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\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
International Baccalaureate · PYP 4

Decimals

⌚ ~234 min ▢ 27 blocks ✓ 119 exercises ➣ Prerequisites : Numbers to 999 999, Fractions
I Decimal Fractions

To measure a length shorter than one unit, the unit must be divided into equal parts. Consider a ribbon shorter than one metre. The unit (one metre) is divided into 10 equal parts; each part is one tenth of the unit, written \(\dfrac{1}{10}\). The ribbon covers 7 of these parts, so its length is \(\dfrac{7}{10}\) of a metre.
Figure
On the number line, the segment from 0 to 1 is the unit. Dividing it into 10 equal parts places \(\dfrac{7}{10}\) exactly:
Figure
Dividing the unit into 10 equal parts matches our base-ten numeration: ten parts make one unit, just as ten ones make one ten. The fractions built on this division are so useful that they receive a name of their own.

Definition — Decimal Fraction
A decimal fraction is a fraction whose denominator is \(10\), \(100\), \(1\,000\), …
  • \(\dfrac{7}{10}\) is read “seven tenths”: the unit is divided into 10 equal parts.
  • \(\dfrac{23}{100}\) is read “twenty-three hundredths”: dividing each tenth into 10 equal parts divides the unit into 100 equal parts.
Example
The unit, from 0 to 1, is divided into 10 equal parts.
  1. Read the decimal fraction located at the point \(A\).
    Figure
  2. The segment from \(0\) to \(\dfrac{1}{10}\) is itself divided into 10 equal parts. Read the decimal fraction located at the point \(B\).
    Figure

  1. The unit is divided into \(\textcolor{colorprop}{10}\) equal parts, so each part is \(\dfrac{1}{10}\). The point \(A\) is located \(\textcolor{colordef}{3}\) parts from 0, so \(A=\dfrac{\textcolor{colordef}{3}}{\textcolor{colorprop}{10}}\).
  2. Each tenth is divided into 10 equal parts, so the unit is divided into \(10\times 10=\textcolor{colorprop}{100}\) equal parts: each part is \(\dfrac{1}{100}\). The point \(B\) is located \(\textcolor{colordef}{7}\) parts from 0, so \(B=\dfrac{\textcolor{colordef}{7}}{\textcolor{colorprop}{100}}\).

Proposition — Whole Number and Decimal Fraction Less than 1
Since 10 tenths make one unit, \(\dfrac{10}{10}=1\). Every decimal fraction can therefore be written as the sum of a whole number and a decimal fraction less than 1, and conversely: $$\dfrac{47}{10} = 4 + \dfrac{7}{10} \qquad\text{and}\qquad 4 + \dfrac{7}{10} = \dfrac{47}{10}.$$
Figure
Example
  1. Write \(\dfrac{35}{10}\) as the sum of a whole number and a decimal fraction less than 1.
  2. Write \(2+\dfrac{6}{10}\) as a single decimal fraction.

  1. \(\begin{aligned}[t] \dfrac{35}{10} &= \dfrac{30}{10}+\dfrac{5}{10} &&\text{(split off the whole tens of tenths)}\\ &= 3+\dfrac{5}{10} &&\left(\dfrac{30}{10}=3\right) \end{aligned}\)
  2. \(\begin{aligned}[t] 2+\dfrac{6}{10} &= \dfrac{20}{10}+\dfrac{6}{10} &&\left(2=\dfrac{20}{10}\right)\\ &= \dfrac{26}{10} &&\text{(add the numerators)} \end{aligned}\)

Example
Decimal fractions with the same denominator are added like any fractions with a common denominator: add the numerators and keep the denominator. Compute:
  1. \(\dfrac{3}{10}+\dfrac{4}{10}\)
  2. \(\dfrac{26}{100}+\dfrac{24}{100}\)

  1. Figure
    \(\begin{aligned}[t] \dfrac{3}{10}+\dfrac{4}{10} &= \dfrac{3+4}{10} &&\text{(same denominator: add the numerators)}\\ &= \dfrac{7}{10} \end{aligned}\)
  2. \(\begin{aligned}[t] \dfrac{26}{100}+\dfrac{24}{100} &= \dfrac{26+24}{100} &&\text{(same denominator: add the numerators)}\\ &= \dfrac{50}{100}\\ &= \dfrac{5}{10} &&\text{(50 hundredths make 5 tenths)} \end{aligned}\)

Because their denominators are \(10\), \(100\), \(1\,000\), …, decimal fractions have a shorter writing that uses a decimal point: for example, \(\dfrac{7}{10}\) is also written \(0.7\). This writing with a decimal point is that of the decimal numbers: a decimal number can be written as a decimal fraction or, more briefly, with a decimal point.
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Skills to practice
  • Reading and Writing Decimal Fractions
  • Adding Decimal Fractions
II What Are Decimals?

In our number system, the position of a digit tells us its value. Consider the number 235.
\(235=\) Figure
It is made of three parts:
  • The 5 is in the Ones place, meaning 5 ones (\(5\)).
  • The 3 is in the Tens place, meaning 3 tens (\(30\)).
  • The 2 is in the Hundreds place, meaning 2 hundreds (\(200\)).
Our number system follows a pattern based on 10.
  • When we move left \(\leftarrow\), each place is 10 times bigger. (10 ones make a ten, 10 tens make a hundred).
  • When we move right \(\rightarrow\), each place is 10 times smaller. (A hundred divided by 10 is a ten, a ten divided by 10 is a one).
What happens if we keep moving right? What is a one divided by 10?
We get a piece called a tenth. A tenth is a fraction, written as \(\dfrac{1}{10}\).
To write numbers that include these smaller pieces, we use a decimal point to separate the whole part from the fractional part.
\(35.4\) = Figure
This number means 3 tens, 5 ones, and 4 tenths.

Definition — Decimal Number
A decimal number is a number that can be written with a whole part and a fractional part separated by a decimal point, using a finite number of digits after the point. We can represent a decimal number in different ways:
  • Standard Form:
    \(35.4\)
  • Expanded Form $$\begin{aligned} 3\text{ tens} &+& 5\text{ ones} &+& 4\text{ tenths} \\ 30 &+& 5 &+& \frac{4}{10} \end{aligned}$$
  • Place Value Table:
    Tens Ones . Tenths
    3 5 . 4
  • Base Ten Cubes:
    Figure
Definition — The Hundredth
The pattern of our number system does not stop at the tenths: each place is still 10 times smaller than the place on its left.
  • A one divided by 10 is a tenth: \(\dfrac{1}{10} = 0.1\).
  • A tenth divided by 10 is a new, smaller piece called a hundredth. A hundredth is written \(\dfrac{1}{100}\) as a fraction, or \(0.01\) as a decimal.
A hundredth is one tenth of one tenth. Going back up:
  • 10 hundredths make 1 tenth: \(\dfrac{10}{100} = \dfrac{1}{10}\).
  • 100 hundredths make 1 one: \(\dfrac{100}{100} = 1\).
The place-value table gains one more column on the right. For the number \(2.64\) (2 ones, 6 tenths, 4 hundredths):
Ones . Tenths Hundredths
2 . 6 4
\(2.64\) = Figure
Example
Write in decimal form:
  1. \(\dfrac{34}{100}\)
  2. \(\dfrac{207}{100}\)

  1. $$\begin{aligned}[t] \frac{34}{100} &= \frac{30}{100} + \frac{4}{100} && \text{(split the hundredths)} \\ &= \frac{3}{10} + \frac{4}{100} && \text{(30 hundredths are 3 tenths)} \\ &= 0.34 && \text{(3 tenths and 4 hundredths)} \end{aligned}$$
    \(\dfrac{34}{100} = \) Figure
  2. $$\begin{aligned}[t] \frac{207}{100} &= \frac{200}{100} + \frac{7}{100} && \text{(split the hundredths)} \\ &= 2 + \frac{7}{100} && \text{(200 hundredths are 2 ones)} \\ &= 2.07 && \text{(0 tenths and 7 hundredths)} \end{aligned}$$
    Ones . Tenths Hundredths
    2 . 0 7

Proposition — Different Writings of One Number
The same decimal number can be written in several ways. All these writings name exactly the same number. For \(42.48\): $$42.48 = \frac{4248}{100} = 40 + 2 + \frac{4}{10} + \frac{8}{100}$$
  • From the standard form to a decomposition: read each digit by its place. In \(42.48\), the 4 on the left is worth \(40\), the 2 is worth \(2\), the 4 after the point is worth \(\frac{4}{10}\), and the 8 is worth \(\frac{8}{100}\).
  • From the standard form to the hundredths: count everything in hundredths. Each one is 100 hundredths and each tenth is 10 hundredths, so \(42.48\) is \(4248\) hundredths: \(42.48 = \frac{4248}{100}\).
  • From a decomposition back to the standard form: place each part in its column of the place-value table, then read the number.
The word name is one more writing of the same number: \(42.48\) is read “forty-two point four eight”.
Example
  1. Write \(36.07\) as a decimal fraction and as an expanded form.
  2. Write as a single decimal number: \(50 + 3 + \dfrac{9}{10} + \dfrac{2}{100}\).

  1. Expanded form: read each digit by its place. The zero means there are no tenths. $$\begin{aligned}[t] 36.07 &= 30 + 6 + \frac{7}{100} && \text{(3 tens, 6 ones, 0 tenths, 7 hundredths)} \end{aligned}$$ Decimal fraction: count everything in hundredths. $$\begin{aligned}[t] 36.07 &= \frac{3600}{100} + \frac{7}{100} && \text{(36 ones are 3600 hundredths)} \\ &= \frac{3607}{100} \end{aligned}$$
  2. $$\begin{aligned}[t] 50 + 3 + \frac{9}{10} + \frac{2}{100} &= 53 + 0.9 + 0.02 && \text{(value of each term)} \\ &= 53.92 \end{aligned}$$
    Tens Ones . Tenths Hundredths
    5 3 . 9 2

Example — Decimal Writing of Usual Fractions
Write in decimal form:
  1. \(\dfrac{1}{2}\)
  2. \(\dfrac{1}{4}\)
  3. \(\dfrac{3}{4}\)

The unit is the big square. It is divided into \(100\) small squares: each small square is one hundredth.
Figure
  1. Half of the \(100\) small squares is \(50\) small squares: $$\begin{aligned}[t] \frac{1}{2} &= \frac{50}{100} && \text{(half of 100 hundredths)} \\ &= \frac{5}{10} && \text{(50 hundredths are 5 tenths)} \\ &= 0.5 \end{aligned}$$
  2. Shared into \(4\) equal parts, the \(100\) small squares give \(25\) small squares each, since \(4\times 25=100\): $$\begin{aligned}[t] \frac{1}{4} &= \frac{25}{100} && \text{(a quarter of 100 hundredths)} \\ &= 0.25 && \text{(2 tenths and 5 hundredths)} \end{aligned}$$
  3. Three quarters are \(3\) times \(25\) small squares, that is \(75\) small squares: $$\begin{aligned}[t] \frac{3}{4} &= \frac{75}{100} && \text{(3 times 25 hundredths)} \\ &= 0.75 && \text{(7 tenths and 5 hundredths)} \end{aligned}$$
These three writings come back so often that they are known by heart: \(\dfrac{1}{2}=0.5\), \(\dfrac{1}{4}=0.25\) and \(\dfrac{3}{4}=0.75\).

Skills to practice
  • Understanding the Role of Zero
  • Identifying Place Values
  • Writing Decimal Numbers
  • Finding the Digit in a Place Value
  • Writing Decimal Numbers from Decimal Fractions
  • Writing Decimal Fractions from Decimal Numbers
  • Writing Decimal Numbers from Expanded Forms
  • Writing Decimal Numbers from Expanded Forms II
  • Writing Expanded Forms from Decimal Numbers
  • Writing a Decimal Fraction as a Whole Number Plus Tenths and Hundredths
  • Distinguishing the Digit from the Number of Tenths or Hundredths
III Decimals on a Number Line

Every decimal number has an exact position on the number line.
  • Finding Tenths
    If we take the space between two whole numbers, like 0 and 1, and divide it into 10 equal parts, each part is one tenth (0.1).
    Figure
  • Finding Hundredths
    If we magnify a small section, such as the one between 0 and 0.1, and divide it into 10 equal parts, each new part is one hundredth (0.01).
    Figure

Method — Placing a Decimal on a Number Line
To place a decimal like 2.27 on a number line:
  1. Find the start and end points. The number 2.27 is between \(2.2 = 2.20\) and \(2.3 = 2.30\).
  2. Draw the line and divide it into 10 equal parts. Each part will represent a hundredth.
  3. Count up from the start. The marks will be 2.21, 2.22, 2.23, and so on. Find the 7th mark to place 2.27.
Figure
Skills to practice
  • Identifying Decimal Numbers on a Number Line
  • Recognising the Point That Marks a Decimal
  • Marking a Decimal on a Number Line
  • Counting by Tenths and by Hundredths
IV Comparing and Ordering Decimals
Method — Comparing Decimals Step-by-Step
To find out which decimal number is bigger, follow these steps:
  1. Line them up: Place the numbers one above the other, making sure the decimal points are perfectly aligned.
  2. Make them the same length: Add zeros to the end of the shorter number until both have the same number of decimal places. This doesn't change their value.
  3. Compare from left to right: Starting from the left, compare the digits in each place value column. The first number with a larger digit is the larger number. If one whole part has more digits than the other, that number is the larger one.
Example
Compare 6.22 and 6.3.

  1. Line them up:
    \(6.22\)
    \(6.3\)
  2. Make them the same length: We add a zero to 6.3 to make it 6.30.
    \(6.22\)
    \(6.30\)
  3. Compare:
    • Ones place: Both have a 6.
    • Tenths place: The top number has a 2, the bottom has a 3.
    Since 3 is greater than 2, the number 6.30 is greater than 6.22. So, \(6.22 < 6.3\).
Figure\(\lt\)Figure

Method — Ordering Several Decimals
To order a list of decimal numbers, place them step by step:
  1. Write all the numbers with the same number of decimal places by adding zeros — this changes nothing and makes them easy to compare.
  2. Find the smallest (or the largest) and set it first.
  3. Repeat with the numbers that remain, until the list is used up.
The order is ascending when it goes from smallest to largest, descending the other way.
Example
Order the numbers \(3.7\), \(3.09\), \(3.71\) and \(3.2\) in ascending order.

$$\begin{aligned}[t] 3.70,\; 3.09,\; 3.71,\; 3.20 && \text{(same number of decimal places)} \\ 3.09 && \text{(smallest: its tenths digit is } 0\text{)} \\ 3.09 < 3.20 < 3.70 < 3.71 && \text{(then } 2 \text{ tenths, then } 70 < 71 \text{ hundredths)} \end{aligned}$$

Method — Comparing and Ordering Decimal Fractions
To compare two decimal fractions, write them with the same denominator, then compare the numerators: the larger numerator gives the larger fraction. Since one tenth is ten hundredths and one hundredth is ten thousandths, \(\dfrac{7}{10}=\dfrac{70}{100}\) and \(\dfrac{7}{100}=\dfrac{70}{1\,000}\). $$\dfrac{7}{10}=\dfrac{70}{100}\quad\text{and}\quad 70>65,\quad\text{so}\quad \dfrac{7}{10}>\dfrac{65}{100}.$$ One can also write them with a decimal point and compare the decimals: \(0.70>0.65\). To order several decimal fractions, write them all with the largest denominator, then order the numerators.
Skills to practice
  • Comparing Numbers
  • Ordering Decimals in Real-World Problems
V Bounding and Inserting Decimals
Method — Bounding and Inserting Decimals
  1. To bound a decimal between two consecutive whole numbers: the whole part, the whole number written before the decimal point, gives the lower bound, and the next whole number gives the upper bound. For \(6.85\): $$6\leq 6.85<7$$ The sign \(\leq\) reads “is less than or equal to”: the number can be equal to its lower bound (\(6\leq 6<7\)), never to its upper bound.
  2. To bound a decimal between two consecutive tenths: keep the digits up to the tenths place: they give the lower bound; the next tenth gives the upper bound. For \(6.85\): $$6.8\leq 6.85<6.9$$ Here too, the number can be equal to its lower bound: \(6.1\leq 6.1<6.2\).
    Figure
  3. To insert a decimal between two given numbers: first write both numbers with the same number of decimal places, then look for the numbers that lie between them: between \(2.3\) and \(2.6\) lie \(2.4\) and \(2.5\). Between two consecutive whole numbers, or between two consecutive tenths, no number with the same number of decimal places fits: write both numbers with one more decimal place. Between \(5.7=5.70\) and \(5.8=5.80\), we can insert \(5.71\), \(5.72\), …, \(5.79\).
Example
  1. Bound \(7.06\) between two consecutive whole numbers.
  2. Bound \(7.06\) between two consecutive tenths.
  3. Find three decimal numbers between \(9.6\) and \(9.7\).

  1. The whole part of \(7.06\) is \(7\), and the next whole number is \(8\), so: $$7\leq 7.06<8$$
  2. The digits of \(7.06\) up to the tenths place give \(7.0\): the tenths digit is \(0\). The next tenth is \(7.1\), so: $$7.0\leq 7.06<7.1$$ that is, \(7\leq 7.06<7.1\). On a half-line graduated in hundredths from \(7\) to \(7.1\), the number \(7.06\) is \(6\) graduations after \(7\):
    Figure
  3. We write both numbers with one more decimal place: \(9.6=9.60\) and \(9.7=9.70\). The new graduations \(9.61\), \(9.62\), …, \(9.69\) all lie between them. For example: $$9.6<9.62<9.65<9.68<9.7$$
    Figure

Example — Bounding and Inserting Decimal Fractions
  1. Bound \(\dfrac{347}{100}\) between two consecutive whole numbers, then between two consecutive tenths.
  2. Give a decimal fraction between \(\dfrac{3}{10}\) and \(\dfrac{4}{10}\).

  1. \(\dfrac{347}{100}=3.47\): its whole part is \(3\), so \(3<\dfrac{347}{100}<4\). In hundredths, the tenths \(\dfrac{34}{10}\) and \(\dfrac{35}{10}\) are \(\dfrac{340}{100}\) and \(\dfrac{350}{100}\), and \(340<347<350\): $$\dfrac{34}{10}<\dfrac{347}{100}<\dfrac{35}{10}.$$
  2. In hundredths, \(\dfrac{3}{10}=\dfrac{30}{100}\) and \(\dfrac{4}{10}=\dfrac{40}{100}\): between them lie \(\dfrac{31}{100}\), \(\dfrac{32}{100}\), …, \(\dfrac{39}{100}\). For example \(\dfrac{3}{10}<\dfrac{35}{100}<\dfrac{4}{10}\): \(\dfrac{35}{100}\) is exactly halfway.

Skills to practice
  • Bounding a Decimal Between Two Consecutive Whole Numbers
  • Bounding a Decimal Between Two Consecutive Tenths
  • Inserting a Decimal Between Two Numbers
  • Comparing, Bounding and Inserting Decimal Fractions
VI Going further
Skills to practice
  • Explaining Decimal Fractions and Digits
  • Justifying a Comparison
  • Reconstructing a Number from Clues
  • Problems
  • Quiz