(1+i) /√2   Arabic Numerals   (1+i) /√2  

If you start with one, and carry on adding one to it, you end up with what I've been calling whole numbers. Their correct name is the natural numbers. All of the ancient number systems defined the natural numbers, although many of the unary systems had a limit or highest number. There is no highest natural number. Any 'biggest' number you can think of, someone else can think of a higher one! Another way of saying this is that there are infinite natural numbers.

When you count a number of objects, you are using cardinal numbers. These are one, two, three, etc. But if you want to specify the third one, this is an ordinal number. The days of the month are ordinals, e.g. March 15th.

Below are the natural numbers on a number line. When you add, you move to the right.

So far, we can add any natural numbers. But what is the opposite of addition? Subtraction, of course. This is fine while we stay in the natural numbers. 2+3=5, so 5-3=2. But we get problems with the natural numbers when we try to take away a number bigger than the original. Children learning to subtract say cheerfully "You can't", and go on to borrow ten from the next column. Yet you can, of course. You end up with a negative number. There are as many negative numbers as natural numbers (the positive numbers), and they are all integers. There is one other number in the set of integers which is neither positive or negative, and this is zero (or 0). At present, it is just one of the integers, but it causes problems later, so keep an eye on it!

The number line of the integers has no starting or finishing point. Both ends go on "for ever". To subtract a positive number, you move to the left. To add a negative number, you also move to the left. We already know that to add a positive number, you move to the right. To subtract a negative number, you also move to the right!
For example: 2 + 3 = 5 (which is bigger than both of them). 2 - (-3) = 5 as well.
One way to understand this is to think about direction. Subtraction means that you go in reverse direction. Adding a positive number means go to the right. Adding a negative number means going in the opposite direction, which is left. But subtracting a negative number means reversing that, which takes you back to the right.

If we add a number to itself several times, we get multiplication.

Multiplying positive numbers is OK, but what about negative numbers? A positive number times a negative number gives a negative result. But a negative number times a negative number gives a positive result.
For example: 2 x -2 = -4 and -2 x -2 = 4
So one minus sign changes the sign of the result to minus, but two minus signs change the sign back to plus.

Division is the opposite of multiplication. So 5x3=15 and 15/3=5. However, the answer to a division may not be an integer. There are two ways of doing a division. The first gives an integer answer and a remainder, and we use this when doing long division. But if you divide using a calculator, you may get an answer which isn't an integer. This is a rational. This word means the ratio of two numbers, which is a fraction, but you can write a rational as a decimal or even a percentage. The decimal 0.4 means the ratio of four to ten.

We know that there are infinite natural numbers. But every natural number has a corresponding rational between zero and one, by dividing it into one. The natural number 3 corresponds to the rational number 1/3. So there must be infinite rationals between zero and one. The number line below obviously can't show all of them! It just shows the tenths as an indication of the rest. The integers are included among the rationals, since 5/1=5.

We have already said that multiplication is adding a number to itself several times. If we multiply a number by itself several times, this is raising it to a power. The simplest example is squaring. 3 squared, or 3², is 3x3=9. The little ² above the number is called the exponent. Arabic numbers use the integer powers of ten to give the positions of the digits of a number.

It is also possible to have a rational exponent. 9^½ means the square root of 9. The square root of a number is the opposite of the square. 3²=9, so 9^½=3. Another way of writing square root is √9=3.

Some numbers have obvious square roots, such as 4 or 9 or 1/25. Are all square roots rational numbers? It was the Ancient Greeks who proved that √2 was not a rational number. Since then it was discovered that there is a set of numbers that can't be written as the ratio of two numbers. These are the irrationals. √2=1.4142135623730951... I have written ... to show that this number goes on for ever. What's more, groups of digits in it never start repeating themselves, the way they do with a fraction like 1/7. Irrationals are not just square roots. Pi is irrational (=3.141592653589793...) , so is e, used in natural logarithms (2.718281828459045...).

If you combine the rationals and the irrationals, you get the set of real numbers. The number line has now got a lot more numbers. In fact, despite the fact that the rationals are packed infinitely densely on the number line, between any two irrationals, there is infinitely many irrationals! This is impossible to show on a number line, so I'm not going to try!

There is a strange property of square roots. We know that 3x3=9. But we also know that (-3)x(-3)=9. So the square root of 9 may be 3 or -3. The other operations, addition, subtraction and so on, give a single answer. But square root can give more than one answer. We write it like this: √9=±3. Now we have to ask, what is the square root of a negative number? What is √-1? Mathematicians decided to invent a new number to cope with this problem. This number is i, where i=√-1. Now we can work out the square root (or indeed any other root) of any number. √-9=±3i    √-2=i√2 (where √2 is irrational). Note that -i is a square root of -1 as well. Multiples of i are called imaginary numbers, because they really don't exist in the real world. Real numbers exist, even irrationals. π is the ratio of the circumference of a circle to its diameter, and circles are real enough. But i is something invented (or discovered) to make numbers work.

If you mix real numbers and imaginary numbers, you get complex numbers. You do arithmetic with complex numbers, by calculating the imaginary and real parts separately, but remember that ixi=-1.

The number line has now become the complex plane. The reals go left to right (like the x axis) and the imaginary numbers go from bottom to top (like the y axis). All complex numbers can be mapped on this plane, like plotting an (x,y) point. Click here for more information on complex numbers.

So far, we have generated all these different numbers by starting from one and adding. You may think that we now have a complete number system, where every operation has its inverse, and all operations on all numbers work. But they don't. There is an operation on a particular number which doesn't give an answer. It's not even a clever number like an irrational or an imaginary number. It's the number zero. (I told you it was going to cause trouble!) I have mentioned that any number multiplied by zero gives zero. What happens if you divide by zero? If you try it on a calculator, it will tell you that you've made an error. Conventionally, we say that a number divided by zero is infinity, as in 1/0=∞. If you divide one by smaller and smaller numbers, the answers get bigger and bigger, so if you divide 1 by 0, which is smaller than any number, you must get infinity. But infinity is not a number. It's just a way to say that you have no answer to this particular problem.