# 1.9: Decimal Notation

Difficulty Level: At Grade Created by: CK-12

## Decimal Notation

Objectives

The lesson objectives for Decimal Notation are:

• Understanding terminating and periodic decimals
• Expressing a given fraction as a decimal number
• Expressing a given decimal number as a fraction

Introduction

In this concept you will learn the difference between a terminating decimal number and a periodic decimal number. You will also learn how to express a given fraction as a decimal number and how to express a given decimal number as a fraction. All periodic decimal numbers can be expressed as common fractions. If the decimal number is non-periodic, it cannot be expressed as a common fraction.

A rational number is any number that be written in the form \begin{align*}\frac{a}{b}\end{align*} where \begin{align*}b \ne 0\end{align*}. Therefore, periodic decimal numbers and terminating decimal numbers are rational numbers.

Watch This

Guidance

To change a fraction to a decimal number, the numerator must be divided by the denominator. The denominator is the divisor, the numerator is the dividend and the decimal number is the quotient. Let’s investigate the long division process by changing \begin{align*}\frac{3}{4}\end{align*} first and then \begin{align*}\frac{3}{13}\end{align*} to decimal numbers.

\begin{align*}\frac{3}{4}=3 \div 4\end{align*}

\begin{align*}& \overset{ \ \ {\color{red}0.75}}{4 \overline{ ) {3.00 \;}}}\\ & \underline{-0 \;\;\;\;\;}\\ & \ \ 3 \ 0\\ & \underline{- 2 \ 8 \; \;}\\ & \ \quad \ 20\\ & \underline{- \;\;\; 20 \;}\\ & \qquad \ 0\\\end{align*}

When the fraction \begin{align*}\frac{3}{4}\end{align*} was divided, the result was a decimal number that ended. There were two digits after the decimal point. The division was complete when the remainder was zero. This decimal number is known as a terminating decimal number.

\begin{align*}\frac{3}{13}=3 \div 13\end{align*}

\begin{align*}& \overset{ \ \ {\color{red}0.2307692307}}{13 \overline{ ) {3.0000000000 \;}}}\\ & \underline{- \ 0 \;}\\ & \quad 3 \ 0\\ & \underline{- \ 2 \ 6 \;}\\ & \qquad 40\\ & \underline{- \;\;\;\;\;39}\\ & \ \qquad \ 10\\ & \underline{- \;\;\;\;\;\;\;\;0}\\ & \ \qquad \ 100\\ & \underline{- \;\;\;\;\;\;\;\;\;91}\\ & \ \qquad \quad \ \ 90\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;78}\\ & \ \qquad \quad \ \ 120\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;117}\\ & \ \qquad \qquad \ \ 30\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;26}\\ & \ \qquad \qquad \quad \ 40\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;39}\\ & \ \qquad \qquad \quad \ \ 10\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0}\\ & \ \qquad \qquad \quad \ \ 100\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;91}\\ & \ \qquad \qquad \qquad \ \ 9\\\end{align*}

When the fraction \begin{align*}\frac{3}{13}\end{align*} was divided, there was no end to the decimal number. However, as the division continued, a pattern developed. This is known as a periodic decimal number with a period length of 6. The period length is the number of digits in the quotient that form a pattern that will repeat itself as the division continues. In the decimal number 0.230 769 230 7... the period is 230 769. If you were to continue dividing, the period would repeat infinitely. The division would never result in a zero remainder.

Example A

What fraction is equal to 0.45454545?

The decimal number is a periodic or repeating decimal. The period has a length of two. This means that the pattern that is repeating consists of 2 digits. To express the decimal number as a fraction, follow these steps:

Step 1: Let \begin{align*}x=0.45454545\end{align*}

Step 2: The repeating digit is 45. Place the repeating digit to the left of the decimal point by moving the decimal point 2 places to the right.

45.454545

To move the decimal point two places to the right, the decimal number was multiplied by 100. If you go back to step 1, you will see the equation \begin{align*}x=0.45454545\end{align*}. If you multiply one side of the equation by 100, you must multiply the other side by 100. You now have \begin{align*}100x=45.454545\end{align*}

Step 3: The repeating digits must be to the right of the decimal point. The digits are to the right of the decimal point which means that no multiplication is necessary to move the decimal point. The two equations that you now have are \begin{align*}x=0.45454545\end{align*} and \begin{align*}100x=45.454545\end{align*}

Step 4: Subtract the two equations and solve for \begin{align*}x\end{align*}.

\begin{align*}&100x=45.454545\\ & \underline{\;\; -x=0.45454545}\\ & \frac{99x}{99}=\frac{45}{99}\\ & \quad \ x=\frac{45}{99}=\frac{5}{11}\end{align*}

Example B

Decimal numbers that terminate can be more readily changed to fractions.

What fraction is equal to 0.125?

This decimal appears to be a terminating decimal number. There is no indication that the division continues and no sign of a repeating pattern. The steps to follow to express 0.125 as a fraction are:

Step 1: Express the decimal number as a whole number by moving the decimal point to the right. In this case, the decimal must be moved three places to the right.

Step 2: \begin{align*}0.125=125\end{align*}

Express 125 as a fraction with a denominator of 1 and three zeros. The three zeros represent the number of places that the decimal point was moved.

\begin{align*}\frac{125}{1000}\end{align*}

Step3: If possible, simplify the fraction. If you are not sure of the simplified form, your calculator, TI83, will do the calculations.

Therefore, the decimal number of 0.125 is equivalent to the fraction \begin{align*}\frac{1}{8}\end{align*}.

The method shown above is one that can be used if you can’t remember the place value associated with the decimal numbers. If you remember the place values, you can simply write the decimal as a fraction and simplify that fraction.

Example C

Are the following decimal numbers terminating or periodic? If they are periodic, what is the period and what is its length?

i) 0.318181818

ii) 0.375

iii) 0.3125

iv) 0.121 122 111 2

i) 0.318181818 A periodic decimal number with a period of 18. The length of the period is 2.

ii) 0.375 A terminating decimal number.

iii) 0.3125 A terminating decimal number

iv) 0.121 122 111 2 This decimal number is not a terminating decimal nor is it a periodic decimal. Therefore, the decimal number is not a rational number. Decimal numbers that are non-periodic belong to the irrational numbers.

Vocabulary

Irrational Numbers
An irrational number is the set of non-periodic decimal numbers. Some examples of irrational numbers are \begin{align*}\sqrt{3},\sqrt{2}\end{align*} and \begin{align*}\pi\end{align*}.
Periodic Decimal
A periodic decimal is a decimal number that has a pattern of digits that repeat. The decimal number 0.146 532 532 5, is a periodic decimal.
Rational Numbers
A rational number is any number that be written in the form \begin{align*}\frac{a}{b}\end{align*} where \begin{align*}b \ne 0\end{align*}. Therefore, periodic decimal numbers and terminating decimal numbers are rational numbers.
Terminating Decimal
A terminating decimal is a decimal number that ends. The process of dividing the fraction ends when the remainder is zero. The decimal number 0.25 is a terminating decimal.

Guided Practice

1. Express 2.018181818 in the form \begin{align*}\frac{a}{b}\end{align*}.

2. Express \begin{align*}\frac{15}{11}\end{align*} in decimal form.

3. If one tablet of micro K contains 0.5 grams of potassium, how much is contained in \begin{align*}2\frac{3}{4}\end{align*} tablets?

1. 2.018181818

Let \begin{align*}x=2.018181818\end{align*} The period is 18.

2018.181818 The period must be placed to the left of the decimal point. This is done by moving the decimal point three places to the right. (Multiply both sides by 1000)

\begin{align*}1000x=2018.181818\end{align*}

2.018181818 The repeating digits must be placed to the right of the decimal point. The decimal point and the repeating digits are separated by a zero. The decimal point must be moved one place to the right. (Multiply both sides by 10)

\begin{align*}10x=20.18181818\end{align*}

\begin{align*}& 1000 x=2018.181818 \qquad \text{These are the two equations that must be subtracted.}\\ & \underline{-10x=20.18181818}\\ & \frac{990x}{990}=\frac{1998}{990} \qquad \qquad \quad \ \ \text{Solve for} \ x.\\ & \quad \ \ x=\frac{1998}{990}\end{align*}

Use your calculator to simplify the fraction.

\begin{align*}x=\frac{1998}{990}\end{align*}

\begin{align*}x=\frac{111}{55}\end{align*}

The fraction is in the form of \begin{align*}\frac{a}{b}\end{align*}. The decimal number 2.018181818 can be expressed as the fraction \begin{align*}\frac{111}{55}\end{align*}.

2. \begin{align*}\frac{15}{11}=15 \div 11\end{align*}

Long division must be done to express the fraction as a decimal number.

\begin{align*}& \overset{ \ \ 1.363636}{11 \overline{ ) {15.000000 \;}}}\\ & \underline{- \ \ 11 \;}\\ & \qquad 40\\ & \underline{- \;\;\;\;\;33}\\ & \ \qquad \ 70\\ & \underline{- \;\;\;\;\;\;\;66}\\ & \ \qquad \quad 40\\ & \underline{- \;\;\;\;\;\;\;\;\;\;33}\\ & \ \qquad \quad \ \ 70\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;66}\\ & \ \qquad \qquad 40\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;\;\;33}\\ & \ \qquad \qquad \ \ 70\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;66}\\ & \ \qquad \qquad \quad \ 4\\\end{align*}

\begin{align*}\boxed{\frac{15}{11}=1.363636}\end{align*}

3. The number of tablets is given as a mixed number.

\begin{align*}& 2 \frac{3}{4}\\ & \frac{3}{4}=3 \div 4\\ & \overset{ \ \ 0.75}{4 \overline{ ) {3.00 \;}}}\\ & \underline{- 2 \ 8 }\\ & \quad \ \ 20\\ & \underline{- \;\;\;20} \qquad \text{Change the fraction to a decimal number.}\\ & \qquad \ 0\end{align*}

The number of grams of potassium was given as a decimal number. The answer should also be expressed as a decimal number.

The number of tablets is 2.75.

The number of grams of potassium in these tablets is:

\begin{align*}2.75 \times 0.5=1.375 \ grams\end{align*}

Summary

In this lesson you learned how to distinguish between a terminating decimal number and a periodic decimal number. You also learned that periodic decimal numbers and terminating decimal numbers are rational numbers. In addition you learned that the set of non-periodic decimal numbers belong to the irrational numbers.

You were also shown that a given fraction can be changed to a decimal number by long division. If the division ended, the decimal number was a terminating decimal. If the division did not end and the decimal number had digits that formed a repeating pattern, then the decimal number was a periodic decimal.

If a fraction can be changed to a decimal number, then the decimal number can be changed to a fraction. You learned to change a periodic decimal number to a fraction by using a set of structured steps. Although this process requires careful calculations, a correct result can be obtained by following the presented steps. You also learned how to change a terminating decimal number to a fraction by changing the decimal number to a whole number and writing the whole number over the place value of the decimal number.

Problem Set

Express the following fractions in decimal form.

1. \begin{align*}\frac{1}{12}\end{align*}
2. \begin{align*}\frac{6}{11}\end{align*}
3. \begin{align*}\frac{3}{20}\end{align*}
4. \begin{align*}\frac{1}{13}\end{align*}
5. \begin{align*}\frac{3}{8}\end{align*}

Express the following decimal numbers in the form \begin{align*}\frac{a}{b}\end{align*}.

1. 0.325
2. 3.72727272
3. 0.245454545
4. 0.618
5. 0.36363636

Complete the following table.

Wire Fraction Thickness (inches) Decimal Thickness (inches)
A \begin{align*}\frac{5}{64}\end{align*}
B \begin{align*}\frac{11}{32}\end{align*}
C \begin{align*}\frac{1}{20}\end{align*}
D \begin{align*}0.0703125\end{align*}
E \begin{align*}0.1875\end{align*}

Express the following fractions...

1. \begin{align*}\frac{1}{12}\end{align*} Do long-division

\begin{align*}& \overset{ \ \ 0.083333}{12 \overline{ ) {1.000000 \;}}}\\ & \underline{- \ \ 0 \;}\\ & \quad \ \ 10\\ & \underline{- \;\;\;\;\;0}\\ & \qquad 100\\ & \underline{- \;\;\;\;\;\;\;96}\\ & \qquad \quad \ 40\\ & \underline{- \;\;\;\;\;\;\;\;\;36}\\ & \qquad \qquad 40\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;\;36}\\ & \qquad \qquad \ \ 40\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;\;\;\;36}\\ & \qquad \qquad \quad \ 40\\ & \underline{- \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ 36}\\ & \qquad \qquad \qquad 4\end{align*}

1. \begin{align*}\frac{3}{20}\end{align*} Do long-division

\begin{align*}& \overset{ \ \ 0.15}{20 \overline{ ) {3.00 \;}}}\\ & \underline{- \ 0 \;}\\ & \quad \ 30\\ & \underline{- \;\;20}\\ & \quad \ 100\\ & \underline{- \;\;100}\\ & \qquad \ 0\end{align*}

1. \begin{align*}\frac{3}{8}\end{align*} Do long-division

\begin{align*}& \overset{ \ \ 0.375}{8 \overline{ ) {3.000 \;}}}\\ & \underline{- \ 0 }\\ & \quad 30\\ & \underline{- \;24}\\ & \quad \ \ 60\\ & \underline{- \;\;\;56}\\ & \qquad 40\\ & \underline{- \;\;\;\;40}\\ & \qquad \ 0\end{align*}

Express the following decimal numbers...

\begin{align*}& 0.325\\ & 0.325 \times 1000 =325 \qquad \text{Change the decimal to a whole number}\\ &=\frac{325}{1000} \qquad \qquad \qquad \quad \text{Express the decimal as a fraction with 1000 as the denominator.}\\ &= \frac{13}{40} \qquad \qquad \qquad \qquad \text{Use your calculator to simplify the fraction}\end{align*}

\begin{align*}& 0.245454545 \qquad \qquad \quad \text{The period is 45 and has a length of 2.}\\ & \text{Let} \ x=0.245454545\\ & 245.454545 \qquad \qquad \quad \ \ \text{Place the period to the left of the decimal point (Multiply by 1000)}\\ & 1000 x =245.454545\\ & 10 x =2.45454545\end{align*}

The repeating digits must be placed to the right of the decimal point. The decimal point and the repeating digits are separated by a zero. The decimal point must be moved one place to the right. (Multiply both sides by 10)

\begin{align*}& 1000 x = 245.454545\\ & \underline{-10 x = 2.45454545}\\ & \frac{990x}{990}=\frac{243}{990} \qquad \qquad \text{Subtract the two equations and solve for}\ x.\\ & x=\frac{27}{110}\end{align*}

Use your calculator to simplify the fraction.

\begin{align*}& 0.36363636 \qquad \qquad \quad \ \ \text{The period is 36 and has a length of 2.}\\ & \text{Let} x=0.363636363\\ & 36.363636 \qquad \qquad \qquad \ \text{Place the period to the left of the decimal point (Multiply by 100)}\\ & 100 x= 36.363636\\ & x=0.36363636 \qquad \qquad \text{The repeating digits must be placed to the right of the decimal point.}\end{align*}

\begin{align*}& 100 x=36.363636\\ & \ \ \underline{-x=0.36363636}\\ & \ \frac{99x}{99}=\frac{36}{99} \qquad \qquad \qquad \ \text{Subtract the two equations and solve for} \ x.\\ & \quad \ x=\frac{4}{11}\end{align*}

Use your calculator to simplify the fraction.

Complete the following...

1. \begin{align*}\frac{5}{64}\end{align*} Do long-division

\begin{align*}& \overset{ \ \ 0.078125}{64 \overline{ ) {5.000000 \;}}}\\ & \underline{- \ 0 }\\ & \quad 500\\ & \underline{- \;448}\\ & \quad \ \ 520\\ & \underline{- \;\;\;512}\\ & \qquad \ \ 80\\ & \underline{- \;\;\;\;\;\;\;64}\\ & \qquad \ \ 160\\ & \underline{- \;\;\;\;\;\;\;128}\\ & \qquad \quad \ 320\\ & \underline{- \;\;\;\;\;\;\;\;\;320}\\ & \qquad \qquad \ 0\end{align*}

1. \begin{align*}\frac{1}{20}\end{align*} Do long-division

\begin{align*}& \overset{ \ \ 0.05}{20 \overline{ ) {1.00 \;}}}\\ & \underline{- \;\; 0 }\\ & \quad \ 100\\ & \underline{- \;\; 100}\\ & \qquad \ 0\end{align*}

1. 0.1875 Move the decimal to the right of the decimal number. 1875 Express the number as 1875 over 1000. \begin{align*}\frac{1875}{1000}\end{align*} Use your calculator to simplify the fraction. \begin{align*}\frac{1875}{1000}=\frac{3}{16}\end{align*}

## Summary

In this lesson you have learned the difference between rational and irrational numbers. Rational numbers are numbers that can be expressed in the form \begin{align*}\frac{a}{b},b \ne 0\end{align*}. These fractions can then be changed to decimal numbers by dividing the denominator into the numerator. The result of the long-division was either a decimal number that ended (terminating decimal number) or a decimal number that had digits that repeated in a definite pattern (Periodic decimal numbers).

Irrational numbers are numbers that are made up of the set of non-terminating, non-periodic decimal numbers.

You also learned how to change a decimal number to a fraction. Changing a terminating decimal number to a fraction involved expressing the decimal as a whole number and writing this number over the denominator that represented the place value of the last digit of the decimal number. You then learned to simplify the fraction by using the TI83 calculator. Changing a periodic decimal number to a fraction involved applying a series of structured steps. If you carefully followed the steps and then used your calculator to simplify the fraction, your answer was always a rational number.

Irrational numbers cannot be expressed in the form \begin{align*}\frac{a}{b},b \ne 0\end{align*}.

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