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"...

***

 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).

***

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).

 

***

 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.

***

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...