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There are many ways to write the same rational number, including fractions, mixed numbers, and decimal numbers.
Decimal numbers can be classified based on the digits after the decimal point. As an example, consider the decimal number corresponding to Is there a repeating digit? How about the decimal equivalent of 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.

## 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

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

## 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

## Identifying Types of Decimals

Determine whether the given number is a repeating decimal, a terminating decimal, or neither. ## 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 or the quotient will be a repeating decimal number. Consider, for example, the following fraction.
It is possible to write this fraction as a decimal in two steps.
1
Use Long Division
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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. 2
Use Bar Notation to Write the Repeating Decimal
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The fraction was found to be a repeating decimal number.
The digit that repeats is so a bar is drawn over that digit.

## 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
Animation
Comedy
Science Fiction
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
Animation
Comedy
Science Fiction
Total
There are students in the class. Of those students, prefer animation. This can be represented by a fraction by writing the part, the students who like animation, over the whole, the students in the class.
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 or a pattern appears. It appears that the remainder will always repeat — in other words, the digit in the quotient will always repeat. This means that the division resulted in a repeating decimal number.
The same value can also be written by drawing a bar over the repeating digit.
b From the table, it can be seen that students like the comedy genre. This means that the fraction of those students is Again, use long division to rewrite it as a decimal. Don't stop until the remainder is or a pattern appears. In this case, more than one digit repeats in the quotient.
Put a bar above those three digits. This repeating decimal corresponds to option C.

## Converting a Repeating Decimal Number Into a Fraction

Terminating decimal numbers can be converted to fractions by dividing the number by where is the number of decimal places.
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.
Even though they have an infinite number of digits, these numbers can be expressed as fractions. Follow these five steps to do so.
1
Assign a Variable for the Repeating Decimal
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Let be a variable to represent the given repeating decimal number. An equation can be written by setting equal to the number. Consider as an example.
2
Count the Number of Repeating Digits and Let It Be
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For the given decimal, the repeating digit is
There is only one digit with a bar above it. This means
3
Multiply the Repeating Decimal by
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Since there is one repeating digit, multiply each side of the equation by or
4
Subtract the Equation in Step From the Equation in Step
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Now subtract the original equation from the equation obtained in Step
Writing the decimals without using bar notation can make subtraction easier to understand.
The zeros can be ignored. It is found that is equal to
5
Solve for the Variable and Express It as a Fraction in Its Simplest Form
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Finally, solve the equation for
Start by eliminating the decimal in the numerator to simplify the fraction. This can be done by multiplying the numerator and denominator by because the numerator has decimal places.
Since the greatest common factor of and is simplify the fraction by dividing both the numerator and denominator by
It is found that is equal to Remember that is also equal to
Therefore, by the Transitive Property of Equality, the fraction is equal to the given repeating decimal number.

## Which Two Genres Did Vincenzo Choose?

Vincenzo is organizing the information he gathered from his survey.

Genre Number of Students
Action
Animation
Comedy
Science Fiction

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

a Write the result as a fraction.
b Which two movie genres did Vincenzo choose?

### Hint

a Let 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
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.
There are five steps to write this repeating decimal number as a fraction.
1. Assign a variable to represent the repeating decimal.
2. Count the number of repeating digits and call it
3. Multiply the repeating decimal by
4. Subtract the equation in Step from the equation in Step
5. Solve for the variable and express it as a fraction in its simplest form.

### Step

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

### Step

The repeated sequence of digits in the decimal is There are three repeating digits, so

### Step

Since multiply each side of the equation by or

### Step

Now subtract the original equation from the equation obtained in Step
Consider the expanded forms of the numbers when performing this subtraction.
Since the zeros can be ignored, is equal to

### Step

Lastly, solve the obtained equation for
Check whether this fraction can be simplified by finding the greatest common factor (GCF) of the numerator and denominator.
Then, is equal to Finally, simplify the fraction by dividing both numerator and denominator by
As a result, is equal to which is equal to the given repeating number.
b In Part A, it was found that is equivalent to Notice that the denominator is the same as the total number of students, so 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

Genre Number of Students
Action
Animation
Comedy
Science Fiction

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

## Practice Converting Between Fractions and Repeating Decimals

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

How to Submit a Repeating Decimal

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

1. Enter the non-repeating part of the number, including the decimal point, into the first box.
2. Enter the repeating part into the second box.
Here are some examples.

## 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
Although this number cannot be expressed exactly as a ratio of two integers, Archimedes claimed that was between and Was Archimedes correct?

### Hint

Write the mixed numbers as decimals.

### Solution

Archimedes believed that was between two rational numbers, and
To check whether Archimedes was correct, both of the numbers will be written as a decimal. Since the first six decimal places of are given, the numbers will be calculated up to decimal places. In this way, the numbers can be compared.

### Rewriting

First, rewrite as an improper fraction.
Write mixed number as a fraction
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 is given. As seen, the mixed number is equal to

### Rewriting

The same steps as before can be followed to write as a decimal. Start by rewriting the mixed number as an improper fraction.
Write mixed number as a fraction
Next, divide the numerator by the denominator Calculate to decimal places. As shown, is equal to

### Comparing the Numbers

Now that the decimal forms of both numbers have been found, the numbers can be compared.
Notice that the first three digits of the numbers are all the same.
This means that the digits in the thousandth place must be compared.
The number with the greatest digit in the thousandth place is the greatest, and the number with the least digit is the least.
Therefore, is between and Archimedes was correct. The answer is B.

### Extra

Archimedes's Method
The Greek mathematician Archimedes devised a technique to approximate His idea was quite simple — he drew regular polygons inside and outside of a circle. He used a circle with diameter of because its circumference is 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 use a shape with sides. His calculations gave a range between and
Of course, he found this result with pen and paper, without the help of a computer or calculator. Amazing!

## Why Is 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,
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,
Golden ratio,

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. {{ subexercise.title }}