Identities and Inverses

Before I go any further in the History of Zero, perhaps I should spend a bit of time elucidating... 

Why is zero so special? And why are Brahmagupta's insights so profound anyway?

In order to explain, I will need to state a few definitions. Bear with me, they are a teensy bit abstract, but lots of fun. News flash: by Identity I do NOT mean "who you think you are"...

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 Identity:

Assume that we have any set "S" and an operation "%" that relates any two members of S (we'll call them "elements of S") to a third element of S. Then the identity of S under % is an element x in S such that x%s = s for any s in S.

Note: It is easy to prove that such an element x is unique for a particular set and operation (I'll show you another time; let's not get sidetracked).

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To be a bit more specific, in the set of all real numbers we can consider either addition or multiplication as our operation.

The multiplicative identity for the real numbers is 1 (because anything times 1 is itself).

The additive identity for the real numbers is 0 (because anything plus 0 is itself).

 

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 Inverse:

Assume that we have any set "S" with identity i, an operation "%" that relates any two members of S (we'll call them "elements of S") to a third element of S and any particular element of S called s. Then the inverse of s in S under % is an element x in S such that x%s = i.

Note: It is easy to prove that such an element x is unique for a particular set, operation and element. We'll label the inverse of s as s^-1.

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To descend to a more concrete level once again...(i.e. the real numbers for our purposes)

The additive inverse of 2 is -2 (because 2+(-2)=0)

The multiplicative inverse of 2 is 1/2 (because 2*1/2=1)

Now I suggest that you go back and re-read Brahmagupta's claims about negative numbers, positive numbers and zero. Hopefully you'll find them a bit more exciting in the light of this (wonderful) information...

The History of Zero (Part I)

0

When you come to think of it, there isn't much we can do without zero. Zero is just zero...the additive identity to some, just plain nothing to others, a placeholder in the units, tens, hundreds (and hopefullly thousands) columns to still others.

But the number zero has an amazingly chequered history, ranging from the ancient Sumerians to the classical Indian and Arabic mathematicians...so here goes:

Sumerians were farmers, so they needed to keep track of numbers of livestock etc... They had a complicated record keeping system of base 60 (a decent explanation can be found here: http://www.crystalinks.com/sumermath.html).

The Sumerian system was positional. In other words, where a particular symbol is in relation to all the other symbols tells you that symbol's value. An example in modern numerical notation is this:

431: 1 has the value 1, because it is in the units place. 3 has the value 30, because it is in the tens column. 4 has the value 400 because it is in the hundreds column and so on...

Any kind of positional system ends up requiring something to mean "empty" - for example, there is nothing in the 10s column in the number 604, so we write "0". So eventually the Babylonians, who had inherited the Sumerians' system, came up with a special symbol to mean exactly that. At first they just left the space empty, but since spaces tend to get squashed and disappear, at length they actually used a symbol as placeholder:

Well, so far so good. As a human race we have "invented" zero, at least as a "punctuation mark between numbers" (http://www.scientificamerican.com/article.cfm?id=history-of-zero). There's still a long way to go.

The next distinct step (that we know about) took place in 6th century India during the Gupta dynasty. The mathematician Brahmagupta began to work with negative and positive numbers ("debts" and "fortunes") and realised that a sum like 3 - 4 was far from meaningless, despite the fact that it is difficult to imagine it concretely.  He started to see numbers as abstract entities, not just representations of quantity (more discussion at http://www.storyofmathematics.com/indian_brahmagupta.html). 

But of course if you're dealing with negative numbers then you have to face the problems of sums like

(-1) + (+1)

 So he came up with the idea of zero, as a special number rather than just a placeholder. And he developed a whole lot of rules to go with it...

Though a lot of these statements may seem obvious, at the time these ideas were revolutionary! They were so exciting that they quickly (relatively quickly anyway) made their way over to Baghdad and the rest of the Middle East.