Computer - Number System

When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. A computer can understand the positional number system where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.

The value of each digit in a number can be determined using −

• The digit
• The position of the digit in the number
• The base of the number system (where the base is defined as the total number of digits available in the number system)

Decimal Number System

The number system that we use in our day-to-day life is the decimal number system. Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands, and so on.

Each position represents a specific power of the base (10). For example, the decimal number 1234 consists of the digit 4 in the units position, 3 in the tens position, 2 in the hundreds position, and 1 in the thousands position. Its value can be written as

(1 x 1000)+ (2 x 100)+ (3 x 10)+ (4 x l)
(1 x 103)+ (2 x 102)+ (3 x 101)+ (4 x l00)
1000 + 200 + 30 + 4
1234
 
As a computer programmer or an IT professional, you should understand the following number systems which are frequently used in computers.

S.No.                           Number System and Description
1                      Binary Number System Base 2. Digits used : 0, 1

2                      Octal Number System Base 8. Digits used : 0 to 7

3                      Hexa Decimal Number System  
                        Base 16. Digits used: 0 to 9, Letters used : A- F

Binary Number System

Characteristics of the binary number system are as follows −

• Uses two digits, 0 and 1
• Also called as base 2 number system
• Each position in a binary number represents a 0 power of the base (2). Example 20
• Last position in a binary number represents a x power of the base (2). Example 2x where x represents the last position - 1.

Example
Binary Number: 101012

Calculating Decimal Equivalent −

Step                      Binary Number                     Decimal Number
Step 1                        101012                       ((1 x 24) + (0 x 23) + (1 x 22) 

                                                                    + (0 x 21) + (1 x 20))10
Step 2                        101012                       (16 + 0 + 4 + 0 + 1)10
Step 3                         101012                                       2110
Note − 101012 is normally written as 10101.

Octal Number System

Characteristics of the octal number system are as follows −

• Uses eight digits, 0,1,2,3,4,5,6,7
• Also called as base 8 number system
• Each position in an octal number represents a 0 power of the base (8). Example 80
• Last position in an octal number represents a x power of the base (8). Example 8x where x represents the last position - 1

Example
Octal Number: 125708

Calculating Decimal Equivalent −

Step                     Octal Number                     Decimal Number
Step 1                     125708                       ((1 x 84) + (2 x 83) + (5 x 82)
                                                                 +(7 x 81) + (0 x 80))10
Step 2                      125708                (4096 + 1024 + 320 + 56 + 0)10
Step 3                       125708                               549610

Note − 125708 is normally written as 12570.

Hexadecimal Number System

Characteristics of hexadecimal number system are as follows −

• Uses 10 digits and 6 letters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Letters represent the numbers starting from 10. A = 10. B = 11, C = 12, D = 13, E = 14, F = 15
• Also called as base 16 number system
• Each position in a hexadecimal number represents a 0 power of the base (16). Example, 160
• Last position in a hexadecimal number represents a x power of the base (16). Example 16x where x represents the last position - 1

Example
Hexadecimal Number: 19FDE16

Calculating Decimal Equivalent −

Step              Binary Number                         Decimal Number
Step 1            19FDE16                     ((1 x 164) + (9 x 163) + 
                                                       (F x 162) + (D x 161) + (E x 160))10
Step 2             19FDE16                     ((1 x 164) + (9 x 163) + (15 x 162)
                                                         + (13 x 161) + (14 x 160))10
Step 3             19FDE16                     (65536+ 36864 + 3840 + 208 + 14)10
Step 4              19FDE16                                       10646210

Note − 19FDE16 is normally written as 19FDE.