RWM101: Foundations of Real World Math (2022.A.01)

As the prefix dec suggests, writing fractions in this way favors the number 10. If you really think about it, our standard way of writing whole numbers also favors the number 10. In fact, every whole number we have ever written down has secretly used 10s. For example, consider the whole number 135. The digits 1, 3, and 5 are arranged in a certain position or place that indicates how much they are worth. Their positions indicate how many 10s they are worth:

$135 = (1 \text{ one hundred}) + (3 \text{ tens }) +(5 \text{ ones})$.

Each of these place values can be expressed using powers of 10 since $100 = 10^{2}$, $10 = 10^{1}$, and $1 = 10^{0}$. In other words:

$135 = (1\times 10^{2}) +(3\times 10^{1}) +(5 \times 10^{0})$.

If we add a tiny amount to the number 135, say $1/10$, we obtain the mixed number $135\,\frac{1}{10}$. Wouldn't it be nice to also express this number using 10s? That is what decimals do!

Indeed, you can think of decimal notation as extending our regular place value notation to the right. We can express the mixed number $135\,\frac{1}{10}$ using nothing but powers of 10s. But for the fractional amount, we need to use a negative power of 10, or you could call it a fraction with 10 in the denominator:

$135\,\frac{1}{10} =(1\times 10^2) + (3\times 10^1) + (5 \times 10^0) + (1 \times 10^{-1})$.

A decimal point lets us rewrite this expression in a way that hides all of these powers of 10 and their various exponents:

$135\,\frac{1}{10} = 135.1$

The digit to the right of the decimal point is the 1/10's place. Digits after that one occupy the 1/100s and 1/1000s places, respectively. For example:

$135.149 = 135 + \frac{1}{10} + \frac{4}{100} + \frac{9}{1000}$.

Before we perform any actual calculations with decimals, we want you to develop a solid understanding or familiarity with exactly how and what decimal notation expresses.

Here is an experiment you have probably encountered or explored: 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 when 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 is lovely if your restaurant bill of $35.48 is rounded down to the nearest whole number so you can save some money and do 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 yield 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 $0.333$.

Decimal expressions often appear in situations that involve measurements and quantification - in various corners of the real world and lots of hard sciences. Fractions or rational numbers are more likely to occur in purely mathematical discussions. We need to know how to convert between these two notations to keep the lines of communication open between those who prefer pure math and those who follow applied math (with real-world applications).

What kind of notation is better? Fraction or decimal? As a pure mathematician, the answer is clearly fraction notation. Obviously! But it really depends on the context. Rational numbers, such as $10/3$, tell us a lot, and we can apply them to many situations.

For example, Let's take the expression $10/3$. It could mean someone is trying to divide 10 objects into three equal piles, or it could refer to an unknown quantity $x$ that makes the equation $3x-10 = 0$ true. However, it might be more convenient to use the decimal approximation $10/3 \approx 3.33$. This expression allows us to quickly understand and estimate its numerical value (a bit beyond 3).

During your decimal work, you may have come across fractions with decimals that "go on forever". Others do not (or do not seem to). Consider the following:

$\begin{align} \frac{1}{4} &= 0.25 \\ \\ \frac{1}{3} &= 0.333333333333333… = 0.\overline{3}\end{align}$

Why do some fractions behave finitely in terms of decimal representation and others infinitely? It is a great question we encourage you to ponder. Hint: it may have something to do with whether the fraction is equivalent to one whose denominator is a power of 10.

Thankfully, we can always convert an expression with digits to the right of the decimal that go on forever with a repeating string of digits into a fraction. There is a fun way to accomplish the conversion. We call these decimal expressions repeating decimals. Let's watch a video and review a quick example before we point you to a video that outlines this approach with lots of examples.

It can be simple to notate adding whole numbers, such as $2+1 = 3$. However, while understanding the computation $4+8 = 12$ is easy, it is more complicated to notate than our first sum. The numbers four and eight each use only one digit, but the sum is two digits. In other words, adding four and eight together requires us to use a new place value: a one in a new tens place.

This example shows us that adding and subtracting can require using new positions to the left of the decimal point. When we add or subtract decimal expressions, we often have to change positions to the right of the decimal point as well. For example, consider this equation which involves fractions:

$\frac{1}{2} + \frac{1}{2} = 1$

When we write this equation using decimals, we have:

$0.5 + 0.5 = 1.0 = 1$

To the left side of the equal sign, we have two decimal expressions, each using one digit to the right of the decimal point, but once we sum them together, we no longer need to use any digits to the right of the point. To keep track of the positions the digits of our final answer use, we need to match the positions of each decimal expression up to make sure we line up the decimal points as well.

We also need to keep track of the decimal point when we multiply decimals. Consider this example that uses some familiar fractions:

$\frac{1}{2}\times \frac{1}{2} = \frac{1}{4}$.

When we write it with decimal expressions, it becomes:

$0.5 \times 0.5 = 0.25$.

This multiplication equation looks simple enough. After all, we know $5\times 5 = 25$, and $25$ appears in our final answer. However, it may seem a bit strange that the decimal point remained in front of the $25$ for the final answer. We can explain this by treating our decimals as fractions with 10s in their denominators. Namely:

$0.5\times0.5 = \frac{5}{10} \times \frac{5}{10} = \frac{25}{100} = 0.25$

but it can be easier to carry this decimal-multiplication process out by thinking in terms of "moving the decimal place".

Finally, we need to understand how to divide decimal expressions. Dividing decimals works in a similar way to multiplying them, where you can understand quick calculations in terms of moving the decimal place.

Here is a simple example. Consider this division calculation (expressed using fractions):

$\frac{1}{2} \div \frac{1}{4} = \frac{4}{2} = 2$

In decimal notation, this equation becomes:

$0.5 \div 0.25 = 2$

This may look a little strange at first, but it might make it easier to think about in terms of money. You can consider the $0.5$ as $0.50$ or fifty cents and the $0.25$ as one quarter. There are two quarters in fifty cents. As with multiplication, when you divide decimal expressions, you can avoid converting them into fractions and follow the helpful rules for moving the decimal point.

Photo by pics_pd

We use decimal notation all the time when we make measurements, especially when using the metric system, where quantities are based on measures of ten. Let's return to your aunt's coffee shop for an example, where she is busy designing a new sign to display in the window.

Your aunt is trying to make sense of the measurements for this rectangle, which is expressed as 1.25 m by 0.75 m. She would like to paint her sign white and knows that one bottle of paint can cover a 50 cm by 50 cm = 2500 cm² area. How many bottles does your aunt need to paint the entire sign? (Give it a try!)

Probably your first step is to express all of your measurements in the same units. We know that one centimeter equals one-hundredth of a meter, or, in decimal form, that 1 cm = 0.01 m.

Next, we figure out that one bottle of paint will cover $0.5 \times 0.5 = 0.25$ m² units of area. The area of the sign is given by $1.25 \times 0.75 = 0.9375$ m², and so the last step in this process is to divide $0.935$ by $0.25$. After doing some work, we find that:

$0.935 \div 0.25 = 3.75$

This tells us that we will need $3 \frac{3}{4}$ bottles of paint. Assuming we can only purchase individual bottles, your aunt will need to buy a total of 4.