Here is an experiment you have probably encountered or explored – there is nothing wrong with retreading interesting ground, right!? Take out a simple calculator and use it to divide 1 \div 3 or 2 \div 13.

Did you notice your calculator displayed a decimal? Moreover, did you notice that the decimals your calculator returned go on and on without an end in sight? Our calculators give the following:

\begin{align*} \frac{1}{3} &= 0.333333333333333\cdots = 0.\overline{3} \\ \\ \frac{2}{13} &= 0.153846153846… = 0.\overline{153846} \end{align*}

We use an over-line bar to indicate the string repeats. The ones in the decimal representation 1/3 keep going on and on forever, but our software tools cut these ongoing expressions off at some point (perhaps to avoid computing forever and depleting their batteries?).

For most practical purposes, we do not need such exact decimal representations of numbers: two or three digits after the decimal point offer sufficient accuracy. Note that some situations, such as making precise measurements, do require this level of accuracy (or more digits after the decimal).

In this section, we discuss how to cut decimal expressions off at a certain point – another name for this is rounding. For example, if you earn a grade of 89.6 on your psychology exam, you would hope your generous professor "rounds up to the nearest whole number" so you get a score of 90. On the other hand, it would be lovely if your restaurant bill of $35.48 were rounded down to the nearest whole number so you could save some money and not need to rummage around for 48 cents!

Here is another example. Let's say you calculate that something costs 0.557 dollars. However, a dollar has 100 cents which means you need to round your decimal to the hundredths place. Standard conventions for rounding your dollar amount yields 0.56 dollars or 56 cents.

A good slogan for this section is more decimal places = more accuracy.

When we round a decimal expression, we limit how accurately it represents a particular quantity. This is necessary for living and working in the real world. For example, you may approximate the value of the fraction 1/3 with the decimal 0.3, but 0.33 is a better approximation. An even better approximation is 0.333.

Fun fact! Did you know that at the time of this writing, some German physicists made the most precise measurements of the mass of an electron to 14 decimal places? Look it up!
Back to '5.2 Rounding Decimals'