6.1: Reading and Writing Decimals

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Learning Objectives

understand the meaning of digits occurring to the right of the ones position
be familiar with the meaning of decimal fractions
be able to read and write a decimal fraction

Digits to the Right of the Ones Position

We began our study of arithmetic 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.

$10 times 100,000 is the millions position. 10 times 10,000 is the hundred thousands position. 10 times 1,000 is the ten thousands position. 10 times 100 is the thousands position. 10 times 10 is the hundreds position. 10 times 1 is the tens position. 1 is the ones position.$

This means that each position has $\dfrac{1}{10}$ the value of the position to its left.

$1,000,000 is the millions position. One tenth of 1,000,000 is the hundred thousands. One tenths of 100,000 is the ten thousands. One tenth of 10,000 is the thousands position. One tenth of 1,000 is the thousands. One tenth of 100 is the tens position. One tenth of 10 is the ones position.$

Thus, a digit written to the right of the units position must have a value of $\dfrac{1}{10}$ of 1. Recalling that the word "of" translates to multiplication $(\cdot)$, we can see that the value of the first position to the right of the units digit is $\dfrac{1}{10}$ of 1, or

$\dfrac{1}{10} \cdot 1 = \dfrac{1}{10}$

The value of the second position to the right of the units digit is $\dfrac{1}{10}$ of $\dfrac{1}{10}$, or

$\dfrac{1}{10} \cdot \dfrac{1}{10} = \dfrac{1}{10^2} = \dfrac{1}{100}$

The value of the third position to the right of the units digit is $\dfrac{1}{10}$ of $\dfrac{1}{100}$, or

$\dfrac{1}{10} \cdot \dfrac{1}{100} = \dfrac{1}{10^3} = \dfrac{1}{1000}$

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 $(10, 10^2, 10^3, ...)$.

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.

$millions, hundred thousands, ten thousands, thousands, hundreds, tens and ones are to the left of the decimal point. tenths, hundredths, thousandths, ten thousandths, hundred thousandths, and millionths are to the right of the decimal point.$

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.

42.6

The 6 is in the tenths position.

$42.6 = 42 \dfrac{6}{10}$
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.

$9.8014 = 9 \dfrac{8014}{10,000}$
0.93

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

$0.93 = \dfrac{93}{100}$

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.

0.7

The 7 is in the tenths position.

$0.7 = \dfrac{7}{10}$

Note

We can insert zeros to the right of the right-most digit in a decimal fraction without changing the value of the number.

$\dfrac{7}{10} = 0.7 = 0.70 = \dfrac{70}{100} = \dfrac{7}{10}$

Reading Decimal Fractions

Reading a Decimal Fraction
To read a decimal fraction,

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

Sample Set A

Read the following numbers.

6.8

$6.8 is six and eight tenths. The 8 is in the tenths position.$

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.

Sample Set A

14.116

$14.116 is fourteen and one hundred sixteen thousandths. The six is in the thousandths position.$

Sample Set A

0.0019

$0.0019 is nineteen ten thousandths. The nine is in the ten thousandths position.$

Sample Set A

Eighty-one

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

$81 = 81.0$

Practice Set A

Read the following decimal fractions.

12.9

Answer: twelve and nine tenths

Practice Set A

4.86

Answer: four and eighty-six hundredths

Practice Set A

7.00002

Answer: seven and two hundred thousandths

Practice Set A

0.030405

Answer: thirty thousand four hundred five millionths

Writing Decimal Fractions

Writing a Decimal Fraction
To write a decimal fraction,

Write the whole number part.
Write a decimal point for the word "and."
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.

Thirty-one and twelve hundredths.

Solution

The decimal position indicated is the hundredths position.

31.12

Sample Set B

Two and three hundred-thousandths.

Solution

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

Sample Set B

Six thousand twenty-seven and one hundred four millionths.

Solution

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

Sample Set B

Seventeen hundredths.

Solution

The decimal position indicated is the hundredths position.

0.17

Practice Set B

Write each decimal fraction.

Three hundred six and forty-nine hundredths.

Answer: 306.49

Practice Set B

Nine and four thousandths.

Answer: 9.004

Practice Set B

Sixty-one millionths.

Answer: 0.000061

Exercises

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

Exercise $\PageIndex{1}$

1. 3.941
9 is in the position.
4 is in the position.
1 is in the position.

Answer: Tenths; hundredths, thousandths

Exercise $\PageIndex{2}$

17.1085
1 is in the position.
0 is in the position.
8 is in the position.
5 is in the position.

Exercise $\PageIndex{3}$

652.3561927
9 is in the position.
7 is in the position.

Answer: Hundred thousandths; ten millionths

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

Exercise $\PageIndex{4}$

9.2

Exercise $\PageIndex{5}$

8.1

Answer: eight and one tenth

Exercise $\PageIndex{6}$

10.15

Exercise $\PageIndex{7}$

55.06

Answer: fifty-five and six hundredths

Exercise $\PageIndex{8}$

0.78

Exercise $\PageIndex{9}$

1.904

Answer: one and nine hundred four thousandths

Exercise $\PageIndex{10}$

10.00011

For the following 10 problems, write each decimal fraction.

Exercise $\PageIndex{11}$

Three and twenty one-hundredths.

Answer: 3.20

Exercise $\PageIndex{12}$

Fourteen and sixty seven-hundredths.

Exercise $\PageIndex{13}$

One and eight tenths.

Answer: 1.8

Exercise $\PageIndex{14}$

Sixty-one and five tenths.

Exercise $\PageIndex{15}$

Five hundred eleven and four thousandths.

Answer: 511.004

Exercise $\PageIndex{16}$

Thirty-three and twelve ten-thousandths.

Exercise $\PageIndex{17}$

Nine hundred forty-seven thousandths.

Answer: 0.947

Exercise $\PageIndex{18}$

Two millionths.

Exercise $\PageIndex{19}$

Seventy-one hundred-thousandths.

Answer: 0.00071

Exercise $\PageIndex{20}$

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 $\PageIndex{21}$

$3 \div 4$

Answer: seventy-five hundredths

Exercise $\PageIndex{22}$

$1 \div 8$

Exercise $\PageIndex{23}$

$4 \div 10$

Answer: four tenths

Exercise $\PageIndex{24}$

$2 \div 5$

Exercise $\PageIndex{25}$

$4 \div 25$

Answer: sixteen hundredths

Exercise $\PageIndex{26}$

$1 \div 50$

Exercise $\PageIndex{27}$

$3 \div 16$

Answer: one thousand eight hundred seventy-five ten thousandths

Exercise $\PageIndex{28}$

$15 \div 8$

Exercise $\PageIndex{29}$

$11 \div 20$

Answer: fifty-five hundredths

Exercise $\PageIndex{30}$

$9 \div 40$

Exercises for Review

Exercise $\PageIndex{31}$

Round 2,614 to the nearest ten.

Answer: 2610

Exercise $\PageIndex{32}$

Is 691,428,471 divisible by 3?

Exercise $\PageIndex{33}$

Determine the missing numerator.

$\dfrac{3}{14} = \dfrac{?}{56}$

Answer: 12

Exercise $\PageIndex{34}$

Find $\dfrac{3}{16}$ of $\dfrac{32}{39}$

Exercise $\PageIndex{35}$

Find the value of $\sqrt{\dfrac{25}{81}} + (\dfrac{2}{3})^2 + \dfrac{1}{9}$

Answer: $\dfrac{10}{9}$ or $1 \dfrac{1}{9}$