What are decimals?

• Many ancient civilisations, including the Harappan, Greeks, Romans, and Egyptians, used a system of numerals called the decimal. Even the Vedas are said to have this system of numbering. It is said that this is because humans have ten fingers. The decimal numbering system uses multiples of 10. Decimals can also express fractions, making the system more accurate and useful. 
• Fractions are said to have been first introduced in decimals at the end of the 4th century BCE by the Chinese. It later acquired the Middle East around the 10th Century and finally made its way to Europe by the 16th Century. 
• The current representation of fractions in decimal format was formalised in 1616 by John Napier, a Scottish Mathematician. He is responsible for the addition of the decimal point. The left part of the decimal point is whole numbers, and the fractional part is represented by the numbers on the right of the decimal point.

What are decimal fractions?

The definition of a decimal fraction is a fraction in which the denominator is a power of ten (such as 10, 100, 1000, etc.). Decimal fractions can be written without a denominator using a decimal point, making it easier to perform calculations on fractions. 

Types of decimal fractions

Decimal fractions are of two types:

1. Terminating decimal fractions:

This decimal fraction is “a decimal fraction in which there is an available or limited number of digits after the decimal point are known as terminating decimal fractions.“ These decimal fractions have a fixed number of digits after the decimal point.

Examples:

56.8476

6.2458674

2. Non-terminating decimal fractions:

This decimal fraction contains digits after the decimal point that repeat infinitely. This definition of this decimal fraction can be worded as “The decimal fractions that have endlessly repeating digits after a decimal point are known as non-terminating decimal fractions.” 

These decimal fractions are of a different two types:

• Recurring Decimal Fractions
• Non-recurring Decimal Fractions

Recurring Decimal Fractions:

• This type of decimal fraction is defined as those decimal fractions with an endless number of digits that repeat at uniform intervals. 

Examples:

10.12121212….

3.3333

• In the case of the above numbers, the digits after the decimal point repeat at uniform intervals following a particular order; hence, they are called recurring decimal fractions. They can be represented by putting a bar sign over the repeating numbers.

Recurring decimals are of two types: pure recurring decimal fractions and mixed recurring decimal fractions.

1. Pure recurring decimal fractions are those decimal fractions in which the digits repeat continuously after the decimal point. They can be represented by putting a bar sign over the repeating numbers.

Examples: 

8.565656

7.55555

2. Mixed decimal fractions are those decimal fractions in which a few digits after the decimal point repeat and which also contain digits that do not repeat. 

Examples:

56.84588888

89.2587777

Non-recurring decimal fractions

These decimal fractions are defined by the number of digits after the decimal point is infinite without any particular order.

Examples:

5.5648135489

Place value of decimals:

• The value of an integer in a number is defined by its position in the number concerning a point. This is known as the place value positional notation system. 
• In the case of decimal fractions, multiplication of the digits on the right side of the decimal point is done with ascending negative powers of 10 from left to right. For the whole numbers on the left side of the decimal point, the multiplication is done with ascending positive powers of 10.

Place value chart

Mathematical operations on decimal fractions

Addition and subtraction of decimal fractions:

The decimal fractions are lined up using the decimal point and added or subtracted like whole numbers. When there is a whole number, it is assumed that its decimal point is to the right. 

Addition:

To Add: 56.15 + 89.11 + 99

   56.15

+ 89.11

+ 99.00

= 244.26

Subtraction:

To Subtract: 24.26 – 5.1

   24.26

–      5.1

= 19.16

Multiplication of decimal fractions:

Decimal fractions are multiplied just like whole numbers, and the decimal point in the answer is placed so that the number of digits after the decimal point in the answer is equal to the sum of digits on the right side of the answer. Decimal fractions multiplied.

To Multiply: 56.152 × 9.2

      56.152

*           9.2

= 516.5984

Division of decimal fractions:

When the divisor has a decimal fraction, the decimal point is moved to the right until it becomes a whole number by multiplying with powers of 10 equal to the number of digits on the right-hand side of the decimal point. The decimal point in the dividend is also moved in the same number of places, and if there are fewer digits in the dividend, zeros are added to it. 

To Divide: 89.11 by 0.456

89.11/0.456=89.11*1000/.456*1000=89110/456=195.4166

Conclusion:

The definition of a decimal fraction is a fraction in which the denominator is a power of ten (such as 10, 100, 1000, etc.). Decimal fractions can be written without a denominator using a decimal point, making it easier to perform calculations on fractions. 

There are two types of decimal fractions, terminating and non-terminating. 

The non-terminating decimal fractions are further subdivided into recurring and non-recurring. 

The recurring decimal fractions are divided into pure recurring and mixed decimal fractions. 

The value of an integer in a number is defined by its position in the number concerning a point. This is known as the place value positional notation system. The operation on decimal fractions is also shown.