Sunday, September 11, 2005

Musings on 1, 2, 3 ...

Kronecker once remarked, "God created the integers, all else is the work of man." Being a Bright I think that integer arithmetic is also the work of man. In the following paragraphs I would explain how this work was done.

Let's begin at the beginning, and assume the concept of unity i.e. there is a thing called 'one' or 'unity' and is normally denoted by the symbol '1'. Another fundamental concept which we need to develop the whole integer arithmetic is that of a successor. So we propose the existence of another entity which is the successor of unity, and denote it by s(1) for the sake of brevity. As a human being, we are not content with 1 and s(1) so we go on and create successor of s(1) which we denote by s(s(1)). We go on creating successors one after another and obtain the following stuff:

1, s(1), s(s(1)), s(s(s(1))), s(s(s(s(1)))), ...

You would have guessed that this is the same as the sequence 1, 2, 3, 4, ..., but for the time being let us not go into decimal symbols and assume that we will use 1 and s (for successor) as our symbols. What I would like to point out is that these two simple concepts, unity and successor, are enough to explain whole of arithmetic.

The numbers obtained using 1 and the successor mechanism will be called natural numbers. If we follow the above sequence of natural numbers carefully, we observe that every term in that sequence (except the first one) is a successor of the previous. Note that unity itself is not the successor of any entity. We note the basic properties of this number system:
  • There is a distinguished entity called unity (which we denote by the symbol 1).
  • Every entity other than unity is a successor of some unique entity.
  • A successor of an entity is not the same as the entity itself (i.e. s(x) != x).
  • Every entity has a unique successor.
  • Unity is not the successor of any entity.
These properties define what we call a system of natural numbers and the individual entities in the system are called natural numbers. The principle that "any two systems of entities with the above properties are essentially equivalent as far as their mathematical use is concerned" is called the Principle of Mathematical Induction.

I guess most of you people have never happened to see the principle stated in this form, but the essence of this statement is the same as the one provided in many textbooks. To see the equivalence of the version given here with the one provided in textbooks, you only need to observe that s(x) (here) is the equivalent of (x + 1) (textbooks).

Now the introduction of operations like addition and multiplication is fairly straightforward. We begin with addition first (because multiplication is defined in terms of addition). The basic idea is: first learn how to add 1 to a natural number. You must have guessed the following definition:
x + 1 = s(x) ... (adding unity is the same as finding successor)

After learning how to add 1 to a number x, we want to add a number y (different from unity) to a given number x. Since y is different from 1, it is a successor of some number say z. We now define:
x + y = x + s(z) = s(x + z)

This completes the inductive definition of addition. To illustrate this with examples, let's bring back the decimal symbols and add 3 to 4. To add 3 (y) to 4 (x), we first find z such that s(z) = 3. Clearly z = 2. So we need to add this 2 to 4 and then find the successor of the result. Thus

4 + 3 = 4 + s(2) = s(4 + 2)
= s(4 + s(1)) = s(s(4 + 1))
= s(s(s(4))) = s(s(5)) = s(6)
= 7.

Thus adding a number to another is like adding a succession of 1's to the number. The reader will now find it easy to grasp the following inductive definition of multiplication:
x * 1 = x
x * y = x * s(z) = x * z + x provided (y != 1)

Notice how addition is used in the definition of multiplication. I would advise the reader to multiply 2 by 3 using this definition. I need not elaborate further on + and *, it is sufficient to remark that using these definitions one can deduce all the properties of addition and multiplication. One can then define subtraction and division as the inverse of addition and multiplication respectively.

The post is getting a little lengthy and so I would stop here. The discussion on integers (positive and negative) will be continued in the next post and till that time the reader can use these definitions of addition and multiplication to prove commutative, associative and distributive laws.


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