Overview: Decimals | Quantitative Techniques for CLAT PDF Download

Decimals in Maths

Initially, when numbers were started being used for calculation, it was being used for very basic calculations, but when trade increased, and nations grew, there was a need to count large units of currency and goods which is why decimals came to exist. Decimal fractions are said to be developed by China in the 4th Century BC, and then they spread to the Middle East and from there to Europe. The Decimal Number System is denoted by a decimal point, consisting of a whole number part and a fractional part separated by a decimal point. The dot in a Decimal number is called a Decimal Point, and the Decimals are referred to as the digits after the Decimal Points.

The Decimal Number System is one of the essential number systems in Mathematics. Students learn Decimals at an early age in order to solve mathematical problems within fractions of seconds.

Definition of Decimals

Decimals represent the integers and non integers numbers. It is denoted by a dot present between whole numbers and fraction part called a decimal point. For example, 12.5 where 12 is whole number, 5 is fractional part and the dot (.) is denoted as decimal point.

Decimals fractions were first developed by Chinese in the 4th century BC. And it is used in our day to day life by representing money, length, mass, weight etc. We all know that all the decimal can be represented as fractions.

For example,

• 1/1000 = 0.001
• 1/100 = 0.01
• 1/10 = 0/1

Properties of Decimals

The properties of Decimal are as follows:

• The quotient will be zero if it is divided by any decimal number.
• The quotient will be one when the decimal number are divided by itself
• The quotient will be same as decimal number if it is divided by 1.
• The product remains the same, if any two decimal numbers are multiplied in any order.
• The product is same as decimal fraction when it is multiplied by 1.
• The product remains the same, if whole numbers and decimal numbers are multiplied by any order.

Types of Decimals

There are three types of Decimals as mentioned below:

• Non- Recurring Decimals: These have finite numbers after the decimal point. For example, 4.52761
• Recurring Decimals: These have one or more repeating numbers after the decimal point. For example, 6.75757575
• Decimal Fraction: The denominators are in tenths, hundredths, thousandths etc. For example, 0.35= 35/100

Arithmetic Operation of Decimals

Apart from integers, you can add, subtract, multiply and divide decimal numbers. They are as follows:

• Addition: Add decimals in the same way as you do for whole numbers.
• Subtraction: The subtraction of decimals can be done by subtracting hundredths from hundredths, tenths from tenths, and ones from ones.
• Multiplication: Multiply the integers thinking that there is no decimal point. Find out the product and count the numbers present after the decimal point.
• Division: Make the decimal number as whole number in order to divide the decimal numbers.

Conversion of Decimals To Fractions

Lets us check the conversion of decimals to fractions or fractions to decimals as given below:

• Decimals as fractions: Decimal points represent the tenths, hundredths, thousandths etc.
  For example, 0.25 can be expanded to write as 25 x 1/100 to get 25/100.
• Fractions as Decimals: We can get decimals by dividing the numerator by denominator.
  For example, 3/5 = 0.6

Examples of Decimals

Some of the examples of decimals are mentioned below:

Example 1: Write each of the following as decimals : (a) Two ones and five-tenths (b) Thirty and one-tenth
Sol: 
(a) Two ones and five-tenths = 2 + 5/10 = 2.5
(b) Thirty and one-tenth = 30 + 1/10 = 30.1

Example 2: Write each of the following as decimals: (a) Seven-tenths (b) Two tens and nine-tenths (c) Fourteen point six (d) One hundred and two ones (e) Six hundred point eight?
Sol:
(a) Seven-tenths= 7 /10 = 0.7
(b) Two tens and nine-tenths = (2 × 10) + (9 × 1 /10 ) = 20 + 0.9 = 20.9
(c) Fourteen point six = 14.6
(d) One hundred and two ones = (1 × 100) + (2 × 1) = 100 + 2 = 102
(e) Six hundred point eight = 600.8

Example 3: Write the following decimals as fractions. Reduce the fractions to lowest form. (a) 0.6 (b) 2.5 (c) 3.8
Sol: 
Given, (a) 0.6 (b) 2.5 (c) 3.8
(a) 0.6 = 6 /10 = 3/5
Hence, 0.6 = 3 /5
(b) 2.5 = 2 + 0.5
= 2 + 1/ 2 = 5/ 2
Hence, 2.5 = 5/ 2
(c) 3.8 = 3 + 8/10
= 3 + 4/ 5 = 19/ 5
Hence, 3.8 = 19/ 5