# Decimals

### Learning Outcomes

- Name a decimal number
- Given the name of a decimal number, write it in decimal notation
- Convert a decimal to a fraction or mixed number

Counting number | Name |
---|---|

[latex]1[/latex] | One |

[latex]10=10[/latex] | Ten |

[latex]10\cdot 10=100[/latex] | One hundred |

[latex]10\cdot 10\cdot 10=1000[/latex] | One thousand |

[latex]10\cdot 10\cdot 10\cdot 10=10,000[/latex] | Ten thousand |

Decimal | Fraction | Name |
---|---|---|

[latex]0.1[/latex] | [latex]{\Large\frac{1}{10}}[/latex] | One tenth |

[latex]0.01[/latex] | [latex]{\Large\frac{1}{100}}[/latex] | One hundredth |

[latex]0.001[/latex] | [latex]{\Large\frac{1}{1,000}}[/latex] | One thousandth |

[latex]0.0001[/latex] | [latex]{\Large\frac{1}{10,000}}[/latex] | One ten-thousandth |

*ten thousand*. Likewise, the names of the decimal places correspond to their fraction values. Notice how the place value names in the first table relate to the names of the fractions from the second table. This chart illustrates place values to the left and right of the decimal point. Notice two important facts shown in the tables.

- The "th" at the end of the name means the number is a fraction. "One thousand" is a number larger than one, but "one thousandth" is a number smaller than one.
- The tenths place is the first place to the right of the decimal, but the tens place is two places to the left of the decimal.

*five dollars and three cents*. Naming decimals (those that don’t represent money) is done in a similar way. We read the number [latex]5.03[/latex] as

*five and three hundredths*. We sometimes need to translate a number written in decimal notation into words. As shown in the image below, we write the amount on a check in both words and numbers. When we write a check, we write the amount as a decimal number as well as in words. The bank looks at the check to make sure both numbers match. This helps prevent errors.

Let’s try naming a decimal, such as [latex]15.68[/latex]. | |

We start by naming the number to the left of the decimal. | fifteen______ |

We use the word "and" to indicate the decimal point. | fifteen and_____ |

Then we name the number to the right of the decimal point as if it were a whole number. | fifteen and sixty-eight_____ |

Last, name the decimal place of the last digit. | fifteen and sixty-eight hundredths |

*fifteen and sixty-eight hundredths*.

### Name a decimal number.

- Name the number to the left of the decimal point.
- Write "and" for the decimal point.
- Name the "number" part to the right of the decimal point as if it were a whole number.
- Name the decimal place of the last digit.

### Example

Name each decimal: 1. [latex]4.3[/latex] 2. [latex]2.45[/latex] 3. [latex]0.009[/latex] 4. [latex]-15.571[/latex] Solution1. | |

[latex]4.3[/latex] | |

Name the number to the left of the decimal point. | four_____ |

Write "and" for the decimal point. | four and_____ |

Name the number to the right of the decimal point as if it were a whole number. | four and three_____ |

Name the decimal place of the last digit. | four and three tenths |

2. | |

[latex]2.45[/latex] | |

Name the number to the left of the decimal point. | two_____ |

Write "and" for the decimal point. | two and_____ |

Name the number to the right of the decimal point as if it were a whole number. | two and forty-five_____ |

Name the decimal place of the last digit. | two and forty-five hundredths |

3. | |

[latex]0.009[/latex] | |

Name the number to the left of the decimal point. | Zero is the number to the left of the decimal; it is not included in the name. |

Name the number to the right of the decimal point as if it were a whole number. | nine_____ |

Name the decimal place of the last digit. | nine thousandths |

4. | |

[latex]-15.571[/latex] | |

Name the number to the left of the decimal point. | negative fifteen |

Write "and" for the decimal point. | negative fifteen and_____ |

Name the number to the right of the decimal point as if it were a whole number. | negative fifteen and five hundred seventy-one_____ |

Name the decimal place of the last digit. | negative fifteen and five hundred seventy-one thousandths |

### TRY It

[ohm_question]146224[/ohm_question] [ohm_question]146225[/ohm_question] [ohm_question]146568[/ohm_question] [ohm_question]146569[/ohm_question]### example

1. Write the number six and seventeen hundredths:six and seventeen hundredths | |

The word and tells us to place a decimal point. |
___.___ |

The word before and is the whole number; write it to the left of the decimal point. |
[latex]6[/latex]._____ |

The decimal part is seventeen hundredths. Mark two places to the right of the decimal point for hundredths. | [latex]6[/latex]._ _ |

Write the numerals for seventeen in the places marked. | [latex]6.17[/latex] |

Answer: Solution

fourteen and thirty-seven hundredths | |

Place a decimal point under the word ‘and’. | ______. _________ |

Translate the words before ‘and’ into the whole number and place it to the left of the decimal point. | [latex]14[/latex]. _________ |

Mark two places to the right of the decimal point for "hundredths". | [latex]14[/latex].__ __ |

Translate the words after "and" and write the number to the right of the decimal point. | [latex]14.37[/latex] |

Fourteen and thirty-seven hundredths is written [latex]14.37[/latex]. |

### try it

[ohm_question]146570[/ohm_question]### Write a decimal number from its name.

- Look for the word "and"—it locates the decimal point.
- Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
- Place a decimal point under the word "and." Translate the words before "and" into the whole number and place it to the left of the decimal point.
- If there is no "and," write a "0" with a decimal point to its right.

- Translate the words after "and" into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
- Fill in zeros for place holders as needed.

### example

Write twenty-four thousandths as a decimal.Answer: Solution

twenty-four thousandths | |

Look for the word "and". | There is no "and" so start with 0 [latex]0[/latex]. |

To the right of the decimal point, put three decimal places for thousandths. | |

Write the number [latex]24[/latex] with the [latex]4[/latex] in the thousandths place. | |

Put zeros as placeholders in the remaining decimal places. | [latex]0.024[/latex] |

So, twenty-four thousandths is written [latex]0.024[/latex] |

### try it

[ohm_question]146571[/ohm_question] [ohm_question]146572[/ohm_question][latex]\begin{array}{cccc}5=5.0\hfill & & & -2=-2.0\hfill \\ 5=5.00\hfill & & & -2=-2.00\hfill \\ 5=5.000\hfill & & & -2=-2.000\hfill \end{array}[/latex] and so on [latex]\dots[/latex]

## Converting decimals to fractions or mixed numbers

We often need to rewrite decimals as fractions or mixed numbers. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. We know that [latex]$5.03[/latex] means [latex]5[/latex] dollars and [latex]3[/latex] cents. Since there are [latex]100[/latex] cents in one dollar, [latex]3[/latex] cents means [latex]{\Large\frac{3}{100}}[/latex] of a dollar, so [latex]0.03={\Large\frac{3}{100}}[/latex]. We convert decimals to fractions by identifying the place value of the farthest right digit. In the decimal [latex]0.03[/latex], the [latex]3[/latex] is in the hundredths place, so [latex]100[/latex] is the denominator of the fraction equivalent to [latex]0.03[/latex].[latex]0.03={\Large\frac{3}{100}}[/latex]

For our [latex]$5.03[/latex] lunch, we can write the decimal [latex]5.03[/latex] as a mixed number.[latex]5.03=5{\Large\frac{3}{100}}[/latex]

Notice that when the number to the left of the decimal is zero, we get a proper fraction. When the number to the left of the decimal is not zero, we get a mixed number.### Convert a decimal number to a fraction or mixed number.

- Look at the number to the left of the decimal.
- If it is zero, the decimal converts to a proper fraction.
- If it is not zero, the decimal converts to a mixed number.
- Write the whole number.

- 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

- Simplify the fraction, if possible.

### example

Write each of the following decimal numbers as a fraction or a mixed number:- [latex]4.09[/latex]
- [latex]3.7[/latex]
- [latex]-0.286[/latex]

1. | |

[latex]4.09[/latex] | |

There is a [latex]4[/latex] to the left of the decimal point. Write "[latex]4[/latex]" as the whole number part of the mixed number. | |

Determine the place value of the final digit. | |

Write the fraction. Write [latex]9[/latex] in the numerator as it is the number to the right of the decimal point. | |

Write [latex]100[/latex] in the denominator as the place value of the final digit, [latex]9[/latex], is hundredth. | [latex]4{\Large\frac{9}{100}}[/latex] |

The fraction is in simplest form. | So, [latex]4.09=4{\Large\frac{9}{100}}[/latex] |

2. | |

[latex]3.7[/latex] | |

There is a [latex]3[/latex] to the left of the decimal point. Write "[latex]3[/latex]" as the whole number part of the mixed number. | |

Determine the place value of the final digit. | |

Write the fraction. Write [latex]7[/latex] in the numerator as it is the number to the right of the decimal point. | |

Write [latex]10[/latex] in the denominator as the place value of the final digit, [latex]7[/latex], is tenths. | [latex]3{\Large\frac{7}{10}}[/latex] |

The fraction is in simplest form. | So, [latex]3.7=3{\Large\frac{7}{10}}[/latex] |

3. | |

[latex]−0.286[/latex] | |

There is a [latex]0[/latex] to the left of the decimal point. Write a negative sign before the fraction. | |

Determine the place value of the final digit and write it in the denominator. | |

Write the fraction. Write [latex]286[/latex] in the numerator as it is the number to the right of the decimal point. Write [latex]1,000[/latex] in the denominator as the place value of the final digit, [latex]6[/latex], is thousandths. | [latex]-{\Large\frac{286}{1000}}[/latex] |

We remove a common factor of [latex]2[/latex] to simplify the fraction. | [latex]-{\Large\frac{143}{500}}[/latex] |

### try it

[ohm_question]146573[/ohm_question] [ohm_question]146574[/ohm_question] [ohm_question]146575[/ohm_question]## Locating and ordering decimals with a number line

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

Locate [latex]0.4[/latex] on a number line. Solution The decimal [latex]0.4[/latex] is equivalent to [latex]{\Large\frac{4}{10}}[/latex], so [latex]0.4[/latex] is located between [latex]0[/latex] and [latex]1[/latex]. On a number line, divide the interval between [latex]0[/latex] and [latex]1[/latex] into [latex]10[/latex] equal parts and place marks to separate the parts. Label the marks [latex]0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0[/latex]. We write [latex]0[/latex] as [latex]0.0[/latex] and [latex]1[/latex] as [latex]1.0[/latex], so that the numbers are consistently in tenths. Finally, mark [latex]0.4[/latex] on the number line.### try it

1. [ohm_question]146228[/ohm_question] 2. Locate [latex]0.6[/latex] on a number line.Answer:

### example

Locate [latex]-0.74[/latex] on a number line.Answer: Solution The decimal [latex]-0.74[/latex] is equivalent to [latex]-{\Large\frac{74}{100}}[/latex], so it is located between [latex]0[/latex] and [latex]-1[/latex]. On a number line, mark off and label the multiples of [latex]-0.10[/latex] in the interval between [latex]0[/latex] and [latex]-1[/latex] ( [latex]-0.10[/latex] , [latex]-0.20[/latex] , etc.) and mark [latex]-0.74[/latex] between [latex]-0.70[/latex] and [latex]-0.80[/latex], a little closer to [latex]-0.70[/latex] .

### try it

[ohm_question]146577[/ohm_question] [ohm_question]146578[/ohm_question]### Order Decimals

Which is larger, [latex]0.04[/latex] or [latex]0.40?[/latex] If you think of this as money, you know that [latex]$0.40[/latex] (forty cents) is greater than [latex]$0.04[/latex] (four cents). So,[latex]0.40>0.04[/latex]

In previous chapters, we used the number line to order numbers.[latex]\begin{array}{}\\ a<b\text{ , }a\text{ is less than }b\text{ when }a\text{ is to the left of }b\text{ on the number line}\hfill \\ a>b\text{ , }a\text{ is greater than }b\text{ when }a\text{ is to the right of }b\text{ on the number line}\hfill \end{array}[/latex]

Where are [latex]0.04[/latex] and [latex]0.40[/latex] located on the number line? We see that [latex]0.40[/latex] is to the right of [latex]0.04[/latex]. So we know [latex]0.40>0.04[/latex]. How does [latex]0.31[/latex] compare to [latex]0.308?[/latex] This doesn’t translate into money to make the comparison easy. But if we convert [latex]0.31[/latex] and [latex]0.308[/latex] to fractions, we can tell which is larger.[latex]0.31[/latex] | [latex]0.308[/latex] | |

Convert to fractions. | [latex]{\Large\frac{31}{100}}[/latex] | [latex]{\Large\frac{308}{1000}}[/latex] |

We need a common denominator to compare them. | [latex]{\Large\frac{31\cdot\color{red}{10}}{100\cdot\color{red}{10}}}[/latex] | [latex]{\Large\frac{308}{1000}}[/latex] |

[latex]{\Large\frac{310}{1000}}[/latex] | [latex]{\Large\frac{308}{1000}}[/latex] |

[latex]{\Large\frac{31}{100}}={\Large\frac{310}{1000}}\text{ and }0.31=0.310[/latex]

If two decimals have the same value, they are said to be equivalent decimals.[latex]0.31=0.310[/latex]

We say [latex]0.31[/latex] and [latex]0.310[/latex] are equivalent decimals.### Equivalent Decimals

Two decimals are equivalent decimals if they convert to equivalent fractions.### Order decimals

- Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match.
- Compare the numbers to the right of the decimal point as if they were whole numbers.
- Order the numbers using the appropriate inequality sign.

### example

Order the following decimals using [latex]<\text{ or }\text{>}[/latex]:- [latex]0.64[/latex] ____ [latex]0.6[/latex]
- [latex]0.83[/latex] ____ [latex]0.803[/latex]

Answer: Solution

1. | |

[latex]0.64[/latex] ____ [latex]0.6[/latex] | |

Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of [latex]0.6[/latex]. | [latex]0.64[/latex] ____ [latex]0.6[/latex] |

Compare the numbers to the right of the decimal point as if they were whole numbers. | [latex]64>60[/latex] |

Order the numbers using the appropriate inequality sign. | [latex]0.64>0.60[/latex] [latex]0.64>0.6[/latex] |

2. | |

[latex]0.83[/latex] ____ [latex]0.803[/latex] | |

Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of [latex]0.83[/latex]. | [latex]0.83[/latex] ____ [latex]0.803[/latex] |

Compare the numbers to the right of the decimal point as if they were whole numbers. | [latex]830>803[/latex] |

Order the numbers using the appropriate inequality sign. | [latex]0.830>0.803[/latex] [latex]0.83>0.803[/latex] |

### try it

[ohm_question]146232[/ohm_question] [ohm_question]146237[/ohm_question] [ohm_question]146238[/ohm_question]### example

Use [latex]<\text{or}>[/latex]; to order. [latex]-0.1[/latex] ____ [latex]- 0.8[/latex].Answer: Solution:

[latex]-0.1[/latex] ____ [latex]- 0.8[/latex] | |

Write the numbers one under the other, lining up the decimal points. | [latex]-0.1[/latex] [latex]-0.8[/latex] |

They have the same number of digits. | |

Since [latex]-1>-8,-1[/latex] tenth is greater than [latex]-8[/latex] tenths. | [latex]-0.1>-0.8[/latex] |

### try it

[ohm_question]146239[/ohm_question]## Rounding decimals

In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at [latex]\$3.279[/latex] per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay [latex]$3.28[/latex] , because the final price would be rounded to the nearest cent. In a lesson about whole numbers, we learned that we can round numbers to get an approximate value when the exact value is not needed. Suppose we wanted to round [latex]$2.72[/latex] to the nearest dollar. Is it closer to [latex]$2[/latex] or to [latex]$3[/latex]? What if we wanted to round [latex]$2.72[/latex] to the nearest ten cents; is it closer to [latex]$2.70[/latex] or to [latex]$2.80[/latex]? The number lines in the image below can help us answer those questions. (a) We see that [latex]2.72[/latex] is closer to [latex]3[/latex] than to [latex]2[/latex]. So, [latex]2.72[/latex] rounded to the nearest whole number is [latex]3[/latex]. (b) We see that [latex]2.72[/latex] is closer to [latex]2.70[/latex] than [latex]2.80[/latex]. So we say that [latex]2.72[/latex] rounded to the nearest tenth is [latex]2.7[/latex]. Can we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.### Round a decimal.

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

- Rewrite the number, removing all digits to the right of the given place value.

### Tip For Success

If the digit to the right of your given place value is 5 or above, give your given place value a shove. If the digit to the right of your given place value is 4 or less, let your given place value rest.### example

Round [latex]18.379[/latex] to the nearest hundredth. Solution[latex]18.379[/latex] | |

Locate the hundredths place and mark it with an arrow. | |

Underline the digit to the right of the [latex]7[/latex]. | |

Because [latex]9[/latex] is greater than or equal to [latex]5[/latex], add [latex]1[/latex] to the [latex]7[/latex]. | |

Rewrite the number, deleting all digits to the right of the hundredths place. | [latex]18.38[/latex] |

[latex]18.38[/latex] is [latex]18.379[/latex] rounded to the nearest hundredth. |

### try it

[ohm_question]146242[/ohm_question]### example

Round [latex]18.379[/latex] to the nearest- tenth
- whole number
Answer:

1. Round [latex]18.379[/latex] to the nearest tenth. | |

[latex]18.379[/latex] | |

Locate the tenths place and mark it with an arrow. | |

Underline the digit to the right of the tenths digit. | |

Because [latex]7[/latex] is greater than or equal to [latex]5[/latex], add [latex]1[/latex] to the [latex]3[/latex]. | |

Rewrite the number, deleting all digits to the right of the tenths place. | [latex]18.4[/latex] |

So, [latex]18.379[/latex] rounded to the nearest tenth is [latex]18.4[/latex]. |

2. Round [latex]18.379[/latex] to the nearest whole number. | |

[latex]18.379[/latex] | |

Locate the ones place and mark it with an arrow. | |

Underline the digit to the right of the ones place. | |

Since [latex]3[/latex] is not greater than or equal to [latex]5[/latex], do not add [latex]1[/latex] to the [latex]8[/latex]. | |

Rewrite the number, deleting all digits to the right of the ones place. | [latex]18[/latex] |

So [latex]18.379[/latex] rounded to the nearest whole number is [latex]18[/latex]. |

### try it

[ohm_question]146244[/ohm_question]## Licenses & Attributions

### CC licensed content, Original

- Question ID 146571, 146572, 146568,146569, 146224, 146225.
**Authored by:**Lumen Learning.**License:**CC BY: Attribution. - Question ID 146574, 146575, 146576.
**Authored by:**Lumen Learning.**License:**CC BY: Attribution. - Question ID 146237, 146238, 146239.
**Authored by:**Lumen Learning.**License:**CC BY: Attribution. - Question ID 146239, 146242, 146244.
**Authored by:**Lumen Learning.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL.

### CC licensed content, Shared previously

- Read and Write Decimals.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution. - Examples: Write a Number in Decimal Notation from Words.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution. - Ex 1: Convert a Decimal to a Fraction.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution. - Example: Identify Decimals on the Number Line.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution. - Decimal Notation: Ordering Decimals.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution. - Examples: Rounding Decimals.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution.

### CC licensed content, Specific attribution

- Prealgebra.
**Provided by:**OpenStax**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected].