# Analysis of a Complex Kind – Week 1

### Brief History of Complex Numbers

\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, no 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)