Musings on 1, 2, 3 ... (contd...)

For those who find my previous two posts a bit abstract, I ensure that this one is the last piece and completes the "Musings" trilogy. And yes, I would keep it short.

We invented the naturals and the integers in the previous posts and we next proceed to the rationals. Let's assume then that a, b, c, d etc. stand for integers. We know that division is not always possible in the system of integers and we wish to get rid of that limitation. The strategy is similar to the one used for creating integers out of rationals. We observe that an integer can be expressed as the ratio of two integers in many different ways. We thus denote an integer as an ordered pair (a, b) where a is divisible by b and b != 0. The integer being represented is a/b. Two ordered pairs are equivalent if they represent the same integer. Thus (a, b) = (c, d) iff a*d = b*c.

We then consider the system of all ordered pairs of integers of the form (a, b) with the only restrictions that b != 0. Thus we may have a case when a is not divisible by b. The definition of equivalence is the same and as before we put all the equivalent pairs in a group. Such a group is called a rational number, and we denote such groups by [a, b] where (a, b) is any pair belonging to the group. The definitions for rationals are now very clear:

[a, b] = [c, d] iff a * d = b * c
[a, b] + [c, d] = [a*d + b*c, b*d]
[a, b] * [c, d] = [a*c, b*d]

The rational number [0, a] with a != 0 is denoted by 0 (zero) and [a, a] is denoted by 1 (one) and they are the additive and multiplicative identities. If [a, b] is such that a != 0 != b, then [b, a] is also rational and then [a, b] * [b, a] = [a*b, b*a] = 1, and so [b, a] is the reciprocal of [a, b] and is denoted by 1/[a, b]. The division is now defined to be:

[a, b]/[c, d] = [a, b]*(1/[c, d])
= [a, b] * [d, c] = [a*d, b*c] provided b != 0 != c, d != 0

The rationals are adequate enough for the needs of arithmetic, but human mind does not seem to be content with it, and it goes on to introduce irrationals. I won't demystify the irrationals here, because that is already done (in the best possible way) in G. H. Hardy's "A Course of Pure Mathematics" (see Chapter 1). I would have a lot more to say about this classic and influential work, but not now.

I know that there are some people out there who might not be satisfied with such an abrupt ending of the "Musings" and would wish to go on with the "reals" and the "complex numbers", but these are best provided in books rather than blogs. For others, I hope this posts completes the trilogy effectively.