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Rational numbers are defined as a ratio of integers:
| (1.1.1.1) |
where n and m are integers and r is a rational number. n is called the numerator and m - the denominator. The requirement:
| (1.1.1.2) |
excludes the entities which do not represent numbers (discussed later).
A reduced form of a rational number is the case when n and m in (1.1.1.1) do not have a common integer factor (except ±1). Any rational number can be reduced to this form (if it is not already so) by the division of n and m by the maximal common factor.
The following link discusses the representation of the rational numbers on an infinite axis: Fig. Rational numbers.
The natural numbers form a set of infinite items, which are countable, meaning that we can order them in a row so that each one is associated to an ordinal number (1,2,3,....). All the integers are also countable, since we can arrange them in an ordered row and associate to each one an ordinal number. One way of such arrangement is:
| (1.1.1.3) |
Notice that after each positive number, the corresponding negative number was added. As an example, the nineth ordinal number belongs to the integer −4 .
Although the integers appear to be only a tiny part of all the rational numbers, it can be shown that the rational numbers are countable as well, as seen from Fig. Countability.
It was shown in Fig. Rational numbers that all the rational numbers have their location on an infinite axis. But not all the points of the axis can be represented as rational numbers. For instance the well known number π has a location on the axis but is not rational.
All the numbers that are located on this axis and are not rational are called irrational. The rational and the irrational numbers together are called real numbers and the axis is called the axis of the real numbers. It can be shown that between two different real numbers, no matter how close they are, there are at least one rational and one irrational number.
As an example of an irrational number, we are going to prove that p , the square-root of 2 () , is irrational:
| (1.1.1.4) |
Let's assume that p is a rational number and is represented in the reduced form:
| (1.1.1.5) |
From (1.1.1.5) it follows that:
| (1.1.1.6) |
If 2 is a factor of n , it means that
| (1.1.1.7) |
where k is another integer, then (1.1.1.6) is true. On the other hand, if 2 is not a factor of n then the only possibility is:
| (1.1.1.8) |
therefore n² does not satisfy (1.1.1.6). From (1.1.1.6) and (1.1.1.7) one obtains:
| (1.1.1.9) |
and therefore 2 is a factor of m , which is inconsistent with the assumption that p is a rational number represented in the reduced form. Such a kind of proof is called "by contradiction".
It can be shown that the set of irrational numbers (and therefore - of the real numbers) is not countable. In a loose way one can say that the axis of real numbers is more densely populated with irrational, than with rational numbers.
Exercise 1. Use the definition of rational numbers {(1.1.1.1) and (1.1.1.2)} to prove that the results of addition (sum), multiplication (product) and division (ratio) of rational numbers are also rational.
Exercise 2. Prove that all the rational numbers r in the interval: are countable. Give the first 12 of them. (hint: Fig. Countability)
Exercise 3. Prove that p, the square-root of 3 (), is irrational ( hint: see the example above for square-root of 2 ).
Exercise 4. By following the method of the proofs that and are irrational, would you expect to be able to prove that is also irrational. If you could, then find your mistake.
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