# Digital Number System

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 usingThe 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).

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

2 | Octal Number SystemBase 8. Digits used: 0 to 7 |

4 | Hexa Decimal Number SystemBase 16. Digits used: 0 to 9, Letters used: A- F |

## 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.

### 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

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.

### 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

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.