Decimals- Definition, Types, Properties and Arithmetic Operations on Decimals

In mathematics, we have many types of numbers. For eg:- real numbers, natural numbers, whole numbers, prime numbers. Decimal numbers are one of its types.

In this blog, we will discuss “ Decimals”, types of decimals, its properties, and arithmetic operations performed on decimals with solved examples.

What are Decimal Numbers?

Definition 1 = Decimals can be defined as a set of numbers lying between the integers on a number line.
Definition 2 = Decimals are used to express the whole number and the fraction together. The whole number and the fractional part is separated by a dot, namely decimal point.

Now, let us understand the definition with an

Example:-

25.8
Here, 25 is the whole number and 8 is the fractional part.

Similarly, in 65.2
65 is the whole number and 2 is the fractional part.

Types of Decimals

In algebra, we divide decimal numbers into 2 main types:-

1. Terminating decimal number
2. Non- terminating decimal numbers

1. Terminating Decimal Number

This is also called non repeating decimal numbers. In this type of decimal numbers, decimal does not reoccur(repeat) and ends after a fixed number of decimal places.

For example

= 5.23, 28.2 , 45.05 etc.

2. Non – Terminating Decimal Number

This is also called repeating decimal numbers.In this type of decimals, decimal numbers have infinite digits after the decimal point.

For example

= 543.23712345678……..17.79235678099….. etc.

Non-terminating decimal numbers can be further sub-divided into 2 parts:

Read More : Multiplication of Fractions

A. Recurring Decimal Numbers

As the name itself suggests, recurring means repeating. In recurring decimal numbers, digits after decimal repeat after a fixed interval.

For example

= 25.125125125……….34.256256256…… etc.

B. Non-Recurring Decimal Numbers

Non-recurring decimal numbers are just opposite of recurring decimal numbers, the digits after decimal never repeat after a fixed interval.

For example

= 345.123457237578………. 45.1265234………… etc.

Read More : Interest – Do you know, what is Simple Interest?

Properties of Decimal Numbers

1. If any two or more than 2 decimal numbers are multiplied in any order, the product always remains the same.This property is called commutative property.
2. When we multiply the whole number and a decimal number in any order, the product still remains the same.It also follows the commutative property.
3. When a decimal number is multiplied by 1, the product is the decimal fraction itself. This property is known as the multiplicative identity for decimal numbers.
4. When a decimal fraction is multiplied by 0, the product is always zero. This property is known as the property of zero.
5. When a decimal number is divided by 1, the quotient is the decimal number.
6. When a decimal number is divided by the same number, the quotient is 1.
7. When 0 is divided by any decimal, the quotient is always 0.
8. We cannot divide a decimal number by 0 is not possible, as reciprocal of 0 is infinity and does not exist.

Read More: Addition of Fraction

Place Value System in Decimals

The place value system in decimals is the same as the whole number for the whole number part. But, after the decimal point, we use decimal fractions to represent the place value.

When we go towards the left, each digit in the place value table has a value that is equal to ten times the value of the next digit on its right.

Let’s consider the number 336.

1. “6” is placed at One’s place, which means 6 ones i.e 6.
2. “3” is in the Tens place, which means 3 tens which is thirty.
3. Again, “3” is placed at the Hundred’s place, which means 3 hundred.
4. That’s why we read it as “Three hundred thirty-six”.

When we move towards the left, each position is 10 times bigger and when we move towards the right every position becomes 10 times smaller from Hundreds to Tens, to Ones.

Now the question is , what is 10 times smaller than ones?

Read More : Basic Concept Of Angles In Geometry

It’s 1/10 referred to as tenths….. Then 1/100 is referred to as hundredths and so on.

Let us take

an example

256.34

Hundreds	Tens	ones	Decimal pt.	tenths	Hundreds
2	5	6	.	3	4

 

Conversion into Decimal to Fraction and vice versa…

In this concept we will learn conversion of:-

1. Decimal numbers into fractions
2. Fractions into decimal numbers

1. Decimal Numbers into Fractions

Firstly, we will convert our decimal numbers into fractions. When converting decimal to fraction, write down the decimal numbers in the expanded form and then simplify the values. Let us understand with

an example

0.25
25 x 1/100 = 25/100 =  ¼

Read More : TRIANGLE: Definition, Parts, Properties,Types and Formulae

2. Fractions into Decimal Number

we can convert our fractions into decimal numbers. When converting fraction to decimal, you can simply divide the numerator by the denominator. Simplify the values if needed.Let us understand with

an example:-

11/2 = 6.5
25/3 = 7.3333….

Arithmetic Operations on Decimals

All the arithmetic operations like addition, subtraction, multiplication, and division operation can be performed on decimal numbers too.

1. Addition
2. Subtraction
3. Multiplication
4. Division

1. Addition

When you are adding the decimal numbers, write the decimals in the columns with the decimal points below each other as the tenths come under the tenths, the hundredths come under the hundredths and so on.

Add in the same way as you add the whole numbers.

Read More: Remember Multiplication Tables of Any Number Using Mental Mathematics

2. Subtraction

When you are subtracting the decimal numbers, write the decimals in the columns with the decimal points below each other as the tenths come under the tenths, the hundredths come under the hundredths and so on.

Now, subtract in the same way as you add the whole numbers.

3. Multiplication

In multiplication, first multiply the given numbers like whole numbers, as if the decimal point is not present here. Then, find the product and count up how many numbers are present after the decimal point in both numbers.

In answer now, put the decimal according to your count.

Read More: How to find HCF in 10 seconds

4. Division

When dividing the decimal numbers, move the decimal point in the numbers so that the number becomes the whole number. Now, perform the division like the integer division. And put the decimal point accordingly.

In the next blog, we will learn the arithmetic operations in detail and compare the decimal numbers.

Practice Questions for you:-

1. Convert 56/3 into decimal number
2. Is 5.323232….. A non terminating decimal? If yes, which type is this?
3. Convert 2.34 into fraction.
4. Convert 3456.87 into fraction
5. What is the place of 9 in the given number – 34567.89 ?