### Unit 5: Decimals

In this unit, we study decimals, which are simply another way to express fractions. We use this way of notating fractions (including improper ones and mixed numbers), most notably when dealing with money or currencies.

For example, if the price of your aunt's amazing vanilla latte is $3.50, we understand this to mean it costs three whole dollar bills and fifty cents. The period or decimal point, as mathematicians call it, separates whole numbers from the fractions in this expression – it serves this purpose in this price and in all other decimal expressions. Our whole numbers (in this case, **3**) sit to the left of the decimal point, and fractional quantities sit to the right of the decimal point.

Since the digit **5** appears immediately to the right of the decimal, it really means **five-tenths**. The zero means **zero hundredths**. Of course, we can reduce the fraction five tenths as

Altogether this means we can translate $3.50 literally or mathematically to **three whole dollars and one-half of a dollar**. That is

Because decimals are just a different way to express factions, they are just as useful and omnipresent. In this unit, we explore how to convert decimals into explicit fractions and how to add, subtract, multiply, and divide them.

**Completing this unit should take you approximately 6 hours.**

Upon successful completion of this unit, you will be able to:

- recognize the place value of decimals;
- apply rounding and estimation concepts with decimals;
- convert between fractions and decimals;
- perform operations with decimals; and
- solve real-world and mathematical problems with decimals.

### 5.1: Decimal Place Value

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:Each of these place values can be expressed using powers of 10 since , , and . In other words:

If we add a tiny amount to the number 135, say , we obtain the mixed number . 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 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:

A decimal point lets us rewrite this expression in a way that

**hides**all of these powers of 10 and their various exponents: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: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.

Watch this video for more examples with other numbers.

Complete this assessment to practice your understanding of decimals and check your answers. If you need more practice, do Problem Sets 2 and 3.

### 5.2: Rounding Decimals

Here is an experiment you have probably encountered or explored: take out a simple calculator and use it to divide or .

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:

We use an over-line bar to indicate the string repeats. The ones in the decimal representation 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 on your psychology exam, you would hope your generous professor

**rounds up to the nearest whole number**so you get a score of . 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 with the decimal , but is a better approximation. An even better approximation is .

Watch this video for examples of how to round decimals to specific decimal places.

Then read this text up to Sample Set A. The second paragraph gives a set of rules for determining how to round a decimal. The first two worked examples under Sample Set A show step-by-step directions for rounding numbers. Complete the practice problems and check your answers.

### 5.3: Converting Between Decimals and Fractions

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 , tell us a lot, and we can apply them to many situations.

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

Regardless of the flavor you prefer for your non-whole numbers, watch this video to learn how to switch between them. The narrator shows examples of how to convert decimals to fractions.

Read this text and complete the practice problems. Be sure to check your answers.

Complete these problems to practice converting decimals and fractions and check your answers. If you need more practice, do Problem Sets 2 and 3.

### 5.4: Converting Repeating Decimals to Fractions

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:

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.Watch this video whose main point is this: every time you multiply a decimal expression by 10, simply move the decimal point one digit to the right.

Now let's consider this never-ending, repeating decimal:

We have used the letter to give this decimal number a name that is shorter and easier to say! Now, what happens when we multiply this number by 10? (Try it yourself before reading ahead!)

We obtain the new decimal:

where we separated the "whole number" part after the second equal sign and the

**fractional part**appearing to the right of the decimal point. In fact, that separated decimal portion is exactly our original number again! This gives us a straightforward equation to manipulate (one that only involves s and whole numbers):where we subtracted the number from both sides of the equation in the third line and then divided both sides of the equation by in the last one.

As long as you have a repeating decimal, this process can be used to convert it into a fraction – no matter how

**long**it takes the decimal to repeat. For instance, the repeating decimal:repeats a chunk or string of three digits, and so we can multiply it by to discover its secret fraction name. Proceeding just as we did above for , one finds:

A similar process can also be carried out for decimal expressions whose "repeating part" takes a while to "kick in". For example, we can manipulate the expression:

to discover:

which we can rearrange and solve to find:

Watch these videos to explore more examples. Many involve repeating decimals.

- Now, read this section on repeating decimals. Pay attention to the notation using a line above the repeating portion of the decimal. Complete the practice problems and check your answers.
Let's go on a brief optional tangent and see if you can identify a decimal situation we have failed to address. We have discussed terminating decimals (decimal expressions that

**stop**at some finite location beyond the decimal point) and repeating decimals (decimal expressions that do not**stop**at some finite location beyond the decimal point. Rather, they begin repeating a fixed string of digits at some finite point beyond the decimal).But what about a third possibility? What happens when you have an unending decimal expression that is not repeating?

For example, consider an unending decimal expression whose first several digits are:

Because there is no evident pattern of repetition, we cannot convert this expression into a fraction. There are also numbers, such as the

**Champernowne Constant**, that have a clear and never-ending pattern, but we cannot express them as a fraction either:We cannot even convert even a seemingly simplistic but non-ending decimal expansion such as this one into a fraction:

What is going on? What types of numbers do they represent If we cannot convert these

**strange decimal expressions**into fractions? These expressions represent (or approximate) a new kind of number we call an**irrational number**. We call them irrational precisely because we cannot express them as a ratio or fraction of whole numbers.In other words, both decimal notation and fraction notation fail to help explain these quantities. But mathematicians have derived these numbers because they help explain other, deeper mathematical operations which are at play. For example, the first irrational expression above approximates the famous number , which we use to measure the lengths of circles. We use the expression to measure certain right triangles. But we cannot express it as a ratio of whole numbers either.

Now that you have a solid understanding of decimals, it is time to

**pull the rug out**from under you. As it turns out, every decimal expression**goes on forever**. In fact, every whole number goes on forever!Let's start with the whole number one. We can represent this quantity in decimal notation as one, as 1.0, 1.00, or 1.000. An infinite string of zeros continues to the right of the decimal point; we have just agreed to rarely, if ever, write them! That is:

It is also true that the whole number one has an infinite string of zeros that continue on the left before the 1 itself! Again, we have merely agreed to rarely (if ever) write these zeros.

This means that we have:

Of course, this also applies to other decimal expressions, such as these:

Since including all of these zeros is cumbersome and distracting, we will not use this notation. We encourage you to similarly avoid it, but it is worth mentioning for two reasons. First, it helps demystify ongoing decimal expressions - they have been with us all along! These observations also help explain what decimal notation, and our usual notation for whole numbers, actually means.

### 5.5: Adding and Subtracting Decimal Expressions

It can be simple to notate adding whole numbers, such as . However, while understanding the computation 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:

When we write this equation using decimals, we have:

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.

Watch this video for a worked example of an addition problem with decimal expressions. Pay attention to how to line up the decimal point when adding decimals.

Complete this assessment to practice adding hundredths. Be sure to check your answers.

- Watch these videos to learn how to subtract with decimal numbers.
- Complete this assessment to practice subtracting hundredths. Be sure to check your answers.
- Read this text for some more examples. Pay attention to the "How To" box for an outline of how to add and subtract decimals. Complete the practice problems and check your answers.

### 5.6: Multiplying Decimal Expressions

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

When we write it with decimal expressions, it becomes:

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

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

Watch this video for a worked example of how to multiply decimal numbers. Pay attention to expressions like "move the decimal point".

Complete this assessment to practice multiplying decimals. Be sure to check your answers.

Read this text. Pay attention to the "How To" overview of the steps needed to multiply decimals. Complete the practice problems and check your answers.

### 5.7: Dividing Decimal Expressions

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):

In decimal notation, this equation becomes:

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 as or fifty cents and the 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.

Watch these videos for worked examples of how to divide with decimal numbers.

Read this text. Pay attention to the "How To" boxes which give brief step-by-step summaries of how to divide decimals. Complete the practice problems and check your answers.

### 5.8: Word Problems Using Decimals

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 cm

^{2}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 m

^{2}units of area. The area of the sign is given by m^{2}, and so the last step in this process is to divide by . After doing some work, we find that:This tells us that we will need bottles of paint. Assuming we can only purchase individual bottles, your aunt will need to buy a total of 4.

Decimals also appear in many other places in the real world, especially in problems involving money. Watch this video for more word problems that use decimals.

Complete this assessment to practice calculating applications with decimals. Be sure to check your answers.