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Section 21.1 Making Sense with Dollars and Cents

Although fractions are important and useful for algebraic manipulations, many of the day-to-day numbers that people encounter are decimals. But it turns out that decimals are just a special way of writing parts of a whole, which means that they're also just fractions. Most students don't have a good sense of the interplay between the two notations, and quite often think of them as mostly unrelated to each other.

Where do decimals even come from? It turns out that this is an extension of the way we write numbers. We use what a place value system which means that our numbers depend on both the symbol we use (the digits 0-9) and the position of that digit within the number itself.

Activity 21.1.1. Representations of Numbers.

Let's think about the number 237. We have been trained to understand that when we put digits side-by-side like this to make larger numbers, that each digit refers to a different group size.

Try it!

Write the number 8367 in expanded form and label each of the parts with the corresponding unit (as shown above).

Solution.

The choice to use ones, tens, and hundreds (and also thousands, ten thousands, and so forth) is because each of those numbers turns out to be a power of 10.

Notice that as we go down the list, the exponents increase and so do the numbers. If we go up the list, the exponents get decrease and so do the numbers. And in the same way we used this pattern to develop negative exponents, we can develop decimals. The naming of these units are a little awkward to say out loud, but the mathematics behind them is simple. Tenths are the size you get when you take one object and break it into ten pieces. Hundredths are the size you get when you take one object and break it into one hundred pieces.

Activity 21.1.2. Expanded Form of a Number.

Try it!

Write the number 35.79 in expanded form and label each of the parts with the corresponding unit.

Solution.

It turns out that decimals have multiple representations. This comes out of thinking about multiple representations of the same fraction.

\begin{equation*} \begin{aligned} 0.3 = \frac{3}{10} \amp = \frac{30}{100} = 0.30 \\ \amp = \frac{300}{1000} = 0.300 \\ \amp = \frac{3000}{10000} = 0.3000 \end{aligned} \end{equation*}

Activity 21.1.3. The Connection Between Fractions and Decimals.

There is a simple connection between the number of decimals and the powers of ten. The number of zeros in the power of in the denominator corresponds to the number of decimal places for that representation of the number. For example, the number 0.0365 (four decimal places) corresponds to \(\frac{365}{10000}\) (four zeros in the denominator). In fact, this number also corresponds to the exponent of the in the denominator: \(\frac{365}{10000} = \frac{365}{10^4}\text{.}\) Thinking about numbers this way will help to avoid certain types of errors.

Try it!

Write the number as a fraction.

Solution.
\begin{equation*} 0.086 = \frac{86}{1000} = \frac{86}{10^3} \end{equation*}

Most of the challenges with addition and subtraction of decimals are resolved by simply ensuring that all your numbers have the same number of decimal places. The best analogy for this is money. American currency is always written with two decimal places (if decimal places are being used). And what this does is that it helps us think about the coins relative to same base unit (1 cent) all the time, and there's no confusion about whether a dime ($0.10) is the same as a penny ($0.01).

Activity 21.1.4. Adding Decimals.

It can be helpful to ensure that all of the numbers are written to the same number of decimals if the calculation is intended to be performed mentally or by hand. This will help to reinforce the underlying concept as well as avoid computational errors.

\begin{equation*} 3.04 + 1.1 = 3.04 + 1.10 = 4.14 \end{equation*}

Try it!

Calculate \(2.5 + 1.22\text{.}\)

Solution.
\begin{equation*} 2.5 + 1.22 = 2.50 + 1.22 = 3.72 \end{equation*}

Activity 21.1.5. Subtracting Decimals.

The exact same trick applies to subtraction.

\begin{equation*} 2.8 - 1.06 = 2.80 - 1.06 = 1.74 \end{equation*}

Try it!

Calculate \(4.77 - 2.3\text{.}\)

Solution.
\begin{equation*} 4.77 - 2.3 = 4.77 - 2.30 = 2.47 \end{equation*}

In practice, most decimal calculations are done by calculator or computer. In fact, certain disciplines (such as physics and chemistry) use the number of decimals as an indication of the quality of a measurement, so that 4.21 is not the same measurement as 4.21000. So we will not be spending a lot of time performing large decimal calculations by hand. Instead, this section should be interpreted as developing a conceptual basis, not a computational basis, for decimals.