Chapter 9 of 14 · Fractions
Chapter 8. Decimal fractions and decimal notation
Fractions with denominator ten, one hundred or one thousand give decimal notation: this chapter makes the move in both directions.
A family of fractions apart
Certain fractions behave differently from the others, not by nature, but through the choice of denominator. They are the ones whose denominator is ten, one hundred, one thousand, and so on. They are called decimal fractions, and they serve as a bridge between fractions and the decimal notation that everybody uses for prices and measurements.
The reason for this special standing lies in our number system. Our way of writing numbers rests on groupings of ten: ten units make one ten, ten tens make one hundred. The same ratio of ten carries on in the other direction: one unit shared into ten gives one tenth, one tenth shared into ten gives one hundredth. Decimal notation is nothing other than the extension of the place value table to the right of the units.
This chapter therefore brings in no new rule about fractions. It applies everything that has come before to one particular family, and it draws a shorter writing from it.
What this chapter makes you able to do
Four gestures, in both directions of conversion.
- Recognise a decimal fraction.
- Write a decimal fraction as a number with a decimal point.
- Write a number with a decimal point as a decimal fraction.
- Add two decimal fractions with different denominators.
The relations between units, tenths and hundredths
Share one unit into ten equal parts: each part is a tenth. Now share each of those tenths into ten: you obtain a hundred parts for one unit, therefore hundredths. Two equalities follow from this directly, and they have to be held before anything else: ten tenths are worth one unit, and ten hundredths are worth one tenth.
The same construction carries on: ten thousandths are worth one hundredth, and a thousand thousandths are worth one unit. It is always the same ratio of ten that links two neighbouring place values, exactly as between units, tens and hundreds.
These equalities make it possible to rewrite any decimal fraction in another place value. Three tenths are worth thirty hundredths, since each tenth is worth ten hundredths. It is the rule of chapter 4 applied to a case where the factor is always ten.
| Place value | What it is worth in units | Written as a fraction |
|---|---|---|
| The ten | Ten units | Not written as a fraction: it is a whole number. |
| The unit | The reference whole | 1, or 10/10, or 100/100. |
| The tenth | One unit shared into ten | 1/10 |
| The hundredth | One unit shared into a hundred, or one tenth shared into ten | 1/100 |
Worked example. Writing 318/100 with a decimal point
The problem: give the decimal notation of the decimal fraction 318/100, justifying each step.
- Read the fraction
The denominator is 100: we count in hundredths. The fraction is therefore worth three hundred and eighteen hundredths. - Take out the whole units
One hundred hundredths are worth one unit. In three hundred and eighteen hundredths, we find three times one hundred hundredths, that is to say three units, and eighteen hundredths are left. - Write the breakdown
The fraction is therefore worth 3 + 18/100, that is to say three units and eighteen hundredths. - Separate the tenths from the hundredths
Eighteen hundredths are worth one tenth plus eight hundredths, since ten hundredths make one tenth. So we have three units, one tenth and eight hundredths. - Set down the decimal notation
Each place value takes its position to the right of the decimal point, in order: the tenths first, the hundredths next. The number is written 3.18. The decimal point does not separate two numbers: it marks the place of the units digit.
The road back, and addition
To write a number with a decimal point as a decimal fraction, we look at the place value of the last digit. In 0.7 the last digit is in the tenths place, so the number is worth 7/10. In 2.45 the last digit occupies the hundredths place, so the number is worth 245/100. The denominator is read off the position, never off the length of the writing.
The addition of two decimal fractions in different place values brings in nothing new: it is the putting over the same denominator of chapter 6, with a factor that is always ten or one hundred. To add 3/10 and 25/100, we write three tenths in hundredths, that is 30/100, then we add: thirty hundredths plus twenty-five hundredths make fifty-five hundredths, that is 55/100.
Finish one off yourself: that is the exercise along the way below, and its answer is at the end of the volume.
One point of vocabulary deserves to be set down, because it prevents a lasting misunderstanding. Every decimal fraction can be written with a decimal point, but not every fraction can be written with a decimal point in a finite way: a third can be written neither in tenths, nor in hundredths, nor in thousandths, since 3 divides none of those powers of ten. Decimal notation therefore does not replace fractions, it covers only a part of them.
Exercise along the way, chapter 8. The calculation 7/10 + 4/100
Set in the course of the chapter: it is the putting over the same denominator of chapter 6, with a factor of ten.
- Work out 7/10 + 4/100: bring both fractions into the same place value, add them, then give the decimal notation of the result.
A trap named by studies on the teaching of decimals: treating the number written to the right of the decimal point as an independent whole number. This habit comes from everyday language, where a price of three euros twenty-five is said in the same way as a clock time of three twenty-five, as though two separate counts were being set side by side. It leads to believing that 31.7 is smaller than 31.28 because 7 is smaller than 28. The remedy is to come back to the place values: 7 tenths are worth 70 hundredths, which goes past 28 hundredths.
Exercise 8.1. Recognising and converting
Answer by justifying through the place value, and not through the number of digits.
- Among 3/10, 4/7 and 25/100, which ones are decimal fractions? Justify your answer.
- Write 9/10 with a decimal point.
- Write 0.63 as a decimal fraction.
Exercise 8.2. Changing place value
This exercise crosses this chapter with chapter 4: state each time which number you are multiplying by.
- Write 4/10 in hundredths.
- Write 7/100 in thousandths.
- How many hundredths are three tenths and two hundredths worth?
Exercise 8.3. Adding and breaking down
This exercise crosses this chapter with chapters 2 and 6. Give the result as a fraction, then in decimal notation.
- Work out 5/10 + 30/100.
- Work out 250/100 + 7/10, then break the result down into a whole number plus a fraction smaller than the whole.
- One bottle holds 75/100 of a litre, another 4/10 of a litre. What total quantity do these two bottles hold, and does it go beyond one litre?
Self-check, chapter 8
Answer without turning back, then compare with the answer.
- What is it that makes a fraction decimal?
- What exactly does the decimal point mark in a number such as 3.18?
- Why can a third not be written exactly with a decimal point and a finite number of digits?
What to remember from this chapter
- A decimal fraction is a fraction with denominator ten, one hundred, one thousand, and so on.
- Ten tenths are worth one unit, and ten hundredths are worth one tenth: it is the same ratio of ten as between the place values of whole numbers.
- Decimal notation is the extension of the place value table to the right of the units, and the decimal point marks the place of the units.
- Adding two decimal fractions amounts to bringing them into the same place value, then adding.
- Every decimal fraction is written with a decimal point, but not every fraction is decimal.