Though infinity (written as ∞* is often spoken of as a number, especially, by schoolchildren, it is really more a concept. Basically it is the furthest possible number from zero. However, numbers go on forever. No matter how big a number you have, you can always add one to it to get a new one. This is entirely possible, since numbers are not real objects, they are simply abstract concepts which allow people to easily describe amounts of stuff. It is therefore impossible to have ∞ of something, unless you happen to be some sort of advanced god. If you are, then it is probably still a bad idea, as where would you put it all?

The Arithmetic of Infinity

As proof that infinity is not a real (or, indeed, an 'imaginary') number, let's have a look at mathematics involving it. If a is any number, then:

• a+∞=∞
• ∞-a=∞
• a - ∞= - ∞
• a*∞=∞
• a/∞=0
• ∞/a=∞
• a^∞=infinity if a is greater than one or less than minus one
• a^∞=0 if a is between minus one and one
• ∞^a=∞
• i^∞=+/-1, where i=√(-1)*
• At present, ∞^i is undefined. Nobody is particularly interested in defining it.

Hilbert's Hotel Paradox

David Hilbert proposed the following paradox to explain just how unlike other numbers infinity is. Suppose some god runs a hotel with ∞ rooms, all of which are somehow full. However, this manager will have no trouble if a new guest asks for a room. The manager simply moves the person in room 1 to room 2, the person in room 2 to room 3, etc., so that the person in room n goes to room n+1. Thus room 1 is freed up for the new guest, and the rest can all get new rooms - since there are ∞ rooms, everyone will always have a higher-numbered room to go to.

What if ten new guests all want rooms? In that case, the person in room n goes to room n+10. The same logic applies.

So what if ∞ guests arrive simultaneously, and they all want rooms? Again, no problem. Everyone in the hotel moves to the room twice their own number, so that the person in room n moves to room 2n. This frees up all the odd-numbered rooms, and there are ∞ odd numbers and even numbers less than ∞- hence, all the travellers have rooms.

Zeno's Paradox

The concept of infinity is considerably older than the concept of some other numbers, such as zero. Infinity is quite important to the motion paradox of the Greek philosopher Zeno. Zeno one day reasoned as follows:

• Suppose a warrior sees an enemy archer about to shoot at him. The warrior turns and runs away at a steady speed in the same direction as the arrow is pointing.
• Suppose the archer is 100m away from the warrior, and the arrow flies ten times faster than the runner. By the time the arrow has reached the original position of the warrior, the warrior has moved on another 10m.
• When the arrow gets to that position, the warrior has moved on another metre.
• When the arrow covers that metre, the warrior has moved on another 10cm.
• etc.

This logic can also be used to show that Hercules cannot beat a turtle in a race if the turtle has a head start. Common sense tells us that this is rubbish - of course Hercules will overtake the turtle, quite quickly is we use equal divisions of time. Similarly, the warrior is better off to just duck.

In order for Zeno's paradox to work out, time and space must be split into infinitely shorter divisions. If this is possible, then a stage will eventually be reached when the warrior and the arrow will not move at all in the allotted time division, since time will have been split up into ∞ pieces and, as the number of divisions gets larger (and the size gets smaller), the distance moved by both the arrow and the warrior decreases. This indicates than any number divided by infinity gives zero. It also indicates that there is a minimum size for divisions of space and time, otherwise how could the arrow ever catch up to the warrior?

Universes and Elementary Particles

Some say that the Universe is infinite in size, others that it is not. An infinitely big Universe supports the Steady State theory, and a finite Universe holds up well with the Big Bang theory. However, could it possibly be both?

Take a piece of paper and imagine that there are a bunch of 2D people living on it. Roll it into a cylinder, and it becomes 3D. Bend the cylinder into a doughnut shape, and the 2D people living on it will never be able to find the edge - hence, it will seem infinite to them, despite being finite. There is a theory in quantum physics that the Universe really is doughnut shaped.

It was once thought that there might be ∞ particles in an atom. However, this is clearly bunk. If that was true, then nothing could possibly exist. If each elementary particle contained even the tiniest bit of mass, then ∞ of them would take up an infinite amount of space and contain an infinite amount of mass. According to the theory of relativity, as soon as something contains infinite mass, it ceases to exist.