- Decimals are numbers with one visible point somewhere in the number. This point is called decimal point.
- The decimal point is denoted by a dot (.).
- One way of representing a part of a whole is through decimal numbers.

**Example:** \(0.33,\;2.05,\;3.71\)

Decimal numbers consist of wholes, a decimal point and parts of whole.

**For example:** \(2.33\)

To the left of the decimal point are the wholes and to the right of the decimal point are the parts of whole.

- Place value refers to the location of any digit in a number.
- In a number, the same digit can have different place values.
- Tabular representation of the location of each digit in a number is called the place value chart.
- Take any number and represent it in place value chart.

**For example:** \(59022.4056\)

**In the place value chart, location of wholes:**

(Starting from left of the decimal point)

\(2\) at ones place

\(2\) at tens place

\(0\) at hundreds place

\(9\) at thousands place

\(5\) at ten thousands place

**Location of parts:**

(Starting from right of the decimal point)

\(4\) at tenths place

\(0\) at hundredths place

\(5\) at thousandths place

\(6\) at ten thousandths place

A Ones

B Tens

C Hundredths

D Tenths

- Grids help us to understand that decimals are parts of the wholes.
- Consider the following situations-

(1) In a grid of \(10\) squares, if all the squares are shaded, then the grid represents the whole number \(1\).

(2) In the same grid of \(10\) squares, if only \(1\) square is shaded, then the shaded square represents one tenth of the grid.

(3) One fourth shaded part of the whole grid represents twenty five hundredths.

(4) Now, in a grid of \(100\) squares, if all the squares are shaded, then the grid represents the whole number \(1\).

(5) In the grid of \(100\) squares, if a column of \(10\) squares is shaded, then the shaded column of squares represents one tenth of the grid.

(6) In the same grid of \(100\) squares, if only \(1\) square is shaded, then the shaded square represents one hundredth of the grid.

**For example:** If we want to represent \(0.26\) using area model, then we will shade \(26\) squares out of the grid of \(100\) squares.

A \(67\)

B \(33\)

C \(0.33\)

D \(0.67\)

- Expanded form of a number is such where the number is stretched out.
- In an expanded form, we write the place values of each digit.

- Suppose, we want to represent the whole number \(240\), in the expanded form.

For this, we first represent the number in the place value chart.

Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
---|---|---|---|---|---|---|

2 | 4 | 0 |

In the place value chart, we put \(2\) at hundreds place, \(4\) at tens place, and \(0\) at ones place.

Now, we write it in words-

\(2\) Hundreds \(+\;4\) Tens \(+\;0\) Ones

\(=2\) times Hundreds \(+\;4\) times Tens \(+\;0\) times Ones

Now, the number of digits \(=200+40+0\)

This is the required form.

**Example:** \(0.24\)

We represent this number in the place value chart.

Hundreds | Tens | Ones | Tenths | Hundredths | Thousandths | |
---|---|---|---|---|---|---|

. |
2 | 4 |

In the place value chart, we put \(2\) at tenths place and \(4\) at hundredths place.

Now, we write it in words-

\(2\) Tenths \(+\,4\) Hundredths

Now, in numbers

\(0.2+0.04\)

This is the required form.

**Note:** Zeros are placed to make sure that the digit has the correct value because if we write \(0.4\), it will show tenths place and not of hundredths.

A \(0.5+0.4+0.8\)

B \(0.5+0.04+0.008\)

C \(5+4+8\)

D \(500+40+8\)

- Decimals can be represented on a number line.
- To represent them, we move in the right direction from zero on the number line.
- First, find the wholes and then the parts of the whole on the number line.

**For example:** We want to represent \(3.7\) on the number line.

- To find the decimal number \(3.7\) on the number line, find the whole \(3.\)
- To find \(3\) on the number line, move in the right direction from zero and mark the whole at \(3.\)

- Now, we have to represent the part \(0.7\) on the number line.
- To represent \(0.7\), divide the interval between \(3\) and \(4\) into \(10\) equal sections, each having a scale of \(0.1\)

- Now, start from \(3\) and count \(7\) slashes by moving in the right direction. Mark the 7th slash.

We have reached the whole \(3\) and the parts \(0.7(7×0.1)\) on the number line.

So, by combining these, we get the decimal number \(3.7\) on the number line.

A \(1.4\)

B \(1.5\)

C \(1.6\)

D \(0.8\)

- To represent decimal numbers on a number line, we first find the wholes and then the parts of the whole by moving in the right direction from zero.
**For example:**We want to represent \(1.45\) on the number line.

- Here, we first find the whole \(1.\) To find \(1\) on the number line, start from \(0\) and move in the forward direction and mark the whole \(1.\)

- So, we have reached the whole \(1.\)
- Now, we need to find the tenth part, i.e. \(0.4\)
- To find \(0.4\) on the number line, divide the interval between \(1\) and \(2\) into \(10\) equal sections having a scale of \(0.10\) each.

- Start from \(1,\) and count \(4\) slashes by moving in the right direction and mark the 4th slash.

- We have reached up to the tenth part, i.e. \(1.4\)
- Now, we need to find the hundredth part, i.e. \(0.45\)
- To find \(0.45\) on the number line, divide the interval between \(0.40\) and \(0.50\) into \(2\) equal sections of scale \(0.5\) each.

- Start from \(0.40\), and count \(1\) slash by moving forward and mark the 1st slash.

- We have reached up to \(1.45\) on the number line.
- So, we have successfully learned to represent a decimal \((1.45)\) up to the hundredths place on a number line.

A Point A

B Point C

C Point D

D Point B

- We can read or write decimals in words with the help of a place value chart.
- Ignore the decimal point and read the number as a whole number.
- Then add the place value of the last digit.
**For example:**Write \(0.4512\) in words.- First, ignore the decimal point and write the number as a whole number, i.e.

Four thousand five hundred twelve

- Write the decimal number in the place value chart.

Decimal point | Tenths | Hundredths | Thousandths | Ten-thousandths |
---|---|---|---|---|

. |
4 | 5 | 1 | 2 |

- Now, add the place value of the last digit.

Four thousand five hundred twelve ten-thousandths

**Example:**(1) \(0.5=\) Five tenths

(2) \(0.045=\) Forty-five thousandths

(3) \(0.05=\) Five hundredths