Rational Numbers

The Natural numbers are easy to understand, because they are so, well, natural. We can count with them and use them to order things. We can go a long way with the natural numbers. But sooner or later, we need something more. Perhaps we want to measure a distance, or figure out the circumference of a circle. We need numbers beyond the naturals.

Negative numbers are one means of extending the negative numbers, and they have an obvious interpretation such as debt. As an example, most of us have at one point in our lives have overdrawn our checking account and seen the statement showing a negative balance. Negative numbers are useful for representing debts, temperature, altitude below sea level—anything that can have values below zero. In our modern construction of number systems, negative numbers and zero are the first things we add to the natural numbers, collectively giving us the set of integers, which we usually represent with the symbol Z. The integers seem to be a very natural extension of the naturals, but they were not the first such extension. Negative numbers first appeared in the Chinese text Nine Chapters on the Mathematical Art, which dates from the Han dynasty (202 BC-220 AD) but likely contained older material. Indian mathematicians developed the rules for the use of negatives, which then spread to the Middle East and from there to Europe.

The oldest extension of the natural numbers is most likely the set of rational numbers. Rational numbers are defined as numbers of the form:

They are called rational because they are ratios of integers. The Egyptians were aware of rational numbers, and had a variety of “recipes” for representing rational numbers as the sum of reciprocals of integers. For example:

These expressions are known as Egyptian fractions, and every rational number can be represented in this way. The ancient Greeks not only knew about rational numbers, they studied them in great detail. Rational numbers formed the basis of Greek music theory, and the religious order of the Pythagoreans believed that all phenomenons in the universe could be reduced to whole numbers or their ratios.

Rational numbers are often denoted by Q. The use of the symbol Q comes from the text Algebre by Nicholas Bourbaki, the pseudonym for a group of mostly French 20^th century mathematicians that wrote a series of books that unsuccessfully attempted to present an exposition of modern advanced mathematics, attempting to set all of mathematics on modern set theory.

Rational numbers are easy to recognize because their decimal expansion is either a terminating (e.g. ¼=0.25) or repeating (e.g. 1/3 = 0.3333…) decimal. It is also true that any repeating or terminating decimal represents a rational number.

The rational numbers have many interesting properties. One of the most important is that the set of rational numbers is densely ordered. Ordered means that for any to numbers a, b ϵ Q, either a≤b, or b≤a. Dense essentially means that for any two rational numbers a<b, we can find another rational number c such that a<c<b. That means that no matter how close together two rational numbers are, we can always squeeze in another rational number in between them.

With the densely ordered property the set of rational numbers seems to be fairly exhaustive and complete. All of the integers are also rational numbers.  But is every number rational?

The answer is of course “No.”