Would you indicate 3 + 3 + 3+3 as 4x3 or as 3x4† Most colleagues and boys that I have asked plump for the former. 4x3 means to them “four threes” or “four times three”, which is emphatically not the meaning “four multiplied by three”. Clearly “four times three” is the same as “three multiplied by four”; (I do not mean that they are equal, I mean that the expressions are two ways of saying the same thing). If we indicate this common meaning by the symbols 4x3, we are in fact doing the opposite of what we do in the case of the other three arithmetical processes. 4 + 3 means (to most people) “take four and add to it three”, or “four increased by three”; 4-3 means “take four and subtract three from it” , or “four diminished by three”; so 4 + 3 means “take four and divide it by three”. Obviously, then, 4x3 must mean “take four, and multiply it by three”; i.e. “four taken three times” or three fours.

“Find the modulus and the argument of each root of the equation:

(University of London A level, part question.)

We were investigating iterative methods of solving the equation, using a micro-package (in BASIC) which performs complex arithmetic and displays the results in an Argand diagram. The facility for “doodling” which this provides led to some interesting conjectures and results, of which this note is an account.