Chapter Notes - Maths - Decimals

What are Decimals?

Decimals are a special way to write fractions as numbers. They use a dot called a decimal point to separate the whole number part from the fractional part.

Understanding Decimals

Imagine you have a whole pizza, and you cut it into 10 equal slices. If you eat 1 slice, you have eaten one-tenth of the pizza. We write this as 1/10 or 0.1 in decimals.

Decimals help us show small parts of a whole. For example:

• 0.5 means half (like 5 out of 10 slices).
• 0.25 means a quarter (like 25 out of 100).

The numbers after the decimal point tell us how much of a whole something is.

We use decimals every day. For example:

• ₹1.50 means 1 rupee and 50 paise.
• 2.5 metres means 2 metres and half a metre more.

• Decimals help us divide things into smaller and more exact parts!

Tenths

There are ten children attending Pooja’s birthday party. Her mother cuts the cake into 10 equal parts. Each part is one-tenth of the whole cake. 

Look at the following figures
Each of the rectangles has been divided into ten equal parts. How many parts are shaded? The fraction is written below each figure.
The denominator of these fractions is 10. These are special types of fractions called decimal fractions.

A fraction with denominator 10, 100, 1000, ... is called a decimal number or simply decimal.

• The fraction 1 / 10 (one-tenth) can be written as 0.1. 
• Similarly, 4 / 10 is written as 0.4 and 9/10 is written as 0.9.
• The numerals 0.1, 0.4 and 0.9 are called decimal numerals or simply decimals. The dot in 0.1, 0.4 and 0.9 is called a decimal point. 0.1 can be read in two ways as under:
  This system of writing numbers with a decimal point is called the decimal system.

EduRev Tips: 

When writing decimals, always place a zero before the decimal point if there’s nothing before it. For example:

0.5 (correct)
.5 (incorrect)

So, 1 / 10 is 0.1, 2 / 10 is 0.2, 3 / 10 is 0.3 and so on.

Hundredths

There are 100 small squares in the large square. 1 small square is one -hundredth or 1 / 100 of the large square. The shaded part consists of 9 small squares. 9 small squares make “nine-hundredths” of the large square. We say that the shaded part is 9 / 100 or 0.09.
Since there are two zeros in the denominator of 9 / 100, therefore, there are two digits to the right of the decimal point in 0.09.

Now, look at the adjacent figure:
The shaded part is “thirty-three” hundredths of the large square.
Notice that there are two digits after the decimal point when writing a decimal to show hundredths.

Reading a Decimal Number

The following diagrams are drawn to help you gain complete understanding:

0.04 = 0 tenth and 4 hundreds

0.10 = 1 tenth and 0 hundredths
0.45 = 4 tenths and 5 hundredths
0.83 = 8 tenths and 3 hundredths

Decimals and Whole Numbers

Whole numbers can also be written using a decimal point.
For example: 1 = 1.0; 3 = 3.0; 5 = 5.0; 7 = 7.0; 9 = 9.0
The zero after the decimal point means there are 0 tenths.

More than One (Mixed Numbers and Mixed Decimals)

You can use a mixed decimal to name a mixed number.  and 3.6 are both read as “three and six-tenths.”

The table below illustrates different ways of reading and writing tenths. The zero after the decimal point means there are 0 tenths.

The decimal numbers can also be shown on a number line. The following picture shows a number line which has been divided into tenths.

Writing Mixed Decimals for a Mixed Number

A mixed decimal can be written for a mixed number as under:

From the above examples, we can see that:

A decimal number is made up of two parts: a whole number part and a decimal part separated by a decimal point. The number of digits after the decimal point is known as the number of decimal places.

For example: 31.9 has one decimal place.
583.28 has two decimal places.

Decimal Fractions in the Place Value Chart

The place value chart that you have studied till now can be shown as follows:Starting from the left and moving towards the right,

As we move towards the right, each place value becomes one-tenth of the previous one.

Thus, the place value chart extended beyond ones to the right is as given below.
For example, the decimal number 683.45 can be shown in the place value chart as:

Example 1: Find the place value of each digit in 625.37.

Expanded Form

Let us write the expanded form for the number 634.95.
634.95 = 6 hundreds + 3 tens + 4 ones + 9 tenths.
This is the expanded form of the number 634.95.
We give some more examples to help you get better understanding.

Example: Write each decimal numeral in expanded form.
(a) 5.86
(b) 65.26

Sol: (a) 5.86 = 5 ones + 8 tenths + 6 hundredths

(b) 65.26 = 6 tens + 5 ones + 2 tenths + 6 hundredths

Equivalent Decimals

The shaded square on the right shows that:
0.2 = 0.20
Thus, we can rename a decimal by writing as many zeros as we like after the last digit in a decimal number.
We have, 0.5 = 0.50 = 0.500 = 0.5000 etc.

Tenths can be renamed as hundredths as well as thousandths: 0.1 = 0.10 = 0.100.
Hundredths can be renamed as thousandths: 0.08 = 0.080; 0.37 = 0.370.