Types of number systems



Types of number systems and representation of numbers are discussed here. Numbers of different types of number systems are written by base number with the number.  Number systems are decimal, binary, octal, and hexadecimal.


Types of number systems

Number
system
Base
Example
Decimal
10
(9421)₁₀

            
Binary
2
(1011)₂

           
Octal
8
(107)₈

         
Hexadecimal
16
(19A)₁₆

          


Decimal number system

The decimal number system is a positional number system. It has 10 symbols or digits. Those are 0,1,2,3,4,5,6,7,8,9.
Hence, its base = 10.
We know that the maximum value of a single digit will be
Number of base -1
So, for this number system, it will be 10-1 =9.

Each position of a digit represents a specific power of 10(the base number).

In our daily life, we use the Decimal number system for counting. And this is the most used number system.

(9421)₁₀ = (9✕10³) + (4✕10²) + (2✕10¹) + (1✕10⁰) 
          
              = 9000+400+20+1
             
              = (9421)₁₀


Positional value for the Decimal number system 

10³,10²,10¹,10⁰ (Base =10)


Binary number system

It is a positional number system. It has only two symbols or digits. These are 0 and 1.

So, it's base = 2.
For this number system, the maximum value of a single digit is 1(base-1=2-1=2). Each position of a digit represents a specific power of 2(The base number=2).

Binary numbers of corresponding decimal numbers

Decimal
Binary
0
0
1
1
2
10
3
11
4
100
5
101
6
110
7
111
8
1000
9
1001
10
1010
11
1011
12
1100
13
1101
14
1110
15
1111
16
10000


(10011)₂ =(1✕2⁴)+(0✕2³)+(0✕2²)+(1✕2¹)+(1✕2⁰)
              
               = 16+0+0+2+1
          
               = (19)₁₀



Positional value for the binary number system

2³,2²2¹,2⁰   .  2⁻¹,2⁻²,2⁻³,2⁻⁴



Use of binary number system in computers

Binary number plays a vital role in computer designing. All the numbers we use are built with 10 decimal numbers. These are 0,1,2,3,4,5,6,7,8,9. 

Computer works by an electrical signal that is why representing these numbers differently are not impossible but very hard. 

But using the binary number system can be represented easily by electrical signals. 

In the binary system, there are two situations 0 or 1. So that easily on and off can be represented. In computers usually, the data representation process is code. These codes are built by a binary system. 

Counting of the decimal system can found by using devices that are mainly built by using the binary number system. 

Bit in computer

A bit in computer terminology means either a 0 or a 1. If any binary number consisting of n number of bits then we can call that, that number is an n-bit binary number. 


Octal Number system

The octal Number system is a positional number system. It has eight symbols or digits. These are 0,1,2,3,4,5,6,7. Hence, it's base = 8. 
For this number system, the maximum value of a single digit is 7. Each position of a digit represents a specific power of 8(base). 

Since there are only 8 digits, 3 bits (2³ = 8) are sufficient for representing any octal number in binary.


Octal numbers of corresponding decimal numbers


Decimal
Octal
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
10
9
11
10
12
11
13
12
14
13
15
14
16
15
17
16
20


(107)₈ = (1✕8²)+(0✕8¹)+(1✕8⁰)
          
           = 64+0+7
           
           = (71)₁₀
       

Positional value for the binary number system

8³ 8² 8¹ 8⁰. 8⁻¹ 8⁻² 8⁻³ 8⁻⁴



Octal number system uses


Octal Number system mainly in the computers for representing the binary values. Because a lot of bits are needed for the binary number system.

Suppose you have a total of 8 fingers in your hand

 So, you have to count like this...


0,1,2,3,4,5,6,7

10,11,12,13,14,15,16,17

and so on.

Hexadecimal number system

The hexadecimal number system is a positional number system. It has 16 symbols or digits. These are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F.
So, it's base = 16.

In the Hexadecimal number system, the maximum value of a single digit is 15 (base-1=16-1=15). Each position of a digit represents a specific power of 16(base). Since there are 16 digits, 4 bits (2⁴=16) are sufficient for representing any hexadecimal number in binary.

Hexadecimal numbers of corresponding decimal numbers


Decimal
Hexadecimal
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
A
11
B
12
C
13
D
14
E
15
F


(19A)₁₆ = (1✕16²) + (9✕16¹) + (A✕16⁰) 
            
             = 256+144+10
            
             = (410)₁₀


Positional value for the Hexadecimal number system

MSB ← 16⁴16³16²16¹16⁰.16⁻¹16⁻²16⁻³16⁻⁴ → LSB

Hexadecimal number system uses

Like the octal number system, this is also used as a short signal of a binary number.

Think that you have 16 fingers total in your hand. Now count , 

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

10,11,12,13,14,15,16,17,18,19,1A,1B,1C,1D,1E,1F

Like this, you have to continue counting. 

Representation of numbers


Different numbers of systems have different unique identities.

For specifying a number system of any number we indicate the base number as a subscript.

All numbers are build by using different types of signs and numbers.

In decimal number system all the numbers are build using 10 numbers(0,1,2,3,4,5,6,7,8,9).

In the binary number system by 0 and 1 all the numbers are structured. 

In the hexadecimal number system 16 representing factors. They are (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F).

In the octal number system by using 0,1,2,3,4,5,6,7 all the numbers are presented.