Sunday, September 11, 2005

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

In the last post I discussed the basic properties of the natural numbers. This approach to natural numbers was first adopted by Peano in 1889, and as such these properties of the natural numbers are known as Peano's Axioms. Since then the natural numbers have been put on sound and rigorous footing.

Let's continue from where we left in the last post. We turn our attention from natural numbers to integers. You might be aware that integers were invented to solve the problem of subtraction. A number can be subtracted only from a greater number and not vice versa. Incidentally subtraction gives us a new way of looking at the natural numbers. Any natural number can be expressed as the difference of two natural numbers in many (actually infinite number of) different ways. For example:

2 = (4 - 2) = (5 - 3) = (100 - 98) = ... (and so on)

We now represent the natural number 2 as an ordered pair of natural numbers (4, 2). This representation is not unique, for we can write 2 as (5, 3) or (100, 98). The idea is that the difference between the numbers in the pair gives the number being represented. Also right now the first number in the pair is greater than the second to make the subtraction possible. Two such representations (a, b) and (c, d) are said to be equivalent if they represent the same natural number. Thus (a, b) = (c, d) if (a - b) = (c - d) which is the same as the condition (a + d) = (b + c).

From this observation we make our advance. We consider all the pairs of natural numbers (a, b) even considering the case when a = b or a < b and not only a > b. Two such pairs (a, b) and (c, d) will be said to be equivalent if (a + d) = (b + d). We club all the equivalent pairs in one group. We represent each such group of equivalent pairs by [a, b] where (a, b) is one member of the group. It does matter which pair is used to represent the group. Such a group is called an integer. The equality of such integers is defined as follows:

[a, b] = [c, d] iff (a + d) = (b + c)

The special integer [a, a] is of fundamental importance is denoted by a special symbol 0 (zero). The integer [a + 1, a] is also very important and is denoted by symbol 1 (one). The addition and multiplication of integers is now defined as follows:

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

The reader can verify that the usual properties of addition and multiplication hold, and further that integers 0 and 1 defined above are respectively additive and multiplicative identity. One curious fact is that [a, b] + [b, a] = [a + b, b + a] = [a + b, a + b] = [a, a] = 0. So [b, a] is the additive inverse of [a, b] and is more properly denoted by -[a, b] (with a minus sign).

The integers of the form [a, b] when a > b behave exactly the same way as the system of natural numbers with [a + 1, a] as unity and [a + 1, b] the successor of [a, b]. If we identify such integers with the natural numbers (as we can in view of Principle of Mathematical Induction) we almost have integers as an extension of the natural numbers. Subtraction is now possible by defining:

[a, b] - [c, d] = [a, b] + (-[c, d])
= [a, b] + [d, c] = [a + d, b + c]

So we have now got integers at hand and the problem of subtraction is solved. I guess you must be asking why the hell did we use the machinery of ordered pairs and equivalence groups. Why didn't we simply added stuff like 0, -1, -2, -3 to our repertoire of natural numbers and done away with it? The point is that in mathematics we build up from the given, and not bring in alien stuff as and when required. The integers had to be created somehow out of the naturals and not by the use of negative ghosts like -1, -2.

The post needs to be concluded to keep it short but the journey into number systems will continue in the next post also. Till then read someone else's blog. Bye.


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