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.