*“A set of values used to represent different quantities is known as Number System“.* For example, a number system can be used to represent the number of students in a class or the number of viewers watching a certain TV program, etc. The digital computer represents all kinds of data and information in binary numbers. It includes audio, graphics, video, text, and numbers. The total number of digits used in a number system is called it’s base or radix. The base is written after the number as subscript such as 51210.

Some important number systems are as follows.

- Decimal number system
- Binary number system
- Octal number system
- Hexadecimal number system

The decimal number system is used in general. However, computers use binary number systems. The octal and hexadecimal number systems are used in the computer.

## Decimal number System

See Also: Convert decimal numbers to binary numbers

The Decimal Number System consists of ten digits from 0 to 9. These digits can be used to represent any numeric value. The base of the decimal number system is 10. It is the most widely used number system. The value represented by the individual digit depends on the weight and position of the digit.

Each number in this system consists of digits that are located at different positions. The position of the first digit towards the left side of the decimal point is 0. The position of the second digit towards the left side of the decimal point is 1. Similarly, the position of the first digit towards the right side of the decimal point is -1. The position of the second digit towards the right side of the decimal point is -2 and so on.

The value of the number is determined by multiplying the digits with the weight of their position and adding the results. This method is known as the expansion method. The rightmost digit of the number has the lowest weight. This digit is called Least Significant Digit (LSD). The leftmost digit of a number has the highest weight. This digit is called the Most Significant Digit (MSD). The digit 7 in the number 724 is the most significant digit and 4 is the least significant digit.

See Also: Number Bases

Example:

The weights and positions of each digit of the number 453 are as follows:

Position | 2 | 1 | 0 |

Weights | 102 | 101 | 100 |

Face value | 4 | 5 | 3 |

The above table indicates that:

The value of digit 4 = 4×102 = 400

The value of digit 4 = 5×10 = 50

The value of digit 3 = 3×10 = 3

The actual number can be found by adding the values obtained by the digits as follows:

400 + 50 + 3 =45310

### Example:

The weights and positions of each digit of the number 139.78 are as follows.

Position | 2 | 1 | 0 | -1 | -2 | |

Weights | 102 | 101 | 100 | . | 10-1 | 10-2 |

Face Value | 1 | 3 | 9 | 7 | 8 |

The above table indicates that:

The value of digit 1 = 1×102 = 100

The value of digit 3 = 3×101 = 30

The value of digit 9 = 9×100 = 9

The value of digit 7 = 7×10-1 = 0.7

The value of digit 8 = 8×10-2 = 0.08

The actual number can be found by adding the values obtained by the digits as follows:

100 + 30 + 9 + 0.7 + 0.8 = 139.78

## Binary Number System

A digital computer represents all kinds of data and information in the binary system. Binary Number System consists of two digits 0 and 1. Its base is 2. Each digit or bit in a binary number system can be 0 or 1. A combination of binary numbers may be used to represent different quantities like 1001. The positional value of each digit in a binary number is twice the place value or face value of the digit of its right side. The weight of each position is a power of 2.

The place value of the digits according to position and weight is as follows:

Position | 3 | 2 | 1 | 0 |

Weights | 23 | 22 | 21 | 20 |

### Example: Convert 101112 decimal number

Position | 2 | 1 | 0 | -1 | -2 |

Weights | 102 | 101 | 100 | 10-1 | 10-2 |

Face Value | 1 | 3 | 9 | 7 | 8 |

101112 = 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20

= 1 x 16 + 0 + 1 x 4 + 1 x 2 + 1 x 1

= 16 + 0 + 4 2 + 1

= 2310

### Example: Convert 101.1012

Position | 2 | 1 | 0 | -1 | -2 | -3 | |

Face Value | 1 | 0 | 1 | . | 1 | 0 | 1 |

Weight | 24 | 21 | 20 | 2-1 | 2-2 | 2-3 |

101.1012 = 1 x 22 + 0x21 + 1 x 20 + 1x 2-1 + 0 x 2-2 + 1 x 2-3

= 1 x 4 + 0 + 1 x 1 + ½ + 0 + 1/8

= 4 + 0 + 1 + 0.5 + 0.125

= 5.62510

## Octal Number System

The octal Number System consists of eight digits from 0 to 7. The base of the octal system is 8. Each digit position in this system represents a power of 8. Any digit in this system is always less than 8. The octal number system is used as a shorthand representation of long binary numbers. The number 6418 is not valid in this number system as 8 is not a valid digit.

The place value of each digit according to position and weight is as follows.

Position | 4 | 3 | 2 | 1 | 0 |

Weight | 84 | 83 | 82 | 81 | 80 |

### Example: convert 458 to decimal number

458 = 4 x 81 + 5 x 80

= 4 x 8 + 5 x 1

= 32 + 5

= 3710

## Hexadecimal number system

The Hexadecimal Number System consists of 16 digits from 0 to 9 and A to F. The alphabets A to F represent decimal numbers from 10 to 15. The base of this number system is 16. Each digit position in the hexadecimal system represents a power of 16. The number 76416 is a valid hexadecimal number. It is different from 76410 which is seven hundred and sixty-four. This number system provides a shortcut method to represent long binary numbers.

The place value of each digit according to position and weight is as follows:

Position | 4 | 3 | 2 | 1 | 0 |

Weights | 164 | 163 | 162 | 161 | 160 |

### Example: Convert 3A16 to decimal number

3A16 = 3 x 161 + A x 160

= 3 x 16 + 10 x 1

= 48 + 10

= 5810

I think it would be easier for reader’s to read those example’s if you make them clear by adding sub script than how they are.