Mathematicians can be pig-headed. Sometimes there are problems that admit of no solution, but mathematician ignores common sense and presses on regardless.One such question would be "WHERE, ON A UNIT CIRCLE, IS A POINT FIVE UNITS TO THE RIGHT FROM THE CENTRE"?

Pythagoras gave an answer. The hypotenuse is a unit, so you square it. The square is still a unit. Now you look at position to right. It is five units. You square it to twenty-five and take it away.

Then you find square root.

But one minus twenty-five is MINUS TWENTY-FOUR. What is root of a negative number?

There is no answer!

But common sense tells you that nothing can be on a circle if it is further from centre than radius. The answer "NOT ON THE CIRCLE" leaps to mind.

Girolamo Cardano was a mathematician who was even consulted by Leonardo da Vinci. He was also a gambler, but used his knowledge of probability to get better of his opponents. He decided that he had solution.

You split up problem into two parts. The root of 36 is 6. However, one can try root of 4 times 9 - in which case one has root of 4 times root of 9. The answer is same.

So you look for root of -1 times root of 24.

The root of any negative number will be root of -1 times root of absolute value of that number.

So root of -36 is root of -1 times six.

The root of -1 is defined as IMAGINARY, so root of -36 is IMAGINARY SIX. And answer to position on a unit circle of point X=5? That will be Y=IMAGINARY ROOT 24.

Cardano was scoffed at. He died. The years went by, and slowly mathematical community began to realise that idea was not quite so bad after all.

It was a NEW TECHNIQUE. To solve problems, you simply follow rules slavishly. These rules began to be taught in universities. People were taught that old numbers - REAL ones - had to be kept separate from imaginary ones. They learned to describe a two-part number having real and imaginary as a COMPLEX NUMBER.

The rules say that whenever you multiply real by real you get real. Multiply real by imaginary (of imaginary by real) and you get imaginary. Multiply imaginary by imaginary and you get MINUS real.

Simple rules, easy to apply.

But how does this impact upon whole of mathematics? What happens to sines, cosines, logarithms and other things when you use complex numbers in place of real ones? A host of great mathematicians - including legendary Leonhard Euler - worked on this question. Complex mathematics was becoming established as a standard tool.

Problems began to become apparent. The simplistic concept of ROOT -1 was actually wrong. It should have been "ONE OF THE ROOTS OF -1". There are actually two.

Then problem of multiple solutions appeared. As I wrote at http://www.wehner.org/euler/ , this problem can be visualised by considering antilogarithm of an imaginary number as a SPIRAL IN COMPLEX SPACE. What is lowest point? One for every turn of spiral. Therefore multiple solutions.