## Basics

Decimal numeral system is a system for writing integer and non integer numbers. Decimal numeral system is
also called as **Base Ten Numeral Positional System**. It means that this system uses 10 as base.

So, decimal refers to notation of a number in decimal numeral system. The symbol used for its notation is
a dot ( . ) or comma ( , ).

English speaking countries use dot ( . ) notations to represent decimal numbers. Other countries use comma ( , ) notation for decimal numbers.

## Parts of a decimal number

A decimal number always has two parts, one is whole number part and the another part is decimal part. Both parts are always seperated by dot ( . ) or comma ( , ).

The part of the number which is written before the decimal is called as whole part and the part written after the decimal is called as the decimal part.

**Example 1: Write the whole part and decimal part of decimal number 1.3**

Here, 1 is the whole number part.

3 is the decimal part.

**Example 2: Write the whole part and decimal part of decimal number 23.14**

Here, the number 23 which is written before the decimal is the whole number part.

The number 41 which is written after the decimal is the decimal part.

## Place value chart of decimal numbers

The figure below is the place value chart which is used to find the value of a digit in a decimal number at a place. This chart also helps in how to read a decimal number and how to write an expanded form of a decimal number.

In the place value chart there are boxes on left and right sides of the decimal box. Each box on left and right side of the decimal box represent a place of a digit. Let’s understand with an example how to a decimal number is represented in using place value chart.

In the above chart, each place is ten times the value of next place value on its right. It means
place
value of a digit increases by 10 times when it moves by one place from right to left.

The value of ten’s place is 10 times the ones place. The value of hundreds place increases by 10
times of tens place.

**Representation of 683.125 on place value chart**

683.125 can be represented using the following place value chart.

The digits 683 which are present on left side of the decimal number 683.125 are shown in the chart in
the boxes which are present on the same left side of the decimal box.

The digits 125 which are present on right side of the decimal number 683.125 are shown in the boxes
which are on the right side of the decimal box.

So, when moving one place to the right inits digit, the place value of a digit decreases. Place value of
the digit will be
$\frac{1}{10}$
or one tenth. So, here, tenths means one whole is divided into ten equal parts and
each part is called one tenth.

Again, when the digit moves one more place right of tenth, place value of the digit will be
$\frac{1}{100}$.

Similarly, when the digit moves another place to right its place value becomes
$\frac{1}{1000}$
or one thousandths.

## How to read decimal numbers?

The decimal numbers can be read in two different ways.

1. The whole part of a decimal number is read as we read out a number in general but the digits present in
decimal part is read out with their **place name** of digits.

**2.7**

2.7 is read as two and seven tenths.

**14.25**

14.25 is read as fourteen and twenty five hundredths.

**4.725**

4.725 is read as four and seven hundred and twenty five hundredths.

2. The whole part of a decimal number is read as we read out a number in general, then read out decimal as
**point** and lastly read out each digit present in decimal part separately.

**2.7**

2.7 is read as two point seven.

**14.25**

14.25 is read as fourteen point two five.

**4.725**

4.725 is read as four point seven hundred two five.

## Writing fractions into decimals

Fractions with denominators 10, 100, 1000 etc. can be written into decimal numbers. So what are the steps to
convert fractions into decimals?
**Step 1:** Count the number of zeros in the denominator.
**Step 2:** Write the value of the numerator.
**Step 3:** Move from right to left of the numerator. Count the number of digits when moving from right to left
direction. Number of digits moved must be equal to the number of zeros in denominator. After moving put
point ‘**.**‘ on the left of the last number reached.
**Step 4:** In case if there are no more digits left on the left of ‘**.**‘, put an additional zero on the
left of the ‘**.**‘.

**Example 1: Convert fraction
$\frac{13}{100}$
into decimals**
**Step 1:** Count the number of zeros in the denominator 100, which are 2.
**Step 2:** Write the value of the numerator which is 13.
**Step 3:** Move from right to left of the numerator 13 by 2 digits, where 2 is the number of zeros in
the denominator. Put point after moving 2 digits. So the number becomes .13
**Step 4:** As .13 has more digits left on the left of ‘**.**‘, so put an additional zero
on the left of the ‘**.**‘. So the number becomes 0.13

∴
$\frac{13}{100}$ can be written in decimal form as 0.13

**Example 2: Convert fraction
$\frac{357}{10}$
into decimals**
**Step 1:** The number of zeros in the denominator 10 is 1.
**Step 2:** Write the value of the numerator which is 357.
**Step 3:** Move from right to left of the numerator 357 by 1 digit, where 1 is the number of
zeros in the denominator. Put point after moving 1 digit. So the number becomes 35.7

∴
$\frac{357}{10}$ can be written in decimal form as 35.7

**Example 3: Convert fraction
$\frac{14568}{1000}$
into decimals**
**Step 1:** The number of zeros in the denominator 1000 are 3.
**Step 2:** Write the value of the numerator which is 14568.
**Step 3:** Move from right to left of the numerator 14568 by 3 digits, where 3 is the number of
zeros in the denominator. Put point after moving 3 digits. So the number becomes 14.568

∴
$\frac{14568}{1000}$ can be written in decimal form as 14.568

## Writing decimals into fractions

Fractions can also be written into decimals. Let’s take a look at the steps, how to convert decimals into
fractions.
**Step 1:** Count the number of decimal places.
**Step 2:** Remove the decimal point and the digits after the decimal. After removing write the number
obtained as the numerator of the fraction.
**Step 3:** To add the denominator to the above fraction, write digit 1 in the denominator and then write
number of zeros after 1 equal to the number of decimal places present in the given decimal number.

**Example 1: Convert decimal 11.9 into fractions**
**Step 1:** Count the number of decimal places in 11.9, which is 1.
**Step 2:** Remove the decimal point. After removing the
number becomes 119 and write this number as the numerator of the fraction.

∴ the fraction becomes
$\frac{119}{\mathrm{\_\_\_\_\_}}$
**Step 3:** To add the denominator to the above fraction, write digit 1 in the denominator.
$\frac{119}{\mathrm{1\_\_\_\_\_}}$

Then write number of zeros after 1 equal to the number of decimal places present in 11.9, which is 1.

So, finally the fraction becomes
$\frac{119}{10}$

**Example 2: Convert decimal 13.72 into fractions**
**Step 1:** Count the number of decimal places in 13.72, which are 2.
**Step 2:** After removing the decimal point, number becomes 1372.

Write 1372 in the numerator.
$\frac{1372}{\mathrm{\_\_\_\_\_}}$
**Step 3:** Write digit 1 in the denominator.
$\frac{1372}{\mathrm{1\_\_\_\_\_}}$

There are 2 decimal places in 13.72, so write 2 number of zeros after 1 in the denominator.

so the fraction is
$\frac{1372}{100}$

**Example 3: Convert decimal 0.01189 into fractions**
**Step 1:** Count the number of decimal places in 0.01189, which are 5.
**Step 2:** After removing the decimal point, number becomes 01189 or it can be written as 1189
after omitting the zero at the start.

Write 1189 in the numerator.
$\frac{1189}{\mathrm{\_\_\_\_\_}}$
**Step 3:** Write digit 1 in the denominator.
$\frac{1189}{\mathrm{1\_\_\_\_\_}}$

There are 5 decimal places in 0.01189, so write 5 number of zeros after 1 in the denominator.
$\frac{1189}{100000}$

## Writing mixed fractions into decimals

Mixed fractions can be converted into decimals also. The very first step is to write the whole number of the mixed fraction and put the decimal point after. Then change the fractional part into decimal and write after the whole part with decimal. Let’s look at the solved examples to know it better.

**Example 1: Convert
$4\frac{5}{10}$
into decimals**

Here, 4 is a whole number and
$\frac{5}{10}$ is a fractional part. whihc is equal to 0.5

∴
$4\frac{5}{10}$
is written as 4.5

**Step 1:** Write the whole number 4 and then decimal after, which is 4.

**Step 2:** Change the fractional part
$\frac{5}{10}$ into decimals, which is 0.5

**Step 3:** write 0.5 after 4., which becomes 4.5

**Example 2: Convert
$128\frac{4}{100}$
into decimals**

**Step 1:** Write the whole number 128 and then decimal after, which is 128.

**Step 2:** Change the fractional part
$\frac{4}{100}$ into decimals, which is 0.04

**Step 3:** write 0.04 after 128., which becomes 128.004

**Example 1: Convert
$1025\frac{1}{10}$
into decimals**

**Step 1:** Write the whole number 1025 and then decimal after, which is 1025.

**Step 2:** Change the fractional part
$\frac{1}{10}$ into decimals, which is 0.1

**Step 3:** write 0.1 after 1025., which becomes 1025.01

## Expanding decimal numbers

Expanding form of a decimal number helps to understand the value of each digit in the number. It can be written in words as well as in numbers. Let’s learn by example how to expand the decimal numbers.

**Example 1: Expand 45.682**

45.682 can be represented as following in a place value chart of decimal numbers.

The above chart shows the place value of 4 at tens, 5 at ones, 6 at tenths, 8 at hundredths and 2 at
thousandths.

So, its expandable form in words can be written as 4 tens + 5 ones + 6 tenths + 8 hundredths + 2
thousandths

Its expandable form in numbers with fractions
will be 40 + 5 +
$\frac{6}{10}$
+
$\frac{8}{100}$
+
$\frac{2}{1000}$

Or, in decimal expansion, it can be written as
40 + 5 + 0.6 + 0.08 + 0.002

**Example 2: Expand 785.243**

Place value of chart of 785.243 is:
numbers.

The above chart shows the place value of 7 at hundreds, 8 at tens, 5 at ones,2 at tenths, 4
at hundredths and 3 at thousandths.

So, its expandable form in words is 7 hundreds + 8 tens + 5 ones + 2 tenths + 4 hundredths + 3
thousandths

Its expandable form in numbers with fractions
is 700 + 80 + 5 +
$\frac{2}{10}$
+
$\frac{4}{100}$
+
$\frac{3}{1000}$

In decimal expansion, it can be written as 700 + 80 + 5 + 0.2 + 0.04 + 0.003