Cartesian Coordinate Systems - 3D Math Primer for Graphics and Game Development - page 3

Figure 1.3. A number line for integers. (Note the ghost sheep for negative numbers.)
At some point in history, it was probably realized that sometimes, par-
ticularly fast talkers could sell sheep that they didn't actually own, thus
simultaneously inventing the important concepts of debt and negative num-
bers. Having sold this putative sheep, the fast talker would in fact own
“negative one” sheep, leading to the discovery of the integers, which consist
of the natural numbers and their negative counterparts. The corresponding
number line for integers is shown in Figure 1.3.
The concept of poverty probably predated that of debt, leading to a
growing number of people who could afford to purchase only half a dead
sheep, or perhaps only a quarter. This led to a burgeoning use of fractional
numbers consisting of one integer divided by another, such as 2/3 or 111/27.
Mathematicians called these rational numbers, and they fit in the number
line in the obvious places between the integers. At some point, people
became lazy and invented decimal notation, writing “3.1415” instead of
the longer and more tedious 31415/10000, for example.
After a while it was noticed that some numbers that appear to turn
up in everyday life were not expressible as rational numbers. The classic
example is the ratio of the circumference of a circle to its diameter, usually
denoted π (the Greek letter pi, pronounced “pie”). These are the so-called
real numbers, which include the rational numbers and numbers such as π
that would, if expressed in decimal notation, require an infinite number of
decimal places. The mathematics of real numbers is regarded by many to
be the most important area of mathematics—indeed, it is the basis of most
forms of engineering, so it can be credited with creating much of modern
civilization. The cool thing about real numbers is that although rational
numbers are countable (that is, can be placed into one-to-one correspon-
dence with the natural numbers), the real numbers are uncountable. The
study of natural numbers and integers is called discrete mathematics, and
the study of real numbers is called continuous mathematics.
The truth is, however, that real numbers are nothing more than a polite
fiction. They are a relatively harmless delusion, as any reputable physicist
will tell you. The universe seems to be not only discrete, but also finite.