5.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 10^{3})+ (2 x 10^{2})+ (3 x 10^{1})+ (4 x l0^{0})

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 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.

**Example**

Binary Number: 10101_{2}

Calculating Decimal Equivalent −

Step |
BinaryNumber |
Decimal Number |

Step1 | 10101_{2} |
((1 x 2^{4}) + (0 x 2^{3}) + (1 x 2^{2}) + (0 x 2^{1}) + (1 x 2^{0}))_{10} |

Step2 | 10101_{2} |
(16 + 0 + 4 + 0 + 1)_{10} |

Step3 | 10101_{2} |
21_{10} |

Note − 10101_{2} 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 8^{0} - Last position in an octal number represents a
**x**power of the base (8). Example 8^{x}where**x**represents the last position – 1

**Example**

Octal Number: 12570_{8}

Calculating Decimal Equivalent −

Step |
OctalNumber |
Decimal Number |

Step1 | 12570_{8} |
((1 x 8^{4}) + (2 x 8^{3}) + (5 x 8^{2}) + (7 x 8^{1}) + (0 x 8^{0}))_{10} |

Step2 | 12570_{8} |
(4096 + 1024 + 320 + 56 + 0)_{10} |

Step3 | 12570_{8} |
5496_{10} |

**Note** − 12570_{8} 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, 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

**Example**

Hexadecimal Number: 19FDE_{16}

Calculating Decimal Equivalent −

Step |
BinaryNumber |
Decimal Number |

Step1 | 19FDE_{16} |
((1 x 16^{4}) + (9 x 16^{3}) + (F x 16^{2}) + (D x 16^{1}) + (E x 16^{0}))_{10} |

Step2 | 19FDE_{16} |
((1 x 16^{4}) + (9 x 16^{3}) + (15 x 16^{2}) + (13 x 16^{1}) + (14 x 16^{0}))_{10} |

Step3 | 19FDE_{16} |
(65536+ 36864 + 3840 + 208 + 14)_{10} |

Step4 | 19FDE_{16} |
106462_{10} |

**Note** − 19FDE_{16} is normally written as 19FDE.