Itmm

8.
Construction of rational numbers:
The division operation is until now, not defined, since division of an integer by integer may not be an integer. We construct rational numbers as follows: Rational numbers constitutes the set of ordered pairs (a,b) such that a,b are integers, b≠0 and where equality, addition and multiplication are defined as shown: 
@ (a,b) = (c,d) if and only if a.d =b.c
@ (a,b) . (c,d) = (a.c,b.d)
@ (a,b)+(c,d) = (ad+bc,bd)
(A smart reader will quickly recognize that (a,b) is the disguised form of a/b)
Define 
1/a := (1,a)
Then, it is clear that
(a,b) =(a,1).(1,b)
Define 
a/b := (a,b)

Construction of natural numbers:
Meaning of ‘set’ and ‘is an element of’ are intuitively clear / assumed to be understood. Peano’s axioms assume the existence ofa set N and a function f : N → N satisfying the axiom:
Whenever Φ belongs to N, then f(Φ) = {Φ} belongs to N.
Then, { N, f, Φ } is the model of natural numbers including zero. The claim becomesclearer when we define
0 := Φ
1 := {Φ}
2 := {Φ,{Φ}}
3 := {Φ,{Φ},{{Φ}}}
…
and so on.
The brace bracket { } is not to confuse, but to distinguish two distinct elements of N, e.g. to distinguish2={horse, cow} from 3={apple, mango, banana}
It can be proved that f(a) = a+1 for all a. Similarly, a+f(b) = f(a+b) and thus, a+b becomes well defined. 
(Perhaps now one can intuitively define product ofa and b as addition of a to a, b-number of times)

A more interesting construction of integers from natural numbers is as shown:

Introduction- Until now, we know operations + and x in naturalnumbers. The operation -(minus) is unknown since it is not well defined for all ordered natural numbers (eg. 1-2 (=-1) is not a natural number).
Construction- Integers are the ordered pair (a,b) (seefootnote) where a & b are natural numbers satisfying some relation among them.
We now have to define when two ordered pairs (a,b) and (c,d) are equal, and operations + & x in them. We do this as...