This is all very well but is Gauss’s map of imaginary numbers a helpful picture? In fact it is very powerful. Each number A+Bi we can think of as a set of directions in this imaginary world: A steps east-west, followed by B moves in the north-south direction. If I gave you two sets of directions A+Bi and C+Di you do one followed by another and end up at a third point which you could have got to by being given the one direction (A+C)+(B+D)i. This number is then the sum of the two original numbers. So for example here is a picture of (6+3i)+(1+2i)=(7+5i):

Multiplication of imaginary numbers also has a nice geometrical interpretation in this map. It depends on another way of telling an explorer in this imaginary world how to get to a number. Instead of specifying how many steps east-west then north-south, one could tell our imaginary traveller to set off at a specified angle to the east-west axis, followed by telling him how far he should go in this direction. For example here is a picture of the number that our explorer would arrive at if he travelled 2 units out at 45 degrees to the east-west axis, i.e in a north-easterly direction. It is the same as the number that he would get to if he travelled to (2)+( 2)i following the old set of instructions.

This is another sort of coordinate for these imaginary numbers which we call polar coordinates. After all if you are at the North Pole it makes more sense to be told how far to go in which direction than talk about east-west and north-south. We write the polar coordinates of the number in our picture as (45º,2). It is much easier now to define how to multiply together two numbers in these coordinates. Where do you have to go if you are told to proceed to the point represented by (A(1),R(1)) multiplied by (A(2),R(2))? Well, the answer is you add the angles and go off in that direction as far as the product of the distances. So this is the point represented by (A(1)+A(2),R(1)xR(2)) in these new polar coordinates. Let's square the number that our polar explorer got to in the last picture: i.e. multiply (45º,2) by (45º,2). We add the angles and multiply the distances to get to the number with polar coordinates (90º,4). This is the same as the imaginary number 4i.

The new number we get will be the same whether you use the algebra or polar coordinates to multiply the original two numbers. For example squaring the north-south-east-west coordinates (2)+( 2)i for the number (45º,2) we get 4i again:

(2+ -2)( 2+ -2)=
(22+ -2-2)+ (2-2+-22)=
(2-2)+(2i+2i)=4i.

The polar coordinates offer a better picture which will help us to understand why some numbers must be indivisible building blocks like the primes in the family of imaginary numbers A+B-6.