**1.4 Decimals**

Topics covered in this section are:

- Round decimals
- Add and subtract decimals
- Multiply and divide decimals
- Convert decimals, fractions, and percents
- Simplify expressions with square roots
- Identify integers, rational numbers, irrational numbers, and real numbers
- Locate fractions and decimals on the number line

**1.4.1 Round decimals**

Decimals are another way of writing fractions whose denominators are powers of ten.

$0.1 = \frac{1}{10}$ | is “one tenth” |

$0.01 = \frac{1}{100}$ | is “one hundreth” |

$0.001 = \frac{1}{1000}$ | is “one thousandth” |

$0.00001 = \frac{1}{10,000}$ | is “one ten-thousandth” |

Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. Figure 1.6 shows the names of the place values to the left and right of the decimal point.

When we work with decimals, it is often necessary to round the number to the nearest required place value. We summarize the steps for rounding a decimal here.

**HOW TO: Round decimals**.

- Locate the given place value and mark it with an arrow.
- Underline the digit to the right of the place value.
- Is the underlined digit greater than or equal to $5$?
- Yes: add $1$ to the digit in the given place value.
- No: do not change the digit in the given place value

- Rewrite the number, deleting all digits to the right of the rounding digit.

**Example 1**

Round $18.379$ to the nearest:

- hundreth
- tenth
- whole number

**Solution**

**Part 1.**

Locate the hundredths place with an arrow. | |

Underline the digit to the right of the given place value. | |

Because $9$ is greater than or equal to $5$, add $1$ to the $7$. | |

Rewrite the number, deleting all digits to the right of the rounding digit. | $18.38$ |

Notice that the deleted digits were NOT replaced with zeros. | So $18.379$ rounded to the nearest hundredth is $18.38$. |

**Part 2.**

Locate the tenths place with an arrow. | |

Underline the digit to the right of the given place value. | |

Because $7$ is greater than or equal to $5$, add $1$ to the $3$. | |

Rewrite the number, deleting all digits to the right of the rounding digit. | $18.4$ |

Notice that the deleted digits were NOT replaced with zeros. | So $18.379$ rounded to the nearest tenth is $18.4$. |

**Part 3.**

Locate the ones place with an arrow. | |

Underline the digit to the right of the given place value. | |

Because $3$ is not greater than or equal to $5$, do not add $1$ to the $8$. | |

Rewrite the number, deleting all digits to the right of the rounding digit. | $18$ |

Notice that the deleted digits were NOT replaced with zeros. | So $18.379$ rounded to the nearest whole number is $18$. |

**1.4.2 Add and Subtract Decimals**

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

**HOW TO: Add or subtract decimals.**

- Determine the sign of the sum or difference.
- Write the numbers so the decimal points line up vertically.
- Use zeros as placeholders, as needed.
- Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.
- Write the sum or difference with the appropriate sign.

**Example 2**

Add or subtract:

- $-23.5-41.38$
- $14.65-20$.

**Solution**

**Part 1.**

$-23.5-41.38$ | |

The difference will be negative. To subtract, we add the numerals. Write the numbers so the decimal points line up vertically. | $\begin{align*} 23&.5 \\ + \ 41&.38 \\ \hline \end{align*}$ |

Put $0$ as a placeholder after the $5$ in $23.5$. Remember, $\frac {5}{10}=\frac{50}{100}$ so $0.5=0.50$. | $\begin{align*} 23&.50 \\ + \ 41&.38 \\ \hline \end{align*}$ |

Add the numbers as if they were whole numbers. Then place the decimal point in the sum. | $\begin{align*} 23&.50 \\ + \ 41&.38 \\ \hline 64&.88\\ \end{align*}$ |

Write the result with the correct sign. | $-23.5-41.38=-64.88$ |

**Part 2.**

$14.65-20$ | |

The difference will be negative. To subtract, we subtract $14.65$ from $20$. | |

Write the numbers so the decimal points line up vertically. | $\begin{align*} 20& \\ – \ 14&.65 \\ \hline \end{align*}$ |

Remember, $20$ is a whole number, so place the decimal point after the $0$. | |

Put in zeros to the right as placeholders. | $\begin{align*} 20&.00 \\ – \ 14&.65 \\ \hline \end{align*}$ |

Subtract and place the decimal point in the answer. | $\begin{align*} 20&.00 \\ – \ 14&.65 \\ \hline 5&.35\\ \end{align*}$ |

Write the result with the correct sign. | $14.65-20=-5.35$ |

**1.4.3 Multiply and Divide Decimals**

When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. We multiply the numbers temporarily ignoring the decimal point and then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product. Finally, we write the product with the appropriate sign.

**HOW TO: Multiply decimals.**

- Determine the sign of the product.
- Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
- Place the decimal point. The number of decimal places in the product is the sum of

the number of decimal places in the factors. - Write the product with the appropriate sign.

**Example 3**

Multiply: $(-3.9)(4.075)$.

**Solution**

$(-3.9)(4.075)$ | |

The signs are different. | The product will be negative. |

Write in vertical format, lining up the numbers on the right. | $\begin{align*} 4.075& \\ \times \ 3.9& \\ \hline \end{align*}$ |

Multiply. | $\begin{align*} 4.075& \\ \times \ 3.9& \\ \hline 36675&\\ + 122250&\\ \hline 158925&\\ \end{align*}$ |

Add the number of decimal places in the factors $(1 + 3)$. Place the decimal point $4$ places from the right. | $\begin{align*} 4.075& \\ \times \ 3.9& \\ \hline 36675&\\ + 122250&\\ \hline 15.8925&\\ \end{align*}$ |

The signs are different, so the product is negative. | $(-3.9)(4.075)=-15.8925$ |

Often, especially in the sciences, you will multiply decimals by powers of $10$ ($10, 100, 1000,$ etc). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of $10$ to number of decimal places we move the decimal point to the right to get the product.

**HOW TO: Multiply a decimal by a power of ten.**

- Move the decimal point to the right the same number of places as the

number of zeros in the power of $10$. - Add zeros at the end of the number as needed.

**Example 4**

Multiply $5.63$ by:

- $10$
- $100$
- $1000$

**Solution**

By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.

**Part 1.**

$5.63(10)$ | |

There is $1$ zero in $10$, so move the decimal point $1$ place to the right. | |

$56.3$ |

**Part 2.**

$5.63(100)$ | |

There is $2$ zeros in $100$, so move the decimal point $2$ places to the right. | |

$563$ |

**Part 3.**

$5.63(1,000)$ | |

There is $3$ zeros in $1,000$, so move the decimal point $3$ places to the right. | |

A zero must be added to the end. | $5,630$ |

Just as with multiplication, division of signed decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed and the sign of the quotient. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.

We review the notation and vocabulary for division:

We’ll write the steps to take when dividing decimals for easy reference.

**HOW TO: Divide decimals.**

- Determine the sign of the quotient.
- Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places—adding zeros as needed.
- Divide. Place the decimal point in the quotient above the decimal point in the dividend.
- Write the quotient with the appropriate sign.

**Example 5**

Divide: $-25.65 \div (-0.06)$.

**Solution**

Remember, you can “move” the decimals in the divisor and dividend because of the Equivalent Fractions Property.

$-25.65 \div (-0.06)$ | |

The signs are the same. | The quotient is positive. |

Make the divisor a whole number by “moving” the decimal point all the way to the right. | |

“Move” the decimal point in the dividend the same number of places. | |

Divide. Place the decimal point in the quotient above the decimal point in the dividend. | |

Write the quotient with the appropriate sign. | $-25.65 \div (-0.06)=427.5$ |

**1.4.4 Convert Decimals**, **Fractions and Percents**

In our work, it is often necessary to change the form of a number. We may have to change fractions to decimals or decimals to percent.

We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal $0.03$, the $3$ is in the hundredths place, so $100$ is the denominator of the fraction equivalent to $0.03$.

$3= \frac{3}{100}$

The steps to take to convert a decimal to a fraction are summarized in the procedure box.

**HOW TO: Convert a decimal to a proper fraction and a fraction to a decimal.**

- To convert a decimal to a proper fraction, determine the place value of the final digit.
- Write the fraction.
- numerator—the “numbers” to the right of the decimal point
- denominator—the place value corresponding to the final digit

- To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

**Example 6**

Write:

- $0.374$ as a fraction
- $- \frac{5}{8}$ as a decimal.

**Solution**

**Part 1.**

$0.374$ | |

Determine the place value of the final digit. | |

Write the fraction for $0.374$: The numerator is $374$. The denominator is $1,000$. | $\frac{374}{1000}$ |

Simplify the fraction. | $\frac {2 \cdot 187} {2 \cdot 500}$ |

Divide out the common factors. | $\frac {187}{500}$ |

so, $0.374=\frac{187}{500}$ |

**Part 2.**

Since a fraction bar means division, we begin by writing the fraction $\frac{5}{8}$ as $8 \overline {)5}$. Now divide.

A **percent** is a ratio whose denominator is $100$. Percent means per hundred. We use the percent symbol, %, to show percent. Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per $100$, so the denominator of the fraction is $100$. We then change the fraction to a decimal by dividing the numerator by the denominator. After doing this many times, you may see the pattern.

*To convert a percent number to a decimal number, we move the decimal point two places to the left.*

To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is $100$, it is easy to change that fraction to a percent. After many conversions, you may recognize the pattern.

*To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.*

**HOW TO: Convert a percent to a decimal and a decimal to a percent.**

- To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.
- To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.

**Example 7**

Convert each:

- a percent to a decimal: $62%$, $135%$, and $13.7%$.
- a decimal to a percent: $0.51$, $1.25$, and $0.093$.

**Solution**

**Part 1.**

Move the decimal point two places to the left. |

**Part 2.**

Move the decimal point two places to the right. |

**1.4.5 Simplify Expressions with Square Roots**

Remember that when a number $n$ is multiplied by itself, we write $n^{2}$ and read it “$n$ squared.” The result is called the **square of a number** $n$. For example, $8^{2}$ is read “$8$ squared” and $64$ is called the *square* of $8$. Similarly, $121$ is the square of $11$ because $11^{2}$ is $121$. It will be helpful to learn to recognize the perfect square numbers.

**SQUARE OF A NUMBER**

If $n^{2}=m$, then $m$ is the **square** of $n$.

What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

$(-3)^{2}=9$ | $(-8)^{2}=64$ | $(-11)^{2}=121$ | $(-15)^{2}=225$ |

Because $10^{2}=100$, we say $100$ is the square of $10$. We also say that $10$ is a *square root* of $100$. A number whose square is $m$ is called a **square root of a number** $m$.

**SQUARE ROOT OF A NUMBER**

If $n^{2}=m$, then $n$ is a **square root **of $m$.

Notice $(10)^{-2}=100$ also, so $-10$ is also a square root of $100$. Therefore, both $10$ and $-10$ are square roots of $100$. So, every positive number has two square roots—one positive and one negative. The radical sign, $\sqrt{m}$, denotes the positive square root. The positive square root is called the **principal square root**. When we use the radical sign that always means we want the principal square root.

**SQUARE ROOT NOTATION**

$\sqrt{m}$ is read “the square root of $m$.”

If $m=n^{2}$, then $\sqrt{m}=n$, for $n≥0$.

The square root of $m$, $\sqrt{m}$, is the positive number whose square is $m$.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write $\sqrt{100}=10$. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $-\sqrt{100}=-10$. We read $-\sqrt{100}$ as “the opposite of the principal square root of $10$.”

**Example 8**

Simplify:

- $\sqrt{25}$
- $\sqrt{121}$
- $-\sqrt{144}$

**Solution**

**Part 1.**

$\sqrt{25}$ | |

Since $5^{2}=25$ | $5$ |

**Part 2.**

$\sqrt{121}$ | |

Since $11^{2}=121$ | $11$ |

**Part 3.**

$-\sqrt{144}$ | |

The negative is in front of the radical sign. | $-12$ |

**1.4.6 Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers**

We have already described numbers as *counting numbers*, *whole numbers*, and *integers*. What is the difference between these types of numbers? Difference could be confused with subtraction. How about asking how we distinguish between these types of numbers?

Counting Numbers | $1,2,3,4,…$ |

Whole Numbers | $0,1,2,3,4,….$ |

Integers | $…,-3,-2,-1,0,1,2,3,4,…$ |

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A **rational number** is a number that can be written as a ratio of two integers.

In general, any decimal that ends after a number of digits (such as $7.3$ or $-1.2684$) is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction. The decimal for $\frac{1}{3}$ is the number $0.\overline 3$. The bar over the $3$ indicates that the number $3$ repeats infinitely. Continuously has an important meaning in calculus. The number(s) under the bar is called the repeating block and it repeats continuously.

Since all integers can be written as a fraction whose denominator is $1$, the integers (and so also the counting and whole numbers are rational numbers).

*Every rational number can be written both as a ratio of integers* $pq$, *where *$p$* and *$q$* are integers and *$q≠0$, *and as a decimal that stops or repeats.*

**RATIONAL NUMBER**

A **rational number** is a number of the form $\frac{p}{q}$ where $p$* *and $q$ are integers and $q≠0$. Its decimal form stops or repeats.

Are there any decimals that do not stop or repeat? Yes! The number $π$ (the Greek letter *pi*, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat. We use three dots (…) to indicate the decimal does not stop or repeat.

$π=3.141592654…$

The square root of a number that is not a perfect square is a decimal that does not stop or repeat.

A numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an **irrational number**.

**IRRATIONAL NUMBER**

An **irrational number** is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

**RATIONAL OR IRRATIONAL**

If the decimal form of a number

*repeats or stops*, the number is a**rational number**.*does not repeat and does not stop*, the number is an**irrational number.**

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of **real number****s**.

**REAL NUMBER**

A **real number** is a number that is either rational or irrational.

Later in this course we will introduce numbers beyond the real numbers. Figure 1.7 illustrates how the number sets we’ve used so far fit together.

Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be? Can we simplify $-\sqrt{25}$? Is there a number whose square is $-25$?

$( \ )^{2}=−25?$

None of the numbers that we have dealt with so far has a square that is $-25$. Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to $\sqrt{-25}$. The square root of a negative number is not a real number.

**Example 9**

Given the numbers $-7%$,$\frac{14}{8}$, $8$, $\sqrt{5}$, $5.9$, $-\sqrt{64}$, list the:

- whole numbers
- integers
- rational numbers
- irrational numbers
- real numbers

**Solution**

- Remember, the whole numbers are $0,1,2,3,…$ so $8$ is the only whole number given.
- The integers are the whole numbers and their opposites (which includes $0$). So the whole number $8$ is an integer, and $-7$ is the opposite of a whole number so it is an integer, too. Also, notice that $64$ is the square of $8$ so $-\sqrt{64}=-8$. So the integers are $-7$, $8$, and $-\sqrt{64}$.
- Since all integers are rational, then $-7$, $8$, and $-\sqrt{64}$ are rational. Rational numbers also include fractions and decimals that repeat or stop, so $\frac{14}{5}$ and $5.9$ are rational. So the list of rational numbers is $-7$, $\frac{14}{5}$, $8$, $5.9$, and $-\sqrt{64}$.
- Remember that $5$ is not a perfect square, so $\sqrt{5}$ is irrational.
- All the numbers listed are real numbers.

**1.4.7 Locate Fractions and Decimals on the Number Line**

We now want to include fractions and decimals on the number line. Let’s start with fractions and locate $\frac{1}{5}$,$-\frac{4}{5}$,$3$,$\frac{7}{4}$,$−\frac{9}{2}$,$−5$ and $\frac{8}{3}$ on the number line.

We’ll start with the whole numbers $3$ and $−5$ because they are the easiest to plot. See Figure 1.8.

The proper fractions listed are $\frac{1}{5}$ and $−\frac{4}{5}$. We know the proper fraction $\frac{1}{5}$ has value less than one and so would be located between $0$ and $1$. The denominator is $5$, so we divide the unit from $0$ to $1$ into $5$ equal parts $\frac{1}{5}$,$\frac{2}{5}$,$\frac{3}{5}$,$\frac{4}{5}$. We plot $\frac{1}{5}$.

Similarly, $−\frac{4}{5}$ is between $0$ and $−1$. After dividing the unit into $5$ equal parts we plot $−\frac{4}{5}$.

Finally, look at the improper fractions $\frac{7}{4}$,$−\frac{9}{2}$, and $\frac{8}{3}$. Locating these points may be easier if you change each of them to a mixed number.

$\frac{7}{4}=1\frac{3}{4}$ | $-\frac{9}{2}=-4\frac{1}{2}$ | $\frac{8}{3}=2\frac{2}{3}$ |

Figure 1.8 shows the number line with all the points plotted.

**Example 10**

Locate and label the following on a number line:

- $4$
- $\frac{3}{4}$
- $-\frac{1}{4}$
- $-3$
- $\frac{6}{5}$
- $-\frac{5}{2}$
- $\frac{7}{3}$

**Solution**

Locate and plot the integers $4$, $-3$.

Locate the proper fraction $\frac{3}{4}$ first. The fraction $\frac{3}{4}$ is between $0$ and $1$. Divide the distance between $0$ and $1$ into four equal parts, then we plot $\frac{3}{4}$. Similarly plot $-\frac{1}{4}$.

Now locate improper fractions $\frac{6}{5}$, $-\frac{5}{2}$, and $\frac{7}{3}$. It is easier to plot them if we convert them to mixed numbers and then plot them as described above: $\frac{6}{5}=1\frac{1}{5}$, $-\frac{5}{2}=-2\frac{1}{2}$, $\frac{7}{3}=2\frac{1}{3}$.

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

**Example 11**

Locate on the number line:

- $0.4$
- $-0.74$

**Solution**

**Part 1. **

The decimal number $0.4$ is equivalent to $\frac{4}{10}$, so $0.4$ is located between $0$ and $1$. On a number line, divide the interval between $0$ and $1$ into $10$ equal parts. Now label the parts $0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0$. We write $0$ as $0.0$ and $1$ as $1.0$, so that the numbers are consistently in tenths. Finally, mark $0.4$ on the number line.

**Part 2.**

The decimal number $-0.74$ is equivalent to $-\frac{74}{100}$, so it is located between $0$ and $1$. On a number line, mark off and label the hundredths in the interval between $0$ and $-1$.

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*Marecek, L., & Mathis, A. H. (2020). Decimals. In Intermediate Algebra 2e. OpenStax. https://openstax.org/books/intermediate-algebra-2e/pages/1-4-decimals. License: CC BY 4.0. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction*