28 May 2017, Manish Padhi
A digital system can understand positional number system only where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.
A 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 base is defined as the total number of digits available in the number system).
1. 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, and its value can be written as
(1x1000)+ (2x100)+ (3x10)+ (4xl)
(1x10^{3})+ (2x10^{2})+ (3x10^{1})+ (4xl0^{0})
1000 + 200 + 30 + 1
1234
As a computer programmer or an IT professional, you should understand the following number systems which are frequently used in computers.
S.N. | Number System & Description |
---|---|
1 | Binary Number System Base 2. Digits used: 0, 1 |
2 | Octal Number System Base 8. Digits used: 0 to 7 |
4 | Hexa Decimal Number System Base 16. Digits used: 0 to 9, Letters used: A- F |
2. Binary Number System
Characteristics
Uses two digits, 0 and 1.
Also called base 2 number system
Each position in a binary number represents a 0 power of the base (2). Example 2^{0}
Last position in a binary number represents a x power of the base (2). Example 2^{x} where x represents the last position - 1.
e.g.
Binary Number: 10101_{2}
Calculating Decimal Equivalent:
Step | Binary Number | Decimal Number |
---|---|---|
Step 1 | 10101_{2} | ((1 x 2^{4}) + (0 x 2^{3}) + (1 x 2^{2}) + (0 x 2^{1}) + (1 x 2^{0}))_{10} |
Step 2 | 10101_{2} | (16 + 0 + 4 + 0 + 1)_{10} |
Step 3 | 10101_{2} | 21_{10} |
Note: 10101_{2} is normally written as 10101.
3. Octal Number System
Characteristics
Uses eight digits, 0,1,2,3,4,5,6,7.
Also called base 8 number system
Each position in a octal number represents a 0 power of the base (8). Example 8^{0}
Last position in a octal number represents a x power of the base (8). Example 8^{x} where x represents the last position - 1.
e.g.
Octal Number: 12570_{8}
Calculating Decimal Equivalent:
Step | Octal Number | Decimal Number |
---|---|---|
Step 1 | 12570_{8} | ((1 x 8^{4}) + (2 x 8^{3}) + (5 x 8^{2}) + (7 x 8^{1}) + (0 x 8^{0}))_{10} |
Step 2 | 12570_{8} | (4096 + 1024 + 320 + 56 + 0)_{10} |
Step 3 | 12570_{8} | 5496_{10} |
Note: 12570_{8} is normally written as 12570.
4. Hexadecimal Number System
Characteristics
Uses 10 digits and 6 letters, 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.
Letters represents numbers starting from 10. A = 10. B = 11, C = 12, D = 13, E = 14, F = 15.
Also called base 16 number system
Each position in a hexadecimal number represents a 0 power of the base (16). Example 16^{0}
Last position in a hexadecimal number represents a x power of the base (16). Example 16^{x}where x represents the last position - 1.
e.g.
Hexadecimal Number: 19FDE_{16}
Calculating Decimal Equivalent:
Step | Hexa Number | Decimal Number |
---|---|---|
Step 1 | 19FDE_{16} | ((1 x 16^{4}) + (9 x 16^{3}) + (F x 16^{2}) + (D x 16^{1}) + (E x 16^{0}))_{10} |
Step 2 | 19FDE_{16} | ((1 x 16^{4}) + (9 x 16^{3}) + (15 x 16^{2}) + (13 x 16^{1}) + (14 x 16^{0}))_{10} |
Step 3 | 19FDE_{16} | (65536+ 36864 + 3840 + 208 + 14)_{10} |
Step 4 | 19FDE_{16} | 106462_{10} |
Note: 19FDE_{16} is normally written as 19FDE.