# Why do we use a number system

## Number systems

Number systems are used to represent numbers. The numbers are represented according to certain rules as a sequence of digits or characters. The number systems best known to us are the decimal system (system of ten), the dual system (system of two) and the hexadecimal system (system of sixteen). There are other number systems, but they do not play a major role in digital technology and computer technology.

Every number system consists of nominal values. The number of nominal values results from the base. The highest nominal value corresponds to the base minus (-) 1. If the highest nominal value is exceeded, the next higher value results from the carryover.

### Decimal number system

**Nominal values:** 0 1 2 3 4 5 6 7 8 9**Base:** 10**Largest face value:** 9**Significance:** 10^{0} = 1, 10^{1} = 10, 10^{2} = 100, etc.

### Dual number system

**Nominal values:** 0 1**Base:** 2**Largest face value:** 1**Significance:** 2^{0} = 1, 2^{1} = 2, 2^{2} = 4, etc.

### Hexadecimal number system

**Nominal values:** 0 1 2 3 4 5 6 7 8 9 A B C D E F**Base:** 16**Largest face value:** F.**Significance:** 16^{0} = 1, 16^{1} = 16, 16^{2} = 256, etc.

### Octal number system

**Nominal values:** 0 1 2 3 4 5 6 7**Base:** 8**Largest face value:** 7**Significance:** 8^{0} = 1, 8^{1} = 8, 8^{2} = 64, etc.

### Number systems in comparison

Binary / dual | Octal | Decimal | Hexadecimal | |||||||
---|---|---|---|---|---|---|---|---|---|---|

16 | 8 | 4 | 2 | 1 | 8 | 1 | 10 | 1 | 16 | 1 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||

0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | |||

0 | 0 | 0 | 1 | 0 | 2 | 2 | 2 | |||

0 | 0 | 0 | 1 | 1 | 3 | 3 | 3 | |||

0 | 0 | 1 | 0 | 0 | 4 | 4 | 4 | |||

0 | 0 | 1 | 0 | 1 | 5 | 5 | 5 | |||

0 | 0 | 1 | 1 | 0 | 6 | 6 | 6 | |||

0 | 0 | 1 | 1 | 1 | 7 | 7 | 7 | |||

0 | 1 | 0 | 0 | 0 | 1 | 0 | 8 | 8 | ||

0 | 1 | 0 | 0 | 1 | 1 | 1 | 9 | 9 | ||

0 | 1 | 0 | 1 | 0 | 1 | 2 | 1 | 0 | A. | |

0 | 1 | 0 | 1 | 1 | 1 | 3 | 1 | 1 | B. | |

0 | 1 | 1 | 0 | 0 | 1 | 4 | 1 | 2 | C. | |

0 | 1 | 1 | 0 | 1 | 1 | 5 | 1 | 3 | D. | |

0 | 1 | 1 | 1 | 0 | 1 | 6 | 1 | 4 | E. | |

0 | 1 | 1 | 1 | 1 | 1 | 7 | 1 | 5 | F. | |

1 | 0 | 0 | 0 | 0 | 2 | 0 | 1 | 6 | 1 | 0 |

1 | 0 | 0 | 0 | 1 | 2 | 1 | 1 | 7 | 1 | 1 |

1 | 0 | 0 | 1 | 0 | 2 | 2 | 1 | 8 | 1 | 2 |

1 | 0 | 0 | 1 | 1 | 2 | 3 | 1 | 9 | 1 | 3 |

1 | 0 | 1 | 0 | 0 | 2 | 4 | 2 | 0 | 1 | 4 |

1 | 0 | 1 | 0 | 1 | 2 | 5 | 2 | 1 | 5 | |

1 | 0 | 1 | 1 | 0 | 2 | 6 | 2 | 2 | 1 | 6 |

1 | 0 | 1 | 1 | 1 | 2 | 7 | 2 | 3 | 1 | 7 |

1 | 1 | 0 | 0 | 0 | 3 | 0 | 2 | 4 | 1 | 8 |

...

### Notation when using different number systems

If you work with numbers from different number systems, then these numbers cannot always be clearly assigned to a number system. The number 100 could belong to the hexadecimal, the dual or the decimal number system. In all number systems, the number 100 would have a different value. That is why numbers are indexed if there is a risk of confusion. Correctly, the index should be represented as a number and correspond to the base of the number. Spellings with letters, regardless of whether they are upper or lower case, are worthy of interpretation because of their similarity to hexadecimal representation and are only valid if the representation system does not allow subscripts.

- Decimal numbers are marked with a small d or 10 (e.g. 100d or 100
_{10}) or not. - Hexadecimal numbers are marked with a lower case h or 16 (e.g. 100h or 100
_{16}) or $ (e.g. $ 100). - Dual numbers are marked with a small b or 2 (e.g. 100b or 100
_{2}) or% (e.g.% 100).

### Conversion of number systems

### Tasks: Converting number systems

### Exercises: Converting number systems

### Numbers in computer science

### Other related topics:

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