# Number Systems

When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. A computer can understand 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.

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).

Contents

## 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 + 4 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 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 binary number system are as follows:

- 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.

### 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 octal number system are as follows:

- Uses eight digits, 0,1,2,3,4,5,6,7.
- Also called 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 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.

### 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.