In 1806, Argand created a device which helps make sense of complex numbers and of one the great mysteries of mathematics, one that pupils fumble with daily: ‘Why does a minus times a minus make a plus?’ The need for complex numbers was identified by Ferro in the 1500s in order to solve cubic equations, then in the 1540s, Cardano solved the problem of how to divide 10 into two parts, the product of which is 40 – ‘putting aside the mental tortures involved’, he said, the solution is: 5 plus the square root of negative 15, multiplied by 5 minus the square root of negative 15. He described his solution as sophistic, as subtle as it was useless, not realising that complex numbers could be used for control theory, improper integrals, fluid dynamics, electromagnetism, electrical engineering, signal analysis, quantum mechanics, relativity, geometry, and number theory – such as why a minus times a minus is a plus (which I usually explain as being ‘for the same reason that reducing your debt is a good thing’).

Jean-Robert Argand, amateur mathematician, managed a bookshop in Paris during which time he published his essay on ‘a method of representing imaginary quantities’ in which the real part of the number is represented on the horizontal axis, the imaginary part on the vertical axis. Using Pythagoras, the length, or modulus, of 5 plus the square root of negative 15 is the square root of 40, as is the length of 5 minus the square root of negative 15, and the product of these two numbers, the square root of 40 and the square root of 40, is equal to 40, visually solving Cardano’s problem. However, on the Argand diagram, the two solutions are imaginary numbers, so how is the product a positive, real number? It transpires that rotations around the Argand Diagram, known as the argument, are summed when multiplying complex numbers, and as the two arguments are equal and opposite, they add to an argument of zero degrees, that is, a positive, real number of 40.

The argument, or rotation, also works for positive and negative real numbers, a positive number having an argument of zero, while a negative number has an argument of pi radians, or 180 degrees. When a negative is multiplied by a negative, the magnitude, or modulus, is multiplied, giving the same answer as if a positive were multiplied by a positive; however, the argument is summed, and two lots of 180 degrees makes 360 degrees. So, next time a pupil asks you, ‘Why is a negative times a negative a positive?’ – you could say, ‘Because it creates a full turn on the complex plane’.

**Keith Doyle**

With thanks to Orlando Merino’s, ‘A short history of complex numbers’, and Wikipedia articles on ‘Complex number’, Complex plane, and ‘Jean-Robert Argand’.