non-real solutions since the graphs of $yx=x^2$ and $y=mx+b$ simply don't intersect in that case.

History

  • Cubic equations were the real reason. Consider

\begin{align*}
x^3=px+q
\end{align*}

  • Represents intersection $y=x^3$ and $y=px+q.$
  • There always must be a solution.

Solution to Cubic

  • Del Ferro (1465-1526) and Tartaglia (1499-1577), followed by Cardano (1501-1576), showed that
    \begin{align*}x^3=px+q\end{align*}
    has a solution given by
    \begin{align*}
    x=\sqrt[3]{\sqrt{\frac{q^2}{4}-\frac{p^3}{27}}+\frac{q}{2}}-\sqrt[3]{\sqrt{\frac{q^2}{4}-\frac{p^3}{27}}-\frac{q}{2}}
    \end{align*}
  • Try it out for $x^3=-6x+20$ (see video)!

Bombelli's Problem

  • About 30 years after the discovery of this formula, Bombelli (1526-1572) considered the equation
    \begin{align*}x^3=15x+4\end{align*}
  • Plugging $p=15$ and $q=4$ into the formula yields
    \begin{align*}
    x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}
    \end{align*}
  • Bombelli had a 'wild thought'...

Bombelli's Idea

  • Bombelli discovered that
    \begin{align*}
    \sqrt[3]{2+\sqrt{-121}}=2+\sqrt{-1} \mbox{ and } \sqrt[3]{2-\sqrt{-121}}=2-\sqrt{-1}
    \end{align*}
  • These clearly add up to 4, the desired solution.
  • Check it out
    \begin{align*}
    (2+\sqrt{-1})^3=2+\sqrt{-121} \mbox{ and } (2-\sqrt{-1})^3=2-\sqrt{-121}
    \end{align*}

The Birth of Complex Analysis

  • Bombelli's discovery is considered the 'Birth of Complex Analysis'.
  • It showed that perfectly real problems require complex arithmetic for their solution.
  • Note: Need to be able to manipulate complex numbers according to the same rules we are used to from real numbers (distributive law, etc)