Decimal Number System

What is the Decimal Number System?

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.

(Image to added soon)


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, rightmost digit means what it says, the middle one is ten times the digit, 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 hexadecimal number by 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)

⅞ = 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

FAQ (Frequently Asked Questions)

1. What are Binary Numbers?

Computers can understand only two conditions “On” and “Off” i.e 1 and 0. The binary number system deals with the study of 0s and 1s.

A binary number system represents a number with the base 2, by using the digits 1 and 0. As it uses only two digits 0 and 1 and has a base of 2, it is called binary.

All electronic devices use a binary number system in their electronic circuit. The input 0 indicates the “OFF” state and the input 1 indicates the “ON” state. Because of these implementations binary number systems are used in modern computer technology.

Each digit is referred to as a bit.

There are no 2, 3, 4, 5, 6, 7, 8, 9 in the binary number system.

Example of the binary number system are:

  • 1012

  • 110012

2. What are Octal Numbers?

The octal number system is the numbers with the base 8 and uses digits from 0 to 7, the digits 8 and 9 are not used in the octal number system. The octal number system is also called the base 8 number system

The octal number system is generally used in a minicomputer.

Example of octal number system are:

  • 4238

  • 718

As we are quite familiar with the binary number system and octal number system,  Now let us study how to convert binary to octal? And binary to octal conversion examples.