What is Zero?

If you look at Roman Numerals, i.e. I for 1, II for 2, V for 5, VIII for 8, X for 10 and so on, you should spot two things. One thing is easy to spot, and the other is quite hard.

It is easy to spot that the complexity of Roman Numerals is not obviously connected to their size. For instance 37 is XXXVII, while 52 is LII. It is quite hard to spot that this is because their system does not place varying values on the symbols (I, X, L, V etc) depending on where they fall in the sequence. Instead the value is always the same for each symbol, X always is 10 and L is always 50.

The whole number being written is calculated by applying a sequence of rules, which basically says, read from left to right adding each symbol's value unless the symbol to the right of it is greater than it, in which case you deduct its value from that symbol. So, XI is 10+1=11 and IX is 10-1 which is 9. Taken a little further, 1998 is MCMXCVIII which is M(1000)+CM(1000-100)+XC(100-10)+V(5)+I(1)+I(1)+I(1). What is important to note is that the two [C] symbols BOTH mean 100 despite them falling in different positions.

Our system of numerals is different. We also use different symbols to represent the constituent parts of big numbers. So far, so Roman. What is different is that the symbols have no relation to each other but their value depends on where in the sequence they fall. In one sense we could be said to read our sequence of symbols from right to left. The right most symbol (if we are dealing with whole numbers only) tells us how many one's are in the total number. The next rightmost symbol tells us how many ten's, the next how many hundred's and so on. So if we see 123, we quickly add 3 one's, two ten's and one hundred.

If it seems odd that we read the number from right to left, remember it is done subconsciously. If read from left to right, one can not know what the first symbol in the number represents (be it millions, hundreds of thousands or so on) until you have counted how MANY symbols are in the number. So logically you start at the right, where you KNOW the symbol means one's, and then add up moving from right to left. In reality though, it doesn't really matter which direction you read the symbols in, as long as you know what value each symbol position has in advance. This is unlike Roman Numerals, where the end result would be very different if you started at one end rather than the other - see XI and IX for example. Also, note that if you have the number 144, the two [4] symbols mean different things. The middle one means 4x10 or 40, and the right most one means 4x1 or 4. So the symbols increase in relative value the further left you go because they are multiplied by larger and larger sums. So the complexity of our numbers is directly proportionate to their numerical size. 237 is a bigger sequence of symbols than 84 and so on. The Romans were no thickies, built loads of stuff, and conquered the majority of their world (just don't tell the Chinese), so why were they using this cumbersome system where thirty seven takes longer to write than fifty two, and both are longer than one thousand and one?

The answer is rather profound. If we want to write the number one hundred and nine, then we start with the symbol [9] at the right, followed by the symbol [0] to show there are no tens, followed by the symbol [1] to show there is one hundred: 109. The Romans could not do this because they had no symbol which meant nothing. They had I, V, X, L, C, D, and M, each of which represents a specific number, but they had nothing which represented zero, because they did not consider that to BE a number. For us it would be like asking "What is the wavelength of black light?" (I do not mean the purpley ultraviolet stuff that shows up washing powder residue and dandruff with equal aplomb). Black light does not have a wavelength because there are no waves, it is the absence of light. Or it would be like asking "How does the sound of a violin not playing differ from the sound of a trumpet not playing?". It is a meaningless question. The absence of something does not need to be measured, and we do not need a symbol to represent it.

You may say that they could represent nothing by just not writing anything, which makes a sort of sense, but the problem is then distinguishing between 10 and 100, or 11, 101 and 1001. No, for a system of numbers based on symbols having different values depending on their position you need a clear symbol to represent "none of this value needed". A Roman would be puzzled by this; it would be like making up a shopping list with one amphora of wine, three buckets of milk, two handfuls of berries, no apples, and five bags of nuts. Why do you even bother to mention the apples if you want none of them?

So we are forced to have the symbol [0] because of the simpler way we write out numbers. But just the fact that we got [0] makes us think differently about the concept. Once it is a symbol, is it a number? I do not propose to try to answer that, but I will say that we can sometimes use it like a number. We can add it to other numbers (we get the number we started with), we can multiply other numbers by it (we get zero every time), we can even multiply other numbers by themselves [0] times - meaning raise them to the power 0 - (oddly we get one and not zero every time). We cannot divide numbers by zero. That breaks things.

For our present purposes then, [0] is as fundamental as [1].