What is unsettling for most when they first encounter imaginary numbers is the concept of the square root of a negative number. One of the things we are taught at school is that when you square a number - positive or negative - then it is always positive. Quite correct. So this number -1 is no ordinary number. That is why mathematicians call it an imaginary number. The other imaginary numbers consist of combinations of this number i and the ordinary numbers, for example 6+3i. Mathematicians make this tantric leap and imagine there is a number whose square is -1.

But what do we actually mean when we are contemplating the square root of 2? It is a number that when you square it you get back the answer 2.

With a number like the square root of 2, you do have the luxury of writing down a decimal expansion which approximates this square root. You may remember from school it starts off 1.41421…. Riemann used to idle away the hours calculating more and more decimal places. His record was 38 places. No mean feat without a computer but it reflects more perhaps on the dull Göttingen nightlife and Riemann's shy persona that this was his evenings' entertainment. But how ever far Riemann calculated the decimal expansion he would never have written down the number. As the Greeks proved, the square root of 2 is something called an irrational number whose decimal expansion never ends or repeats itself. So, this number is frankly as imaginary as the number mathematicians have dreamt up which when multiplied by itself gives the answer -1.

Well, there is one difference. We do at least feel that there is some point on a ruler which will correspond to the square root of 2 even if we can't write down a decimal expansion. It's a point somewhere between 1.4 and 1.5. If you wanted to mark off this point exactly then Pythagoras's Theorem tells us that a right angled triangle with its two smallest sides of length 1 will have a longest side of length exactly 2. So this looks like a very practical way of building this number.

But even this very concrete representation will probably still end up being an approximation. It's not clear whether space at the quantum level satisfies Pythagoras's Theorem. It could well bend and look like a triangle drawn on a ball rather than a flat piece of paper. In which case the length of the longest side won't necessarily be the square root of 2. The number is beginning to look as imaginary as the square root of a negative number. And the name irrational number provides us with the clue that once these numbers unnerved people too.

Even if quantum physics undermines our ability to realise physically the square root of 2, this picture of numbers on a ruler turns out to be quite a powerful model. The point on the ruler isn't exactly the number. But the picture works quite well. To add two numbers A and B we just put two lines of length A and B together on our ruler and the point we reach by combining these two lengths is the sum A+B. For example adding 1 and the square root of 2 gets us to a point which is 1+2, a point somewhere between 2.4 and 2.5.

But Gauss’s map of imagianry is an equally good picture that mathematicians use to see these imaginary numbers. It is just a picture but it helps amazingly in getting to grips with these new sorts of numbers. We now have a picture where we can "see" these new numbers like 6+3i. This number is the point reached by moving 6 steps east-west followed by 3 moves in the north-south direction. So aren’t these numbers just as real as the square root of 2?