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 20th
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.”
