Natural Numbers, Integers, Rationals

Definition: Peano Axioms

• 1 is a natural number
• For every natural number x there exists another natural number x’ called the successor of x.
• 1 # x’ for every natural number x (x’ being the successor of x)
• If x’ = y’ then x = y
• If Q is a property such that:
  1. 1 has the property Q
  2. if x has property Q then x’ has property Q
• then the property Q holds for all natural numbers.

Theorem: The Integers

• Let A be the set N x N and define a relation r on N x N by saying that (a,b) is related to (a’,b’) if a + b’ = a’ + b. Then this relation is an equivalence relation.
• If [(a,b)] and [(a’, b’)] denote the equivalence classes containing (a, b) and (a’, b’), respectively, and if we define addition and multiplication of those equivalence classes as:
  1. [(a,b)] + [(a',b')] = [(a + a', b + b')]
  2. [(a,b)] * [(a’, b’)] = [(a * b’ + b * a’, a * a’ + b * b’)]
• then these operations are well-defined and the resulting set of all equivalence classes has all of the familiar properties of the integers (it therefore serves to define the integers based only on the natural numbers).

When defining operations on equivalent classes, one must prove that the operation is well-defined. That means, because the operation is usually defined by picking particular representatives of a class, one needs to show that the result is independent of these particular representatives. Check the above proof for details.

The above proof is in fact rather abstract. The basic question, however, is: why would anyone even get this idea of defining those classes and their operations, and what does this really mean, if anything ? In particular, why is the theorem entitled ‘The Integers’ ? Try to go through the few examples below:

Examples:

• Which elements are contained in the equivalence classes of, say, [(1, 2)], [(0,0)] and of [(1, 0)] ? Which of the pairs (1, 5), (5, 1), (10, 14), (7, 3) are in the same equivalence classes ?
• What do you get when adding, say, [(1,2)] + [(4, 6)] ? How about [(3,1)] + [(1,3)] ? What about multiplying [(5,4)] * [(7, 4)] and [(1,2)] * [(2,1)] ?
• Can you think of a better notation for denoting the classes containing, say, (5, 7) ? How about the class containing (7, 5) ? Do you see why the above theorem is called 'The Integers' ?

• [(a, b)] * [(a’, b’)] = [(a * a’, b * b’)]

• (b - a) * (b’ - a’) = b * b’ + a * a’ - (a * b’ + a’ * b)

Incidentally, now it is clear why multiplying two negative numbers together does give a positive number: it is precisely the above operation on equivalent classes that induces this interpretation. For example:

• (-2) * (-4) = 8

• (-2) is a representative of the equivalence class of pair of natural numbers (a. b) (which we understand completely) whose difference b - a = -2.

• To represent the class that might be denoted by -4, we might choose the pair (8, 4).

• [(4, 2)] * [(8, 4)] = [ (4 * 4 + 2 * 8) , (4 * 8 + 2 * 4)] = [(32, 40)]

Next, we will define the rational numbers in much the same way, leaving the proof as an exercise. To motivate the next theorem, think about the following:

• a / b = a’ / b’ if and only if a * b’ = a’ * b
• a / b + a’ / b’ = (a b’ + b * a’) / (b * b’)
• a / b * a’ / b’ = (a * a’) / (b * b’)

Theorem: The Rationals

• Let A be the set N x N - {0} and define a relation r on N x N - {0} by saying that (a, b) is related to (a’, b’) if a * b’ = a’ * b. Then this relation is an equivalence relation.
• If [(a, b)] and [(a’, b’)] denotes the equivalence classes containing (a, b) and (a’, b’), respectively, and if we define the operations
  1. [(a, b)] + [(a', b')] = [(a * b' + a’ * b, b * b')]
  2. [(a, b)] * [(a’, b’)] = [(a * a’, b * b’)]
• then these operations are well-defined and the resulting set of all equivalence classes has all of the familiar properties of the rational numbers (it therefore serves to define the rationals based only on the natural numbers).

Note that the second component of a pair of integers can not be zero (otherwise the relation would not be an equivalence relation). As before, this will yield, in a mathematically rigorous way, a new set of equivalence classes commonly called the 'rational numbers'. The individual equivalent classes [(a,b)] are commonly denoted by the symbol a / b, and are often called a 'fraction'. The requirement that the second component should not be zero is the familiar restriction on fractions that their denominator not be zero.

As a matter of fact, the rational numbers are much nicer than the integers or the natural numbers:

• A natural number has no inverse with respect to addition or multiplication
• An integer has an inverse with respect to addition, but none with respect to multiplication.
• A rational number has an inverse with respect to both addition and multiplication.

Example:

• So, why would anyone bother introducing more complicated numbers, such as the real (or even complex) numbers ? Find as many reasons as you can.