2. Infinity and Induction

2.4. The Real Number System

In the previous chapter we have defined the integers and rational numbers based on the natural numbers and equivalence relations. We have also used the real numbers as our prime example of an uncountable set. In this section we will actually define - mathematically correct - the 'real numbers' and establish their most important properties. There are actually several convenient ways to define R. Two possible methods of construction are: Right now, however, it will be more important to describe those properties of R that we will need for the remainder of this class.

The first question is: why do we need the real numbers ? Arenít the rationals good enough ?

Theorem 2.4.1: No Square Roots inQ
  There is no rational number x such that x2 = x * x = 2.

Thus, we see that even simple equations have no solution if all we knew were rational numbers. We therefore need to expand our number system to contain numbers which do provide a solution to equations such as the above.

There is another reason for preferring real over rational numbers: Informally speaking, while the rational numbers are all 'over the place', they contain plenty of holes (namely the irrationals). The real numbers, on the other hand, contain no holes. A little bit more formal, we could say that the rational numbers are not closed under the limit operations, while the real numbers are. More formally speaking, we need some definitions.

Definition 2.4.2: Upper and Least Upper Bound
  Let A be an ordered set and X a subset of A. An element b is called an upper bound for the set X if every element in X is less than or equal to b. If such an upper bound exists, the set X is called bounded above.

Let A be an ordered set, and X a subset of A. An element b in A is called a least upper bound (or supremum) for X if b is an upper bound for X and there is no other upper bound b' for X that is less than b. We write b = sup(X). By its definition, if a least upper bound exists, it is unique.

Examples 2.4.3:
 
  • Consider the set S of all rational numbers strictly between 0 and 1.
    1. Find 5 different upper bounds for S
    2. Find the least upper bound for S.
    3. Is there any difference for the set [0, 1] ?
  • Consider the set of rational numbers {1, 1.4, 1.41, 1.414, 1.4142, ...} converging to the square root of 2.
    1. What is the least upper bound of this set, if all we knew were the rational numbers?
    2. What is the least upper bound of this set, allowing real numbers ?
  • Can you define Lower Bound and Greatest Lower Bound (called Infimum) ?
In the above example we have seen several facts: In fact, this last fact is exactly the property which distinguishes the real numbers from the other number systems, and makes it useful to us. We will state this as a theorem:
Theorem 2.4.4: Least Upper Bound Property
  There exists an ordered field of numbers R with the properties:
  1. it contains all rational numbers
  2. it has the property that any non-empty subset which has an upper bound has a least upper bound.

Note that we have not defined 'ordered field', but that is not so important for us right now. The importance for us is that this property is one of the most basic properties of the real numbers, and it distinguishes the real from the rational numbers (which do not have this property).

In order to prove this theorem we need to know what exactly the real numbers are, and we have indeed given two possible constructions at the beginning of this section.. However, it is more important to understand these properties of R, and to know about the differences between R and the other number systems N, Z, and Q.

We can use this theorem to illustrate another property of the real numbers that makes them more useful than the rational numbers:

Theorem 2.4.5: Square Roots inR
  There is a positive real number x such that x2 = 2

There are several other properties that will be of importance later on. Two of those are the Archimedean and the Density property. Again, as for the Least Upper Bound property, it is more important to understand what these properties mean than to follow the proof exactly.

Theorem 2.4.6: Properties ofR and Q
  The set of real numbers satisfies the Archimedean Property:
  • Let a and b be positive real numbers. Then there is a natural number n such that n * a > b

The set of rational numbers satisfies the following Density Property:

  • Let c < d be real numbers. Then there is a rational number q with c < q < d.

This concludes the 'elementary' part of this text. We have now defined much of our basic notation, learned how to count to infinity, introduced all number systems that we will be using, and we have seen several different types of proofs. We are now ready for some more complicated topics.