Section 1.2 Decimal Representations
A real number like \(3.1459\text{,}\) which can be written with a finite number of digits, is shorthand for a sum of rational numbers:
\begin{align*}
3.1459 \amp= \underbrace{3}_{\text{integer part}}
+ \underbrace{\frac{1}{10}}_{\text{tenths}}
+ \underbrace{\frac{4}{100}}_{\text{hundreths}}
+ \underbrace{\frac{5}{1000}}_{\text{thousandths}}
+ \underbrace{\frac{9}{10000}}_{\text{ten thousandths}}\\
\amp= 3 + \frac{1459}{10000} = \frac{31459}{10000}.
\end{align*}
In fact, every rational number has a decimal representation provided we allow the digits to eventually repeat a regular pattern infinitely often. We write a bar over the digits that repeat.
Example 1.4. Decimal Representations of Rationals.
The following examples provide decimal representations of some rational numbers.
\begin{equation*}
\frac{1}{2} = 0.5000\ldots = 0.5\overline{0} = 0.5
\end{equation*}
\begin{equation*}
\frac{2}{3} = 0.6666\ldots = 0.\overline{6}
\end{equation*}
\begin{equation*}
\frac{12}{37} = 0.324\,324\,324\ldots = 0.\overline{324}
\end{equation*}
\begin{equation*}
\frac{381}{148} = 2.57\,432\,432\,432\ldots = 2.57\overline{432}
\end{equation*}
The relationship between the decimal representation of a rational number (and what it even means to have an infinitely long decimal representation) requires some concepts we won’t discuss until calculus. For now, I’d like you to be aware of the following fact.
Fact 1.5. Decimal Representation of Rational Numbers.
A real number is rational if and only if it has a decimal representation that eventually repeats a regular sequence of digits.
Accordingly, a real number is irrational if and only if it’s impossible to express it in a decimal form with a regularly repeating sequence of digits.
Example 1.6. \(\pi\) is irrational.
Draw a perfect circle of any size. Measure the length of its diameter (the distance across the circle through the center) and its circumference (the distance around the circle).
If you divide the circumference
\(C\) by the diameter
\(d\) you will always obtain the same number called
\(\pi\) ("pi")
\begin{equation*}
\pi = \frac{C}{d}.
\end{equation*}
In fact, \(\pi\) is an irrational number. The decimal representation of \(\pi\) begins with
\begin{equation*}
\pi = 3.1459\ldots,
\end{equation*}
but never terminates or infinitely repeats a fixed sequence of digits. We can at best approximate its value with a finite number of digits which we could indicate as
\begin{equation*}
\pi \approx 3.1459
\end{equation*}
where \(\approx\) is used to indicate an approximation.
Example 1.7. Mathematical Reasoning.
Given that \(\pi\) is irrational, what about other expressions that involve \(\pi\text{?}\) For instance, is it possible that \(1/\pi\) is suddenly rational? Prove that can’t be the case!
Solution.
Let’s assume that \(1/\pi\) it is rational and see what happens. This would mean that there are integers \(p\) and \(q\neq 0\text{,}\) such that
\begin{equation*}
\frac{1}{\pi} = \frac{p}{q}.
\end{equation*}
Note that \(p \neq 0\) because \(1/\pi\) is not zero. Taking the reciprocal of both sides gives
\begin{equation*}
\pi = \frac{q}{p}
\end{equation*}
from which we could conclude that \(\pi\) is rational, a contradiction. Thus, \(1/\pi\) must be irrational after all.
Checkpoint 1.8.
Express \(4.321\) as a ratio of two integers.
What is the circumference of a circle whose radius is 4 inches long?
Given that \(\sqrt{2}\) is irrational, do you think \(\sqrt{3}\) is irrational? What about \(\sqrt{4}\text{?}\)
