# 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 |
Base 2. Digits used : 0, 1 |

2 |
Base 8. Digits used : 0 to 7 |

3 |
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 | 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.

## 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 | 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.

## 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 | Binary 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.