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Section 5.2 Fractions, Decimals and Percentages

Subsection 5.2.1 Expression of Numbers

Figure 5.2.1. Fractions, Decimals and Percentages

When we write a number, we often have choice of how to represent it. Consider all of these numbers:

\begin{equation*} 4 \text{ or } 4.0 \text{ or } 4.\bar{0} \text{ or } \frac{4}{1} \text{ or } \sqrt{16} \text{ or } \frac{12}{3} \end{equation*}

These are all ways of writing the same number. One of the subtle and implicit skills in mathematics is switching between the different ways of writing numbers. We aren't going to get into all the detail of this switching in these notes, but I'll mentioned a few points in this section.

Subsection 5.2.2 Fractions and Decimals

Decimal expansion carry meaning using powers of \(10\text{.}\) The decimal

\begin{equation*} 5.098 \end{equation*}

exactly means the fraction

\begin{equation*} \frac{5098}{1000}\text{.} \end{equation*}

From that point, we could reduce the fraction to lowest terms if we wish. \(2\) is a common factor, so we can divide by 2.

\begin{equation*} \frac{5098}{1000} = \frac{2599}{500} \end{equation*}

There are no more common factors, so this is lowest terms.

\begin{equation*} 5.098 = \frac{2599}{500} \end{equation*}

In general, for a finite decimal expansions, we can write it as a fraction over a power of \(10\text{.}\) The power of \(10\) is determined by the number of digits after the decimal point: if there are \(n\) digits, then the denominator of the fraction is \(10^n\text{.}\) In the example above, there were three digits, so the denominator was \(10^3 = 1000\text{.}\)

Some numbers are expressed as infinite repeating decimals.

\begin{equation*} 0.36363636\ldots \end{equation*}

These are often written with a bar over the repeating piece to indicated repetition.

\begin{equation*} 0.36363636\ldots = 0.\overline{36} \end{equation*}

All repeating decimals can be expressed as fractions. There is a general algorithm for doing so, but I've decided not to put that algorithm in these notes.

Likewise, there is a general algorithm for writing a fraction as a repeated decimal, essentially doing long division until we produce a repeating string of digits. Again, I've decided not to put that algorithm in these notes.

Finally, some numbers can only be written as non-repeating decimals. Here is the start of the decimal expansion for \(\sqrt{2}\text{.}\)

\begin{equation*} \sqrt{2} \doteq 1.41421356237 \ldots \end{equation*}

There is no regular pattern to these digits. Since there is no regular pattern, \(\sqrt{2}\) cannot be written as a fraction. It is an irrational number (by definition, a number which cannot be expressed as a fraction). All fractions correspond to repeating decimals expansions and any non-repeating decimal expansion corresponds to an irrational number.

Subsection 5.2.3 How Should I Write a Number?

If we have all these different ways to write numbers, which ones should we choose? The answers, of course, is that it depends on the context and the need.

Though much could be said on this topic, the major point I want to make in this section is the difference between exact and approximate values. By exact values, I mean any symbol or expression that tells us exactly which number we are talking about. For example, \(\pi\) is an exact value, where \(3.1415\) is an approxmation. Similarly, \(\sqrt{7}\) is an exact value, where \(2.65\) is an approximation.

When writing approximate values, it's not strictly correct to use the equals sign. An approximation is not equal to the original number -- it's just approximately equal. There are (unfortunately) several common notations for approximately equal. In these notes, I use an equals sign with a dot to indicate approximation.

\begin{equation*} \pi \doteq 3.1415 \end{equation*}

The other most common notation is an equals sign with a tidle-like symbol on top.

\begin{equation*} \pi \cong 3.1415 \end{equation*}

You should use one of these symbols whenever you write an approxmation, to indicate that you've moved frome exact values to approximate values.

Exact values are good for any kind of mathematical procedure. Usually when we are doing calculations, proofs or other mathematical constructions, we want to work with exact values. In pure mathematics, we work almost exclusively with exact values.

However, there are limitations. Perhaps we do a long calculation and end up with the following exact value.

\begin{equation*} \pi^2 + 7\sqrt{\frac{53}{2}} \end{equation*}

That's a fine exact value, but at a glance is it very hard to know, roughly, what size of number this is. That's where approximation is more valuable.

\begin{equation*} \pi^2 + 7\sqrt{\frac{53}{2}} \doteq 45.904 \end{equation*}

Writing an approximate decimal value gives us a better sense of this number: it's nearly 46. It we needed to use this in some application, we'd probably want to know this approximate value.

So why don't we work with approximate values all the way through? Sometimes we do, but there is a problem. Approximate values have an error. If I write \(\pi \doteq 3.14\text{,}\) this has an error of a little less than \(\frac{1}{500}\text{.}\) For whatever you are doing, that might be a perfectly reasonable amount of error. However, error propogates through calculate. As you do arithmetic with approximate values, the error gets works. As an example, \(3.14^2 = 9.8596\) but a \(\pi^2 \doteq 9.8696\text{.}\) The error is now a little more than \(\frac{1}{100}\text{.}\) If we continued to do more arithmetic, the error will grow. After a few of arithmetic steps, the error might be larger than the number itself, making the calculation meaningless. The study of error and error propogation is a branch of mathematics itself, which I'm not going to cover. My point here is that we use exact values when we can in order to limit the propogation or error. If we want an approximate decimal answer, it is best to do the approximates as late as possible in the calculation.

Subsection 5.2.4 Fractions and Percentages

A percentage is a fraction of \(100\) (literally, from the Latin per centum, meaning from a hundred). When we write \(47\%\text{,}\) we mean this fraction and decimal:

\begin{equation*} \frac{47}{100} = 0.47\text{.} \end{equation*}

To convert from fractions to decimals, we multiply by \(100\text{.}\) The fraction \(3/17\) has a decimal approximation of \(0.1764\text{.}\) Multiplying by \(100\) gives \(17.64\%\text{.}\)

To convert from decimals to fraction, we divide by \(100\text{.}\) \(48\%\) becomes the fraction \(\frac{48}{100}\text{,}\) which can be writen in lowest terms as \(\frac{12}{25}\text{.}\)

There are many common operations with percentages in a variety of application. For these notes, I've chosen to simply include one: percentage changes. If we have two values, how do we calculate the percentage change from one value to the other? If the values are \(a_1\) and \(a_2\text{,}\) then the percentage change from \(a_1\) to \(a_2\) is calculate by

\begin{equation*} \frac{a_2 - a_1}{a_1} \times 100 \text{.} \end{equation*}

For an example, say a population increases from \(154,244\) to \(157,934\text{.}\) I'll just apply the previous expression.

\begin{equation*} \frac{157934 - 154244}{154244} \times 100 \doteq 2.39 \end{equation*}

The percentage increase in this population is approximately \(2.39\%\text{.}\)