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There are many ways to write the same rational number, including fractions, mixed numbers, and decimal numbers.

Fraction \frac{1 4}{5} Mixed Number 2 \frac{4}{5} Decimal 2.8

Decimal numbers can be classified based on the digits after the decimal point. As an example, consider the decimal number corresponding to

\frac{1}{3} .

Is there a repeating digit? How about the decimal equivalent of

\frac{1}{7} ?

This lesson explores decimal number types and develops methods for converting them into fractions.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Long Division
Equivalent Fractions
Dividing Decimals

Explore

Decimal Forms of Fractions

A rational number can be represented as a decimal number with the help of the long division method. Rewrite the fractions shown in the table as decimals.

Fraction	Decimal
$\frac{2}{9}$
$\frac{1}{3}$
$\frac{3}{4}$
$\frac{4}{1 1}$
$\frac{3}{8}$

Consider the following questions as a guide to classify decimals.

Which fractions have a remainder of zero?
Which fractions do not have a remainder of zero?
Which fractions have the same operations repeated in the long division after a certain point?

Extra

Applet for Calculating the Decimal Numbers

The applet below will help convert the fractions to decimal numbers. It is recommended to calculate at least

4

decimal places.

Discussion

Repeating Decimals

A repeating decimal number, or recurring decimal number, is a number in decimal form in which some digits after the decimal point repeat infinitely. The digits repeat their values at regular intervals and the infinitely repeated part is not zero. When writing the decimal, a line is drawn over the repeating portion to express such a number.

Repeating Decimal Numbers
Number	Notation	Fraction
$0.666666 \dots$	$0 . \overline{6}$	$\frac{2}{3}$
$1.533333 \dots$	$1.5 \overline{3}$	$\frac{2 3}{1 5}$
$5.373737 \dots$	$5 . \overline{37}$	$\frac{5 3 2}{9 9}$

Since they can be expressed as fractions, repeating decimal numbers are rational numbers.

Discussion

Terminating Decimals

A terminating decimal number is a number in decimal form with a finite number of digits. Terminating numbers can be written as fractions, which means that they are rational numbers.

Terminating Decimal Numbers
Number	Fraction
$0.5$	$\frac{1}{2}$
$1.53$	$\frac{1 5 3}{1 0 0}$
$52.372$	$\frac{1 3 0 9 3}{2 5 0}$

Pop Quiz

Identifying Types of Decimals

Determine whether the given number is a repeating decimal, a terminating decimal, or neither.

Discussion

Converting a Fraction Into a Repeating Decimal Number

Fractions can be written as decimal numbers with the help of the long division method. However, when the denominator of a fraction has a factor other than

2

5,

the quotient will be a repeating decimal number. Consider, for example, the following fraction.

\frac{7}{1 5}

It is possible to write this fraction as a decimal in two steps.

Use Long Division

Divide the numerator by the denominator using long division.

After some point, the remainder does not become zero and a particular digit or set of digits in the quotient keeps repeating. This means that the quotient is a repeating decimal number.

Use Bar Notation to Write the Repeating Decimal

The fraction was found to be a repeating decimal number.

\frac{7}{1 5} = 0.4666 \dots

The digit that repeats is

6,

so a bar is drawn over that digit.

\frac{7}{1 5} = 0.4 \overline{6}

Example

Favorite Movie Genres

Vincenzo conducts a survey to find out what kind of movies his classmates like. The table shows the information he gathered from the survey.

Genre	Number of Students
Action	$6$
Animation	$12$
Comedy	$4$
Science Fiction	$5$

a What fraction of the students prefer animation?

b What fraction of the students prefer comedy?

Hint

a How many students are there? Write a fraction using the number of students that prefer animation and the total number of the students. Then, convert the fraction into a decimal number.

b Write a fraction using the number of students that prefer comedy and the total number of the students. Convert the fraction into a decimal number.

Solution

a Start by finding the total number of students in the class.

Genre	Number of Students
Action	$6$
Animation	$12$
Comedy	$4$
Science Fiction	$5$
Total	$27$

There are

27

students in the class. Of those students,

12

prefer animation. This can be represented by a fraction by writing the part, the

12

students who like animation, over the whole, the

27

students in the class.

\frac{1 2}{2 7}

Since the given options are decimals, this fraction must be converted into a decimal number. To do so, use long division. Don't stop until the remainder is

0

or a pattern appears.

It appears that the remainder will always repeat — in other words, the digit

4

in the quotient will always repeat. This means that the division resulted in a repeating decimal number.

\frac{1 2}{2 7} = 0.444 \dots

The same value can also be written by drawing a bar over the repeating digit.

\frac{1 2}{2 7} = 0 . \overline{4}

The answer is A.

b From the table, it can be seen that

4

students like the comedy genre. This means that the fraction of those students is

\frac{4}{2 7} .

Again, use long division to rewrite it as a decimal. Don't stop until the remainder is

0

or a pattern appears.

In this case, more than one digit repeats in the quotient.

\frac{4}{2 7} = 0.148148 \dots

Put a bar above those three digits. This repeating decimal corresponds to option C.

\frac{4}{2 7} = 0 . \overline{148}

Discussion

Converting a Repeating Decimal Number Into a Fraction

Terminating decimal numbers can be converted to fractions by dividing the number by

1 0^{n},

where

n

is the number of decimal places.

Terminating Decimal 0.56 Fraction \frac{5 6}{1 0 0}

However, it is impossible to count the number of decimal places for repeating decimals because they have an infinite number of digits after the decimal point.

Repeating Decimal 0.08 \overline{3} Fraction ?

Even though they have an infinite number of digits, these numbers can be expressed as fractions. Follow these five steps to do so.

Assign a Variable for the Repeating Decimal

Let

x

be a variable to represent the given repeating decimal number. An equation can be written by setting

x

equal to the number. Consider

0.08 \overline{3}

as an example.

x = 0.08 \overline{3}

Count the Number of Repeating Digits and Let It Be

n

For the given decimal, the repeating digit is

3 .

x = 0.08 \overline{3}

There is only one digit with a bar above it. This means

n = 1 .

Multiply the Repeating Decimal by

1 0^{n}

Since there is one repeating digit, multiply each side of the equation by

1 0^{1},

10 .

x = 0.08 \overline{3}

MultEqn

$LHS \cdot 10 = RHS \cdot 10$

x \cdot 10 = 0.08 \overline{3} \cdot 10

Multiply

10 x = 0.8 \overline{3}

Subtract the Equation in Step

1

From the Equation in Step

3

Now subtract the original equation from the equation obtained in Step

3 .

- 10 x = 0.8 \overline{3} x = 0.08 \overline{3}

Writing the decimals without using bar notation can make subtraction easier to understand.

- 10 x = 0.83333 \dots x = 0.08333 \dots 9 x = 0.75000 \dots

The zeros can be ignored. It is found that

9 x

is equal to

0.75 .

Solve for the Variable and Express It as a Fraction in Its Simplest Form

Finally, solve the equation for

x .

9 x = 0.75

DivEqn

$LHS / 9 = RHS / 9$

\frac{9 x}{9} = \frac{0 . 7 5}{9}

CancelCommonFac

Cancel out common factors

\frac{9 x}{9} = \frac{0 . 7 5}{9}

SimpQuot

Simplify quotient

x = \frac{0 . 7 5}{9}

Start by eliminating the decimal in the numerator to simplify the fraction. This can be done by multiplying the numerator and denominator by

1 0^{2}

because the numerator has

2

decimal places.

x = \frac{0 . 7 5}{9}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 1 0 0}{b \cdot 1 0 0}$

x = \frac{0 . 7 5 \cdot 1 0 0}{9 \cdot 1 0 0}

Multiply

x = \frac{7 5}{9 0 0}

Since the greatest common factor of

900

and

75

75,

simplify the fraction by dividing both the numerator and denominator by

75 .

x = \frac{7 5}{9 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 7 5}{b / 7 5}$

x = \frac{7 5 / 7 5}{9 0 0 / 7 5}

CalcQuot

Calculate quotient

x = \frac{1}{1 2}

It is found that

x

is equal to

\frac{1}{1 2} .

Remember that

x

is also equal to

0.08 \overline{3} .

x = 0.8 \overline{3} and x = \frac{1}{1 2}

Therefore, by the Transitive Property of Equality, the fraction is equal to the given repeating decimal number.

0.08 \overline{3} = \frac{1}{1 2}

Example

Which Two Genres Did Vincenzo Choose?

Vincenzo is organizing the information he gathered from his survey.

Genre	Number of Students
Action	$6$
Animation	$12$
Comedy	$4$
Science Fiction	$5$

He selected two genres at random. He used a calculator to divide the number of students who prefer the two genres by the total number of students.

His calculator showed the result as $0.629629629 \dots .$

a Write the result as a fraction.

b Which two movie genres did Vincenzo choose?

Hint

a Let

x

represent the repeating decimal and write an equation. How many repeating digits are there in the decimal number? Multiply the equation by a power of

10 .

b What is the numerator of the fraction found in Part A? Determine which of the two types gives a sum equal to this numerator.

Solution

a The number shown on the calculator is a repeating decimal number because some digits after the decimal point repeat themselves.

0.629629629 \dots \Leftrightarrow 0 . \overline{629}

There are five steps to write this repeating decimal number as a fraction.

Assign a variable to represent the repeating decimal.
Count the number of repeating digits and call it $n .$
Multiply the repeating decimal by $1 0^{n} .$
Subtract the equation in Step $1$ from the equation in Step $3 .$
Solve for the variable and express it as a fraction in its simplest form.

Step $1$

Let

x

be a variable that represents the given repeating decimal number. Write an equation by setting

x

equal to

0 . \overline{629} .

x = 0 . \overline{629}

Step $2$

The repeated sequence of digits in the decimal is $629 .$ There are three repeating digits, so $n = 3 .$

Step $3$

Since

n = 3,

multiply each side of the equation by

1 0^{3},

1000 .

x = 0 . \overline{629}

MultEqn

$LHS \cdot 1000 = RHS \cdot 1000$

x \cdot 1000 = 0 . \overline{629} \cdot 1000

Multiply

1000 x = 629 . \overline{629}

Step $4$

Now subtract the original equation from the equation obtained in Step

3 .

- 1000 x = 629 . \overline{629} x = 0 . \overline{629}

Consider the expanded forms of the numbers when performing this subtraction.

- 1000 x = 629.629629 \dots x = 0.629629 \dots 999 x = 629.000000 \dots

Since the zeros can be ignored,

999 x

is equal to

629 .

Step $5$

Lastly, solve the obtained equation for

x .

999 x = 629

DivEqn

$LHS / 999 = RHS / 999$

\frac{9 9 9 x}{9 9 9} = \frac{6 2 9}{9 9 9}

CancelCommonFac

Cancel out common factors

\frac{9 9 9 x}{9 9 9} = \frac{6 2 9}{9 9 9}

SimpQuot

Simplify quotient

x = \frac{6 2 9}{9 9 9}

Check whether this fraction can be simplified by finding the greatest common factor (GCF) of the numerator and denominator.

629 = 17 \cdot 37 999 = 3^{3} \cdot 37

Then,

GCF (629, 999)

is equal to

37 .

Finally, simplify the fraction by dividing both numerator and denominator by

37 .

x = \frac{6 2 9}{9 9 9}

ReduceFrac

$\frac{a}{b} = \frac{a / 3 7}{b / 3 7}$

x = \frac{6 2 9 / 3 7}{9 9 9 / 3 7}

CalcQuot

Calculate quotient

x = \frac{1 7}{2 7}

As a result,

x

is equal to

\frac{1 7}{2 7},

which is equal to the given repeating number.

0 . \overline{629} = \frac{1 7}{2 7}

b In Part A, it was found that

0 . \overline{629}

is equivalent to

\frac{1 7}{2 7} .

Notice that the denominator is the same as the total number of students, so

\frac{1 7}{2 7}

should be the fraction that Vincenzo entered into the calculator. The numerator of this fraction is the sum of the students who prefer two specific genres.

Now take a look at the table and determine which of the two genres gives a sum equal to $17 .$

Genre	Number of Students
Action	$6$
Animation	$12$
Comedy	$4$
Science Fiction	$5$

The number of students who prefer animation and science fiction is $17 .$ Therefore, Vincenzo selected the animation and science fiction genres.

Pop Quiz

Practice Converting Between Fractions and Repeating Decimals

Write the given fraction as a repeating decimal and vice versa.

$Randomly generated fraction or decimal number$

Extra

How to Submit a Repeating Decimal

If the answer is a repeating decimal, do the following to submit the answer.

Enter the non-repeating part of the number, including the decimal point, into the first box.
Enter the repeating part into the second box.

Here are some examples.

Repeating Decimal 0.4 \overline{09} 0 . \overline{9} \Rightarrow \Rightarrow How to Submit 0.4 \overline{09} 0 . \overline{9}

Example

Was Archimedes Correct?

Vincenzo is watching a movie after school. In one scene, the leading character was explaining some of Archimedes's contributions to mathematics. One of his contributions was a method he used to approximate the value of

π .

π = 3.141592 \dots

Although this number cannot be expressed exactly as a ratio of two integers, Archimedes claimed that

π

was between

3 \frac{1}{7}

and

3 \frac{1 0}{7 1} .

Was Archimedes correct?

Hint

Write the mixed numbers as decimals.

Solution

Archimedes believed that

π

was between two rational numbers,

3 \frac{1}{7}

and

3 \frac{1 0}{7 1} .

π = 3.141592 \dots

To check whether Archimedes was correct, both of the numbers will be written as a decimal. Since the first six decimal places of

π

are given, the numbers will be calculated up to

6

decimal places. In this way, the numbers can be compared.

Rewriting $3 \frac{1}{7}$

First, rewrite

3 \frac{1}{7}

as an improper fraction.

3 \frac{1}{7}

Write mixed number as a fraction

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

\frac{3 \cdot 7 + 1}{7}

Multiply

\frac{2 1 + 1}{7}

AddTerms

Add terms

\frac{2 2}{7}

Next, use long division to divide the numerator by the denominator of the fraction. Divide up to six decimal places because the first six decimal places of

π

is given.

As seen, the mixed number is equal to

3.142857 \dots .

3 \frac{1}{7} = 3.142857 \dots

Rewriting $3 \frac{1 0}{7 1}$

The same steps as before can be followed to write

3 \frac{1 0}{7 1}

as a decimal. Start by rewriting the mixed number as an improper fraction.

3 \frac{1 0}{7 1}

Write mixed number as a fraction

MixedToFrac

$a \frac{b}{c} = \frac{a \cdot c + b}{c}$

\frac{3 \cdot 7 1 + 1 0}{7 1}

Multiply

\frac{2 1 3 + 1 0}{7 1}

AddTerms

Add terms

\frac{2 2 3}{7 1}

Next, divide the numerator

223

by the denominator

71 .

Calculate to

6

decimal places.

As shown,

3 \frac{1 0}{7 1}

is equal to

3.140845 \dots .

3 \frac{1 0}{7 1} = 3.140845 \dots

Comparing the Numbers

Now that the decimal forms of both numbers have been found, the numbers can be compared.

3 \frac{1}{7} 3 \frac{1 0}{7 1} π = 3.142857 \dots = 3.140845 \dots = 3.141592 \dots

Notice that the first three digits of the numbers are all the same.

3 \frac{1}{7} 3 \frac{1 0}{7 1} π = 3.14 2857 \dots = 3.14 0845 \dots = 3.14 1592 \dots

This means that the digits in the thousandth place must be compared.

3 \frac{1}{7} 3 \frac{1 0}{7 1} π = 3.142857 \dots = 3.140845 \dots = 3.141592 \dots

The number with the greatest digit in the thousandth place is the greatest, and the number with the least digit is the least.

3.140845 \dots < 3.141592 \dots < 3.142857 \dots ⇓ 3 \frac{1 0}{7 1} < π < 3 \frac{1}{7}

Therefore,

π

is between

3 \frac{1 0}{7 1}

and

3 \frac{1}{7} .

Archimedes was correct. The answer is B.

Extra

Archimedes's Method

The Greek mathematician Archimedes devised a technique to approximate

π .

His idea was quite simple — he drew regular polygons inside and outside of a circle. He used a circle with diameter of

1

because its circumference is

π .

The circumference of the circle would be somewhere between the perimeters of the polygons. Archimedes calculated the perimeters of the polygons by means of geometric arguments.

His final estimate for

π

use a shape with

96

sides. His calculations gave a range between

3 \frac{1 0}{7 1}

and

3 \frac{1}{7} .

3 \frac{1 0}{7 1} < π < 3 \frac{1}{7}

Of course, he found this result with pen and paper, without the help of a computer or calculator. Amazing!

Closure

Why Is $π$ So Special?

This lesson presented two types of decimal numbers, terminating and repeating decimals. The lesson also introduced methods for converting both fractions. Now comes a tough question! Are there decimal numbers that have an infinite number of places but do not repeat? The answer is yes and they are called irrational numbers. Take, for example,

π .

π = 3.141592653 \dots

No repeating patterns are seen in this type of decimal number, so their decimal representations have no end. Here are a few other commonly used irrational numbers.

Some Irrational Numbers
Euler's number, $e$	$2.718281 \dots$
Golden ratio, $ϕ$	$1.618033 \dots$
$2$	$1.414213 \dots$

Irrational numbers cannot be converted into rational numbers. All that can be done is to get better and better approximations. The diagram shows how decimal numbers are classified.

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Catch-Up and Review

Extra

Hint

Solution

Hint

Solution

Step 1

Step 2

Step 3

Step 4

Step 5

Extra

Hint

Solution

Rewriting 371​

Rewriting 37110​

Comparing the Numbers

Extra

Step $1$

Step $2$

Step $3$

Step $4$

Step $5$

Rewriting $3 \frac{1}{7}$

Rewriting $3 \frac{1 0}{7 1}$