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• Preface
• Acknowledgements

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Textbook by: Ron Stewart. E-mail the author

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to read and write decimals. By the end of the module students should understand the meaning of digits occurring to the right of the ones position, be familiar with the meaning of decimal fractions and be able to read and write a decimal fraction.

## Section Overview

• Digits to the Right of the Ones Position
• Decimal Fractions
• Writing Decimal Fractions

## Digits to the Right of the Ones Position

We began our study of arithmetic ((Reference)) by noting that our number system is called a positional number system with base ten. We also noted that each position has a particular value. We observed that each position has ten times the value of the position to its right.

This means that each position has 110110 the value of the position to its left.

Thus, a digit written to the right of the units position must have a value of 110110 size 12{ { {1} over {"10"} } } {} of 1. Recalling that the word "of" translates to multiplication , we can see that the value of the first position to the right of the units digit is 110110 size 12{ { {1} over {"10"} } } {} of 1, or

1 10 1 = 1 10 1 10 1 = 1 10 size 12{ { {1} over {"10"} } cdot 1= { {1} over {"10"} } } {}

The value of the second position to the right of the units digit is 110110 size 12{ { {1} over {"10"} } } {} of 110110 size 12{ { {1} over {"10"} } } {}, or

1 10 1 10 = 1 10 2 = 1 100 1 10 1 10 = 1 10 2 = 1 100 size 12{ { {1} over {"10"} } cdot { {1} over {"10"} } = { {1} over {"10" rSup { size 8{2} } } } = { {1} over {"100"} } } {}

The value of the third position to the right of the units digit is 110110 size 12{ { {1} over {"10"} } } {} of 11001100 size 12{ { {1} over {"100"} } } {}, or

1 10 1 100 = 1 10 3 = 1 1000 1 10 1 100 = 1 10 3 = 1 1000 size 12{ { {1} over {"10"} } cdot { {1} over {"10"} } = { {1} over {"10" rSup { size 8{3} } } } = { {1} over {"1000"} } } {}

This pattern continues.

We can now see that if we were to write digits in positions to the right of the units positions, those positions have values that are fractions. Not only do the positions have fractional values, but the fractional values are all powers of 10 10,102,103,10,102,103, size 12{ left ("10","10" rSup { size 8{2} } ,"10" rSup { size 8{3} } , dotslow right )} {}.

## Decimal Fractions

### Decimal Point, Decimal

If we are to write numbers with digits appearing to the right of the units digit, we must have a way of denoting where the whole number part ends and the fractional part begins. Mathematicians denote the separation point of the units digit and the tenths digit by writing a decimal point. The word decimal comes from the Latin prefix "deci" which means ten, and we use it because we use a base ten number system. Numbers written in this form are called decimal fractions, or more simply, decimals.

Notice that decimal numbers have the suffix "th."

### Decimal Fraction

A decimal fraction is a fraction in which the denominator is a power of 10.

The following numbers are examples of decimals.

1. 42.6

The 6 is in the tenths position.

42 . 6 = 42 6 10 42 . 6 = 42 6 10 size 12{"42" "." 6="42" { {6} over {"10"} } } {}

2. 9.8014

The 8 is in the tenths position.
The 0 is in the hundredths position.
The 1 is in the thousandths position.
The 4 is in the ten thousandths position.

9 . 8014 = 9 8014 10 , 000 9 . 8014 = 9 8014 10 , 000 size 12{9 "." "8014"=9 { {"8014"} over {"10","000"} } } {}

3. 0.93

The 9 is in the tenths position.
The 3 is in the hundredths position.

0 . 93 = 93 100 0 . 93 = 93 100 size 12{0 "." "93"= { {"93"} over {"100"} } } {}

### Note:

Quite often a zero is inserted in front of a decimal point (in the units position) of a decimal fraction that has a value less than one. This zero helps keep us from overlooking the decimal point.
4. 0.7

The 7 is in the tenths position.

0 . 7 = 7 10 0 . 7 = 7 10 size 12{0 "." 7= { {7} over {"10"} } } {}

### Note:

We can insert zeros to the right of the right-most digit in a decimal fraction without changing the value of the number.
7 10 = 0 . 7 = 0 . 70 = 70 100 = 7 10 7 10 = 0 . 7 = 0 . 70 = 70 100 = 7 10 size 12{ { {7} over {"10"} } =0 "." 7=0 "." "70"= { {"70"} over {"100"} } = { {7} over {"10"} } } {}

1. Read the whole number part as usual. (If the whole number is less than 1, omit steps 1 and 2.)
2. Read the decimal point as the word "and."
3. Read the number to the right of the decimal point as if it were a whole number.
4. Say the name of the position of the last digit.

### Sample Set A

#### Example 1

6.8

##### Note:
Some people read this as "six point eight." This phrasing gets the message across, but technically, "six and eight tenths" is the correct phrasing.

14.116

0.0019

#### Example 4

81

Eighty-one

In this problem, the indication is that any whole number is a decimal fraction. Whole numbers are often called decimal numbers.

81 = 81 . 0 81 = 81 . 0 size 12{"81"="81" "." 0} {}

### Practice Set A

#### Exercise 1

12.9

##### Solution

twelve and nine tenths

#### Exercise 2

4.86

##### Solution

four and eighty-six hundredths

#### Exercise 3

7.00002

##### Solution

seven and two hundred thousandths

#### Exercise 4

0.030405

##### Solution

thirty thousand four hundred five millionths

## Writing Decimal Fractions

### Writing a Decimal Fraction

To write a decimal fraction,

1. Write the whole number part.
2. Write a decimal point for the word "and."
3. Write the decimal part of the number so that the right-most digit appears in the position indicated in the word name. If necessary, insert zeros to the right of the decimal point in order that the right-most digit appears in the correct position.

### Sample Set B

Write each number.

#### Example 5

Thirty-one and twelve hundredths.

The decimal position indicated is the hundredths position.

31.12

#### Example 6

Two and three hundred-thousandths.

The decimal position indicated is the hundred thousandths. We'll need to insert enough zeros to the immediate right of the decimal point in order to locate the 3 in the correct position.

2.00003

#### Example 7

Six thousand twenty-seven and one hundred four millionths.

The decimal position indicated is the millionths position. We'll need to insert enough zeros to the immediate right of the decimal point in order to locate the 4 in the correct position.

6,027.000104

#### Example 8

Seventeen hundredths.

The decimal position indicated is the hundredths position.

0.17

### Practice Set B

Write each decimal fraction.

#### Exercise 5

Three hundred six and forty-nine hundredths.

306.49

#### Exercise 6

Nine and four thousandths.

9.004

#### Exercise 7

Sixty-one millionths.

0.000061

## Exercises

For the following three problems, give the decimal name of the posi­tion of the given number in each decimal fraction.

### Exercise 8

1. 3.941
9 is in the


position.
4 is in the

position.
1 is in the

position.

#### Solution

Tenths; hundredths, thousandths

### Exercise 9

17.1085
1 is in the


position.
0 is in the

position.
8 is in the

position.
5 is in the

position.

### Exercise 10

652.3561927
9 is in the


position.
7 is in the

position.

#### Solution

Hundred thousandths; ten millionths

For the following 7 problems, read each decimal fraction by writing it.

9.2

### Exercise 12

8.1

#### Solution

eight and one tenth

10.15

### Exercise 14

55.06

#### Solution

fifty-five and six hundredths

0.78

### Exercise 16

1.904

#### Solution

one and nine hundred four thousandths

### Exercise 17

10.00011

For the following 10 problems, write each decimal fraction.

### Exercise 18

Three and twenty one-hundredths.

3.20

### Exercise 19

Fourteen and sixty seven-hundredths.

### Exercise 20

One and eight tenths.

1.8

### Exercise 21

Sixty-one and five tenths.

### Exercise 22

Five hundred eleven and four thousandths.

511.004

### Exercise 23

Thirty-three and twelve ten-thousandths.

### Exercise 24

Nine hundred forty-seven thousandths.

0.947

Two millionths.

### Exercise 26

Seventy-one hundred-thousandths.

0.00071

### Exercise 27

One and ten ten-millionths.

### Calculator Problems

For the following 10 problems, perform each division using a calculator. Then write the resulting decimal using words.

### Exercise 28

3÷43÷4 size 12{3¸4} {}

#### Solution

seventy-five hundredths

### Exercise 29

1÷81÷8 size 12{1¸8} {}

### Exercise 30

4÷104÷10 size 12{4¸"10"} {}

four tenths

### Exercise 31

2÷52÷5 size 12{2¸5} {}

### Exercise 32

4÷254÷25 size 12{4¸"25"} {}

#### Solution

sixteen hundredths

### Exercise 33

1÷501÷50 size 12{1¸"50"} {}

### Exercise 34

3÷163÷16 size 12{3¸"16"} {}

#### Solution

one thousand eight hundred seventy-five ten thousandths

### Exercise 35

15÷815÷8 size 12{"15"¸8} {}

### Exercise 36

11÷2011÷20 size 12{"11"¸"20"} {}

#### Solution

fifty-five hundredths

### Exercise 37

9÷409÷40 size 12{9¸"40"} {}

### Exercises for Review

#### Exercise 38

((Reference)) Round 2,614 to the nearest ten.

2610

#### Exercise 39

((Reference)) Is 691,428,471 divisible by 3?

#### Exercise 40

((Reference)) Determine the missing numerator.

3 14 = ? 56 3 14 = ? 56 size 12{ { {3} over {14} } = { {?} over {"56"} } } {}

12

#### Exercise 41

((Reference)) Find 316 of 3239316 of 3239 size 12{ { {3} over {"16"} } " of " { {"32"} over {"39"} } } {}

#### Exercise 42

((Reference)) Find the value of 2581+232+192581+232+19 size 12{ sqrt { { {"25"} over {"81"} } } + left ( { {2} over {3} } right ) rSup { size 8{2} } + { {1} over {9} } } {}

##### Solution

109   or   119109   or   119 size 12{ { {"10"} over {9} } " or "1 { {1} over {9} } } {}

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