When we are first taught about squares and square roots we are usually told that:

- The square of a positive number is positive.
- The square of a negative number is positive.
- Therefore the square root of a positive number has two solutions (one negative and one positive)
- And the square root of a negative number does not have a solution.

Is this just a convention? Could we have chosen that:

- The square of a positive number is negative.
- The square of a negative number is negative.
- Therefore the square root of a negative number has two solutions (one negative and one positive)
- And the square root of a positive number does not have a solution.

It is valid to choose these conventions, numbers with these properties are known as 'imaginary' numbers as opposed to 'real' numbers (although I'm not sure they are any less real, its just a convention that we use).

But this still does not have solutions to all square roots. There does not seem to be a one dimensional number system which has a solution for all possible square roots. There are multidimensional algebras which have solutions to all square roots, such as matrices and complex numbers, here we are discussing complex numbers.

A complex number is a two dimensional number with both a 'real' and an 'imaginary' part, as described above, so that it can have solutions to both positive and negative numbers.

## Complex Number Notation

Complex numbers could be written as two numbers, like this: (a,b), since they are a bit like 2D vectors with different multiplication rules. However the conventional way to denote them is in the form a + i b

where 'i' is the imaginary operator which represents the square root of minus one.

i = √-1

The advantage of this notation is that the numbers behave with all the usual rules of arithmetic, except whenever we get two 'i' operators multiplied together i*i then this is replaced with -1.