Section 1.1 Integers, Rationals, and Reals
A
real number is a number that represents a location on the continuous number line. We refer to the collection of all real numbers as the
set of real numbers and denote this by the symbol
\(\mathbb{R}\text{.}\) Within the set of real numbers are special categories of numbers you should be familiar with. The
set of natural numbers includes the counting numbers and zero:
\begin{equation*}
0, 1, 2, 3, 4, 5, \ldots .
\end{equation*}
If we also include their opposite, we obtain the set of integers:
\begin{equation*}
\ldots, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, \ldots .
\end{equation*}
If we form ratios of integers, i.e. fractions involving integers, we obtain the set of rational numbers.
Example 1.1. Rational Numbers.
The following real numbers are rational numbers:
\begin{equation*}
\frac{7}{10} = 0.7,\quad \quad\frac{2}{3} = 0.6666\ldots, \quad \quad \frac{-20}{5} = - 4.
\end{equation*}
Every integer is a rational number as it can be written as a ratio in many ways. For example,
\begin{equation*}
-4 = \frac{-4}{1} = \frac{4}{-1} = \frac{-20}{5}.
\end{equation*}
Even zero is a rational number, since dividing zero by any non-zero integer results in zero:
\begin{equation*}
0 = \frac{0}{1} = \frac{0}{-50}.
\end{equation*}
A real number which is not rational is called an irrational number. Irrational numbers are everywhere on the real line, filling up the gaps between the rationals numbers. Many famous constants are irrational, such as \(\sqrt{2}\text{,}\) \(\pi\text{,}\) and \(e\text{.}\)
Checkpoint 1.3.
What is an integer? What is a rational number? What is an irrational number? Is every rational number an integer? Is every integer a rational number?
