# Math Lesson 3.5.2 - The Meaning of Decimals. Converting Fractions into Decimals without a Calculator

Welcome to our Math lesson on The Meaning of Decimals. Converting Fractions into Decimals without a Calculator, this is the second lesson of our suite of math lessons covering the topic of Converting Fractions to Decimals and Vice-versa, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

## The Meaning of Decimals. Converting Fractions into Decimals without a Calculator

Decimals are an alternative representation of decimal mixed numbers. They are made of two parts: the left part which shows the whole part of the corresponding decimal mixed number and the right part which shows the numerator of the fractional part of the corresponding decimal mixed number. A dot known as decimal point separates the whole and decimal part of such numbers.

In other words, the decimal mixed number 3 7/10 is alternatively written as 3.7 where 3 shows the whole part of the corresponding mixed number and 7 the numerator of the fractional part of this number. In symbols, we write

a b/10 = a.b
a bc/100 = a.bc
a bcd/1000 = a.bcd

and so on.

The number of zeroes in the denominator of the decimal mixed number determines the number of digits after the decimal point (we say "the number of decimal places"). Thus, if the denominator of the mixed number is 10, the corresponding decimal has one decimal place, if the denominator of the mixed number is 100, the corresponding decimal has two decimal places and so on.

### Example 3

Express the following mixed numbers as decimals.

a. 15 9/10
b.7 29/50
c. 100 1/8

### Solution 3

1. Since the denominator of this mixed number is 10, the corresponding decimal has one digit after the decimal point. Thus, we have
15 9/10 = 15.9
2. It is impossible to write the denominator of this mixed number as 10 but we can write it as 100 by multiplying up and down by 2. As a result, the corresponding decimal will have two digits after the decimal place. Thus, we have
7 29/50 = 7 (29 × 2)/(50 × 2) = 7 58/100 = 7.58
3. This time the denominator can neither be 10 nor 100 but only 1000. Hence, the corresponding decimal will have three decimal places. We have
100 1/8 = 100 (1 × 125)/(8 × 125) = 100 125/1000 = 100.125

As for the other fractions that cannot turn into decimal fractions, we apply the normal division of their numerator and denominator using the division method explained in tutorial 1.3. In most cases, this division gives an infinite number of digits after the decimal place but if you look them carefully, you will detect a kind of recurrence, i.e. the pattern is repeated after a number of digits. For example, when written as decimal, the fraction 3/7 becomes

3/7 = 3 ÷ 7 = 0.428571428571428571

As you see, the recurring part highlighted in yellow is repeated an infinity number of times as this decimal has no end. When we are completing the division by using the method described in tutorial 1.3 and we notice the repetition of two digits, it is quite probable that the recurrence has already began to appear. Let's consider an example in this regard.

### Example 4

Express the following fractions as decimals.

a. 20/6
b. 34/11

### Solution 4

1. We have Hence, the result is a recurrent decimal, 3.333333. which continues to infinity. We use a horizontal line above the recurring part to indicate that that part of the number is repeated continuously. Therefore, instead of writing 3.33333. we write 3.3 to indicate the recurrence.
2. We have Hence, the result is a recurrent decimal, 3.0909. which continues up to infinity. We use a horizontal line above the recurring part to indicate that that part of the number is repeated continuously. Therefore, instead of writing 3.0909. we write 3.09 to indicate the recurrence.

## More Converting Fractions to Decimals and Vice-versa Lessons and Learning Resources

Fractions Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
3.5Converting Fractions to Decimals and Vice-versa
Lesson IDMath Lesson TitleLessonVideo
Lesson
3.5.1Decimal Fractions
3.5.2The Meaning of Decimals. Converting Fractions into Decimals without a Calculator
3.5.3Converting Decimals to Fractions
3.5.4Converting Recurrent Fractions into Decimals

## Whats next?

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