1.3 Working With Decimals

Learning Objectives

By the end of this section, you will be able to:

• interpret decimal notation and round decimals correctly
• add, subtract, and multiply decimals
• convert decimals to and from common fractions
• use decimals to calculate percents

Decimals are just another way we can express real numbers that are in between the whole numbers, and in fact they are basically shorthand for fractions whose denominators are powers of 10. See what I mean below...

Decimal	Fraction Equivalent	In Words
$0.1$	$\frac{1}{10}$	"one-tenth"
$0.01$	$\frac{1}{100}$	"one-hundredth"
$0.001$	$\frac{1}{1000}$	"one-thousandth"

When we learn about whole numbers, we talk about place values in terms of powers of 10 as well. For a number like 1,357 we know that the 5 actually represents 5 tens, the 3 actually represents 3 hundreds, etc. Here's an example:

So you see we can think of the number in the image as listing out a bunch of components,

$$30000 + 1000 + 400 + 20 + 6 + \frac{2}{10} + \frac{8}{100} + \frac{7}{1000} + \frac{9}{10000} = 31426 + 0.2 + 0.08 + 0.007 + 0.0009 = 31426.2879 \notag$$

which can all be written in a single decimal representation of the value.

When I calculate final course grades for students, I generally round each number to the nearest integer, meaning if you got an 87.6, I just consider it an 88. In that case, I'm rounding to the ones place, but you can round to any place value you want. For example, when performing calculations about money, we round to two decimal places, because we only work with dollars and cents.

Rounding to a desired place value: If you want to round to the $n^{th}$ place value, look at the digit directly to the right of that position. If it is less than 5, keep the $n^{th}$ place digit as is and rewrite the number, deleting all digits that come after. If it is greater than or equal to 5, add 1 to the $n^{th}$ place digit before deleting all digits that come after.

Exercise $\PageIndex{1}$

Round $201.765832904$ to the...

1. nearest integer
2. nearest hundred
3. nearest tenth
4. nearest hundredth
5. nearest ten-thousandth (four decimal places)
6. sixth decimal place

Answer

1. Find the ones digit (it's 1) and look to the right. We see a 7 next to it, which is bigger than 5, so we round up and give the answer as 202.
2. Look at the hundreds digit (it's 2) and look to the right. We see a 0, which is less than 5, so we keep the 2 and write 200.
3. Look at the tenths place (it's a 7). To the right, there is a 6, which is bigger than 5, so we round up to 201.8 as our answer.
4. The hundredths place is the 6, and next to it we see a 5, so we round up and give 201.77 as our answer.
5. The fourth decimal place is the 8, and next to it we see a 3, so we give 201.7658 as our answer.
6. The sixth decimal place is the 2, and next to it is a 9, so we round up to 201.765833 as our answer.

Decimals are great, because in real life problems, the numbers don't generally stay cute and clean and whole. Decimals also, in my opinion, are way better to work with than mixed numbers, while giving a similar sort of intuition about the size of a quantity. However, I don't generally expect students to work with decimals much without using a calculator. I want you to know how to do reasonably simple arithmetic in your head, so we'll go over it briefly, but in your science classes you'll use technology to perform calculations with decimals.

Arithmetic With Decimals

To add or subtract two decimals,

1. Determine the sign of the answer (positive or negative),
2. Write them vertically with the decimal points aligned, using placeholder 0s if necessary, then...
3. Add or subtract down columns as usual.

For example, to add $21.5 + 0.25$, line them up by tacking a 0 on the end of $21.5$ (this doesn't change its value as a number), and add.

To subtract $17.503 - 45.001$, we will have a negative answer since we are subtracting a larger number.

To multiply two decimals,

1. Determine the sign of the answer (positive or negative),
2. Multiply them as if they were whole numbers, ignoring the decimal points for the moment, then...
3. The new decimal point position is the sum of the decimal places in the two factors.

This sounds weird, but all I want you to take away from this is common sense like the following examples:

Example $\PageIndex{1}$

Multiply.

1. $2(3.01)$
2. $0.5 (0.5)$
3. $1.5(30)$

Solution

1. I just want to double $3.01$. Intuitively, I double the whole part ( $3 \rightarrow 6$) and double the decimal part ( $0.01 \rightarrow 0.02$), and then add them back together to get $6.02$. I could also ignore the decimal point and say $2 \cdot 301 = 602$, and then since there are two decimal places in $3.01$ and none in $2$, I write the point two places in to get the $6.02$.

2. Ignoring the decimal points, $5 \cdot 5 = 25$. Both decimals have a single decimal place, so the new point goes a total of two decimal places in to yield $0.25$. Or you can realize that $0.5 = \frac{1}{2}$, so I'm looking for "half of one-half," which we know is $\frac{1}{4} = 0.25$.

3. I can think of this as wanting "one-and-a-half of a 30." So $30 + 15 = 45$ is the right answer. Or, I can ignore the decimal point, multiply $15 \cdot 30 = 450$, and then put the new point one place in to get $45.0$.

There is also a special trick you can use when multiplying or dividing by a power of 10 that you should be aware of.

When multiplying a decimal by a power of 10, $10^k$ for some whole number $k$, the result will be the original decimal with the point moved $k$ times to the right, with zeros inserted in as needed!

Example $\PageIndex{2}$

Compute.

1. $(1.23)(10)$
2. $(1.23)(100)$
3. $(1.23)(1000)$

Solution

1. We multiply $123 \cdot 10 = 1230$ and see a total of two decimal places from the factor $(1.23)$, so the final answer is $12.30 = 12.3$. We could have just moved the decimal point to the right by one jump!

2. We get $123.00$, which is the same as if we moved the point two places right to get $123$. The factor $(100)$ is $10^2$ so two jumps makes sense!

3. The factor $(1000)$ is $10^3$, so we move the decimal place three jumps to the right, filling in with a 0 after the 3 to get $1230$. The long method will give you $1230.00$ which is the same!

When dividing a decimal by a $10^k$, the result will be the same number with the decimal point moved $k$ times to the left, with zeros inserted in as needed.

Example $\PageIndex{3}$

Compute.

1. $\frac{1.23}{10}$

2. $\frac{1.23}{100}$

3. $\frac{1.23}{1000}$

Solution

1. You don't really have to know anything about dividing here actually, if you remember that $0.1 = \frac{1}{10}$. That means $\frac{1.23}{10} = 1.23\cdot \frac{1}{10} = (1.23)(0.1)$ can be seen as a multiplication. In that case, we notice there are two decimal places from the first factor and one from the second, for a total of three. We take the product $123 \cdot 1 = 123$ and put the point three places in to get $0.123$, which is the same as if we moved the point one position to the left from the beginning! Let's do that instead for the other two, now that we've convinced ourselves the trick works.

2. Again, $100 = 10^2$, so we move the point two places to the left this time, to get $0.0123$.

3. Here we need to move the point three places to the left, to get $0.00123$ as the answer.

Converting Between Fractions and Decimals

For your health and happiness in daily life and future math and science courses, you should have a grasp of the decimal versions of commonly used fractions. Your average welder (I'm told by my brother, who builds rocket pads) knows all of his powers-of-2 fractions by heart: $\frac{1}{2} = 0.5, \frac{1}{4} = 0.25, \frac{1}{8} = 0.125,$ etc. So have a gander at the list below, and then try to answer the exercise without looking!

Fraction	Decimal
$\frac{1}{2}$	$0.5$
$\frac{1}{3}$	$0.\overline{3}$, which means the 3s go on forever like $0.33333...$
$\frac{2}{3}$	$0.\overline{6}$, which due to rounding rules rounds to things like $0.6667$
$\frac{1}{4}$	$0.25$, like how 25 cents is a quarter of a dollar
$\frac{3}{4}$	$0.75$
$\frac{1}{5}$	$0.2$, which you can remember as $0.20$ because 20 is one-fifth of a 100
$\frac{1}{8}$	$0.125$
$\frac{3}{2}, \frac{5}{2}, \frac{7}{2}, ...$	$1.5 , 2.5, 3.5, ...$, which are just like finding mixed numbers from improper fractions

Exercise $\PageIndex{2}$

Translate from decimal to fraction or vice versa, as appropriate.

1. $1.5 =$
2. $\frac{7}{2} =$
3. $0.75 =$
4. $\frac{2}{100}$
5. $0.2 =$
6. $0.4 =$
7. $\frac{1}{3}$

Answer

1. $\frac{3}{2}$
2. $3.5$
3. $\frac{3}{4}$
4. $0.02$ because that's "two one-hundredths" if you say it in words.
5. $\frac{1}{5}$ or if you forget, think "two tenths," which is the fractions $\frac{2}{10}$, and then simplify!
6. $\frac{2}{5}$ I know, this one isn't on the list, but notice that it's just two copies of $0.2$ and proceed...
7. $0.3333... = 0.\overline{3}$

While we're here, let's expand a bit on that "0.3 repeating" situation for $\frac{1}{3}$. It turns out that some rational numbers have a decimal version that goes on forever, but it follows a pattern. Some examples:

$$\frac{1}{3} = 0.333333..., \quad \frac{1}{6} = 0.16666666..., \quad \frac{1}{7} = 0.142857142857142857…, \quad \frac{1}{11} = 0.09090909... \notag$$

This fun fact works the other direction too, meaning that if you have a repeating decimal, it can be written as a ratio of two integers! If you want to learn how to translate back and forth, check out the end of the exercises section.

If a decimal never stops and there is no repeating pattern to its digits continuing forever, that's an irrational number.

Decimals and Percents

You paid for a math course, but you're getting some Latin today for free. The ending "-cent" comes from the Latin centum, which means a hundred, so the word "percent" means "out of a hundred." If I say 85% of my students like my jokes, that means out of every hundred students I teach, 85 of them like my jokes. This proportion can also be written as a fraction, $\frac{85}{100}$, or its equivalent decimal, $0.85$. Whenever you want to compute a percentage, you will want to convert it to a decimal by following this pattern and moving the decimal point two places to the left. (Where was the decimal point in 85? It was hiding, right there in 85.0...)

Here are the basic types of percentage computation you might be expected to know how to do. You would probably use a calculator for this type of problem.

Example $\PageIndex{4}$

I had 32 students last semester, and I know 85% of my students like my jokes. How many students liked my jokes last semester?

Solution

I need to answer the question, "What is 85% of 32?" We already know how to translate "85%" to math, and the word "of" is usually code for multiplication. I calculate $0.85 \cdot 32 = 27.2$ with my calculator. I can't have 0.2 of a student (without a mess, anyway) so I round down to 27 students.

Example $\PageIndex{5}$

I took a poll of my 32 students, and actually only 22 thought my jokes were funny. :( What percentage of my students like my jokes?

Solution

I know that $\frac{22}{32}$ of my students liked my jokes. My calculator says the decimal equivalent of that is $0.6875$. To find the percent, I just move the decimal point two places to the right to get 68.75%. Yikes.

Okay, that's the gist on what you should be able to do with decimals! Take a stab at the exercises in the next section.