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Computer Number Systems

What are the number systems ?

Number systems are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system.

Computer architecture supports following number systems.

  • Binary number system
  • Octal number system
  • Decimal number system
  • Hexadecimal (hex) number system

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1) Binary Number System

A Binary number system has only two digits that are 0 and 1. Every number (value) represents with 0 and 1 in this number system. The base of binary number system is 2, because it has only two digits.

2) Octal number system

Octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with 0,1,2,3,4,5,6 and 7 in this number system. The base of octal number system is 8, because it has only 8 digits.

3) Decimal number system

Decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits.

4) Hexadecimal number system

A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The base of hexadecimal number system is 16, because it has 16 alphanumeric values. Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15.

Table of the Numbers Systems with Base, Used Digits, Representation, C language representation:

Number system Base Used digits Example C Language assignment
Binary 2 0,1 (11110000)2 int val=0b11110000;
Octal 8 0,1,2,3,4,5,6,7 (360)8 int val=0360;
Decimal 10 0,1,2,3,4,5,6,7,8,9 (240)10 int val=240;
Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,
A,B,C,D,E,F
(F0)16 int val=0xF0;

Number System Conversions

There are three types of conversion:
  • Decimal Number System to Other Base
    [for example: Decimal Number System to Binary Number System]
  • Other Base to Decimal Number System
    [for example: Binary Number System to Decimal Number System]
  • Other Base to Other Base
    [for example: Binary Number System to Hexadecimal Number System]

Decimal Number System to Other Base

To convert Number system from Decimal Number System to Any Other Base is quite easy; you have to follow just two steps:
A) Divide the Number (Decimal Number) by the base of target base system (in which you want to convert the number: Binary (2), octal (8) and Hexadecimal (16)).
B) Write the remainder from step 1 as a Least Signification Bit (LSB) to Step last as a Most Significant Bit (MSB).

Decimal to Binary Conversion Result
Decimal Number is : (12345)10
decimal to binary conversion
Binary Number is
(11000000111001)2

Decimal to Octal Conversion Result
Decimal Number is : (12345)10
decimal to octal conversion
Octal Number is
(30071)8

Decimal to Hexadecimal Conversion Result
Example 1
Decimal Number is : (12345)10
decimal to octal hexadecimal
Hexadecimal Number is
(3039)16
Example 2
Decimal Number is : (725)10
decimal to octal hexadecimal
Hexadecimal Number is
(2D5)16
Convert
10, 11, 12, 13, 14, 15
to its equivalent...
A, B, C, D, E, F

Other Base System to Decimal Number Base

To convert Number System from Any Other Base System to Decimal Number System, you have to follow just three steps:
A) Determine the base value of source Number System (that you want to convert), and also determine the position of digits from LSB (first digit’s position – 0, second digit’s position – 1 and so on).
B) Multiply each digit with its corresponding multiplication of position value and Base of Source Number System’s Base.
C) Add the resulted value in step-B.

Explanation regarding examples:
Below given exams contains the following rows:
A) Row 1 contains the DIGITs of number (that is going to be converted).
B) Row 2 contains the POSITION of each digit in the number system.
C) Row 3 contains the multiplication: DIGIT* BASE^POSITION.
D) Row 4 contains the calculated result of step C.
E) And then add each value of step D, resulted value is the Decimal Number.

Binary to Decimal Conversion
Binary Number is : (11000000111001)2
binary to decimal conversion


Octal to Decimal Conversion Result
Octal Number is : (30071)8
                octal to decimal conversion
                
=12288+0+0+56+1
=12345
Decimal Number is: (12345)10

Hexadecimal to Decimal Conversion Result
Hexadecimal Number is : (2D5)16
hexadecimal to decimal conversion
=512+208+5
=725
Decimal Number is: (725)10




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