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# Secondary Magazine - Issue 35: Focus on

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 01 June 2009 by ncetm_administrator
Updated on 09 June 2009 by ncetm_administrator

# Focus on...$\sqrt{-1}$

Complex numbers were first used in the 16th century when the Italian mathematician Girolamo Cardano published Ars Magna. In this he was the first mathematician to give an algebraic solution to the general cubic equation  + ax² + bx + c = 0 by substituting x for x- a and thus reducing the original to the depressed cubic + px + q = 0. Cardano used a technique which he was shown by Niccolo Fontana to solve any depressed cubic, finding the root at

x= $\sqrt[3]{\frac{-q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27} } } + \sqrt[3]{\frac{-q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27} } }$

This work was built on by Rafael Bombelli in 1572 when solving a depressed cubic such as – 15X – 4 = 0  which, using Cardano’s method, requires calculation of $\sqrt{-121}$. Bombelli knew that this equation had real roots and after what he called a ‘wild thought’, gave a purpose and a consistency to $\sqrt{-1}$, a concept that, until that point, mathematicians had been happy to ignore! You can read more about the beginning of complex numbers on the University of North Dakota website.

“One thing puzzles me Betty," mused John. "Where does 'i' go on the number line?"
A quizzical look came over Betty's face. They tried putting it in all sorts of places on the number line but it just didn't fit. No matter where they placed 'i' it was different to the number that was already there.
(from John and Betty’s Journey into Complex Numbers by Matt Bower).

Descartes was the first to use the term imaginary number in 1637, but they were not widely accepted by the mathematics community until the work of Euler and Gauss in the 1700s.

The Mandelbrot Set is the set obtained from the iterative equation $z_n+1 = z\frac{2}{n} + c$  where c is in the complex plane. A complex number is in the Mandelbrot Set if, when z0 = 0, zn never becomes greater than a given number (dependant on c).
The Mandelbrot set below (from Wolfram Mathworld) shows the values of c in the complex pane coloured according to the number of steps required to reach rmax + 2.

In The Da Vinci Code, the character Robert Langdon jokes that Sophie Neveu "believes in the imaginary number i because it helps her break code".

De Moivre, whose formula links trigonometry and complex numbers, is said to have predicted the date of his own death. He noted that he was sleeping for 15 minutes longer each day and hypothesised that he would die on 27 November 1754, the day in which he calculated he would sleep for 24 hours. He was right!

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