[Maths Class Notes] on Decimal Number System Pdf for Exam

We know the number system and how every number is represented by a base. When the base number is 2 then we call it a binary number. When the base is 8 then we call it an octal number. If the base is 10 then we call it a Decimal Number System and when the base is 16, we call it the hexadecimal system. We can convert the decimal numbers into other forms of number systems easily. This requires practice. On , you will learn about these numbers in detail and also learn important methods for conversions. 

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The Decimal Number System has the base number 10 and this notation is called the decimal notation. This number system has its applications in computer and computer science. This is the base 10 number system as it has 10 digits in its number systems. It can be written as 0, 1, 2,3,4,5,6,7,8 and 9. Every digit in this number system has a position and every digit is 10 times more significant than the previous one. Examples of a decimal number are (12)10 and (300)10.  Thus, we can say that the number system which uses the numbers 0 to 9 and has a base of 10 is the Decimal Number System and every digit in this system has the power value of 10.

We have studied different types of numbers, like real numbers, whole numbers, rational numbers, etc.There are four different types of number systems. They are

• Binary number system which has the base 2, represent any number using 2 digits 0 – 1

• Octal number system which has the base 8, represents any number using 8 digits 0 – 7.

• Decimal Number System which has the base 10, represent any number using 10 digits 0 – 9

• HexaDecimal Number System which has the base 16, represent any number using 10 digits and 6 characters 0 – 9,A,B,C,D,E,F

In this article let us study what is Decimal Number System, Decimal Number System definition, Decimal Number System example, and conversion of the Decimal Number System to different types of number system.

 

Decimal Number System Definition

The Decimal Number System comprises digits from 0-9 that are 0, 1, 2, 3, 4, 5, 6, 7, 8 & 9. The base or radix of the Decimal Number System is 10 because the total number of digits available in the Decimal Number System is 10. All the other digits can be expressed with the help of these 10 digit numbers.

 

Decimal Number System is the most common and easiest number system used in our daily lives. Some of the Decimal Number System examples are:

34110, 5610, 678910, 7810.

Now as we know to write decimal numbers till 10 let us use the 3 rules on a decimal system,to write further numbers.

• Write numbers 0–9.

• Once you reach 9, make the rightmost digit 0 and add 1 to the left which becomes 10.

• Then on the right digit, we write until 9 and when we reach 19 we use 0 on the rightmost digit and add 1 to the left, so we get 20.

• Similarly, when we reach 99, we use 0s in both of these digits’ places and add 1 to the left which gives us 100.

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How to read Decimal Numbers?

In single digits from 0 to 9 the numbers are read as it is. But in the case of two digits, the right digit says what it means, but the left digit means ten times what it says. That is in number 24, 4 is 4, 2 is 20. Altogether forms 24.

 

If we take a three digit number, the rightmost digit means what it says, the middle one is ten times the digit, and the leftmost digit 100 times the digit. Simply if we take number 546, it means (5 x 100) + (4 x 10) + 8  = 54810 

 

(5 x 102) + ( 4 x 101) + 8 = 54810

 

Conversion in Decimal Number System

How to convert Binary to Decimal

For binary number with n digits:

dn-1 … d3 d2 d1 d0

 

The conversion of binary to decimal number can be obtained by the sum of the product of binary digits (dn) and their power of 2 (2n):

 

decimal number = d0×20 + d1×21 + d2×22 + d3 x 23+ ……

 

Example

Convert (1110012)2 in decimal numeral system

 

Binary number:  1   1    1    0    0    1

And their power of 2: 25   24   23   22   21   20

 

(1110012)2  =  1 x 25 + 1 x 24 + 1 x 23 + 0 x 22 + 0 x 21 + 1 x 20  = 5710

 

How to convert Octal to Decimal?

In octal to decimal conversion, a number with base 8 is converted into a number with base 10 by multiplying each digit of octal number by decreasing power of 8. 

 

Example:

Convert (123)8 in decimal numeral system

 

Solution: 

Multiplying each digit with decreasing power of 8

(123)8 = 1 x 82 + 2 x 81 + 3 x 80

= 64 + 16 + 3

=83

 

How to convert Hexadecimal to Decimal?

In hexadecimal to decimal conversion, a number with the base 16 is converted into a number with base 16 by multiplying each digit of the hexadecimal number by a decreasing power of 16. 

 

Example: 

Convert 1516 in decimal numeral system

 

Solution: 

Multiplying each digit with decreasing power of 16

1 x 161 + 5 x 160

=16 + 5

= 21

 

Conversion of Decimal to Other System

How to Convert Decimal to Binary?

Steps to convert decimal number to binary number

1.    Divide the given number by 2.

2.    Take the quotient for the next iteration.

3.    And the remainder for the binary digit.

4.    Divide the obtained quotient again by 2

5.    Repeat the steps until we get a quotient equal to 0.

Example

Convert 1310 to binary:

Solution: 

Divide 13 by 2 

       13/2 = 6 and remainder 1

       6/2 = 3 and remainder is 0

       3/2 = 1 and remainder is 1

       1/2 = 0 and remainder is 1

So we collect the remainders in the order we get 10112

1310 = 10112

 

How to convert Decimal to Octal?

Decimal to Octal conversion is the same like decimal to binary just instead of 2 the number should be divided by 8

 

Example:

Convert 6010 into octal number system

 

Solution: 

Divide 60 by 8

60/8 = 7 and remainder is 4(MSB)

7/8 = 0 remainder is 7(LSB)

we count the remainder from LSB to MSB

So we collect the remainders we get 748

6010 = 748

 

How to convert Decimal to Hexadecimal?

Decimal to Hexadecimal conversion is the same as decimal to binary just instead of 2 the number should be divided by 16.

 

Example:

Convert 11010 to a hexaDecimal Number System. 

 

Solution: 

Divide the given number by 16

110/16 = 6 remainder is 14

6/16 = 0 remainder is 6

(replace 10, 11, 12, 13, 14, 15 by A, B, C, D, E, F respectively)

Hence 14 is replace by E

So 11010 = 6E

 

Fun Facts:

• As there are ten fingers on two hands people started counting by using their fingers, many numeral systems of ancient civilizations use ten and its powers for representing numbers,

• In these old numeral systems large numbers were difficult to multiply and divide, hence these difficulties were solved with the introduction of the Hindu–Arabic numeral system.

• Decimal, also called Hindu-Arabic, or Arabic, number system