Sets of Numbers
A set is probably the most basic building block in the foundation of mathematics. It is the concept
of a set that lets us get started in defining what arithmetic is and how it works. A set is nothing
more than a collection of objects. In mathematics we often deal with collections of numbers, but a
set can be a collection of anything. In fact, the numerals we use (number symbols) were developed
to describe sets. But first things first!
The most common way to denote a set is simply to write down the members of the set enclosed in
curly braces and separated by commas. For example, the set of all natural numbers that are
multiples of 5 could be written by the list method this way: {5, 10, 15, …}. The ellipsis (three dots)
indicates that the list of members of this set goes on forever; we cannot possibly write down all the
multiples of 5, but the list does continue indefinitely—so this set is an example of an infinite set.
Another method to denote this set would be to write
x | x is a natural number and x is a multiple of 5 . This method is called set builder notation
method. The vertical bar is read ―such that,‖ so the set just described is ―the set of all x such that
x is a natural number and x is a multiple of five.‖ This set could also be written this way:
5x | x is a natural number or {x | x  5n
for n  N}. The description method would simply be
{the set of all natural numbers that are multiples of five}
Most of the sets we deal with in basic mathematics are sets of numbers. The most basic of these

sets is the set of natural numbers, N = {1, 2, 3, …}. This set is often referred to as the counting
numbers. Putting in just one other element, zero, we arrive at the set of whole numbers: W = {0, 1,
2, …}. The set Z = {…, -3, -2, -1, 0, 1, 2, 3, …} is called the set of integers, and it is simply the union
of the set W and the set of the opposites of the natural numbers. The set Q =
a

 a and b are integers and b  0  is the set of rational numbers (any number that can be written
b

as a fraction – this includes decimals that terminate and decimals that repeat).
, the set of
irrational numbers, are any real numbers that cannot be written as a fraction. The irrational
numbers include non-terminating decimals that do not repeat (some examples include π, e, or square
roots of non-perfect squares).