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6. Terminating and Repeating Decimals
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eCourses /
Pre-Algebra /
Chapter 3
6. 

Terminating and Repeating Decimals

This lesson provides a deep dive into the world of decimals, specifically focusing on terminating and repeating decimals. It explains how these types of decimals can be understood through the use of long division and their relation to rational numbers. For example, if one is trying to convert a fraction to a decimal, understanding whether the decimal will terminate or repeat can be crucial. This knowledge is not just theoretical; it has practical applications in various fields such as finance for calculating interest rates, in engineering for precise measurements, and even in everyday scenarios like splitting a bill. The aim is to equip the reader with the tools needed to tackle real-world problems that involve decimals.
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Lesson Settings & Tools
12 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Terminating and Repeating Decimals
Slide of 12
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.

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

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.
Applet to compute the division of two numbers
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
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
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 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.
Dividing 7 by 15
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.
Dividing 7 by 15
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.
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
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.
Dividing 12 by 27
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.
The answer is A.
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.
Dividing 4 by 27
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.
Discussion

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
MultEqn

Multiply

Multiply

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
expand_more
Finally, solve the equation for
DivEqn

CancelCommonFac

Cancel out common factors

SimpQuot

Simplify quotient

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.
ExpandFrac

Multiply

Multiply

Since the greatest common factor of and is simplify the fraction by dividing both the numerator and denominator by
ReduceFrac

CalcQuot

Calculate quotient

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.
Example

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
MultEqn

Multiply

Multiply

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
DivEqn

CancelCommonFac

Cancel out common factors

SimpQuot

Simplify quotient

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
ReduceFrac

CalcQuot

Calculate quotient

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.

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.

  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.


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
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
MixedToFrac

Multiply

Multiply

AddTerms

Add terms

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.
Dividing 22 by 7
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
MixedToFrac

Multiply

Multiply

AddTerms

Add terms

Next, divide the numerator by the denominator Calculate to decimal places.
Dividing 223 by 71
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!
Closure

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.



Level 1
Level 2
Level 3
Terminating and Repeating Decimals
Exercises
Zosia knows that three one-thirds is equal to one.
She then remembers that can also be written as a repeating decimal. She adds these numbers as decimals again and gets
Consider the following question to determine whether Zosia's calculations are correct. Write in the form of a fraction.
Are Zosia's calculations correct?

Let's write the given repeating decimal as a fraction. 0.999999 ... We will follow five steps to rewrite it.

  1. Assign a variable to represent the repeating decimal.
  2. Count the number of repeating digits and call it n.
  3. Multiply the repeating decimal by 10^n.
  4. Subtract the equation in Step 1 from the equation in Step 3.
  5. Solve for the variable and express it as a fraction in its simplest form.

Let's do it!

Steps 1 and 2

Let x represent the given repeating decimal number. x = 0.999999 ... Notice that there is only one repeating digit, 9. This means that n= 1.

Step 3

We will multiply both sides of the equation by 10^1, or 10, because n= 1.

Step 4

Now we will subtract the original equation from the equation written in Step 3. cr & 10x = 9.999999 ... - & x = 0.999999 ... & 9x = 9.000000 ... We found that 9x = 9.

Step 5

Finally, we will divide both sides of the equation by 9 to find the value of x.

We found that x is equal to 99, or simply 1. We can conclude that 0.999999 ... is equal to 1. 0.999999 ... = 1

Alternatively, we can use the decimal expansion of 19, which is equal to 0.111111.... 1/9 = 0.111111... When we multiply both sides by 9, we get the following. 9 * 1/9 = 0.999999... On the left hand side, we get 99, or 1.


In Part A, we found that 0.999999 ... is equal to 1. 0.999999 ... = 1 At first glance, it would be easy for us to say that the left-hand side will never equal the right. This is because we cannot easily get a sense of infinitely many 9s. However, the equation is actually true because of the infinite number of 9s on the left. We showed how this is possible with our calculations in Part A. cr & 0.333333 ... & 0.333333 ... +& 0.333333 ... & 0.999999 ... cr & 0.333333 ... & 0.333333 ... +& 0.333333 ... & 1.000000 ... Therefore, all of Zosia's calculations are correct. Three one-thirds, or 33, can be written as 0.999999... or as 1.000000 ....

Terminating and Repeating Decimals

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