# Number system

A number system (more rarely numbering system called) is a system for representation of numbers . The presentation format is not mandatory, many number systems are implemented in a number font or number words. A number is represented according to syntactic rules as a sequence of numerals , also called digits. Modern research differentiates between additive , hybrid and positional ( place value ) number systems.

In an addition system, a number is represented as the sum of the values ​​of its digits. The position of the individual digits does not matter.

One example is the line system ( unary system ), which is useful when something is to be counted in writing (such as the drinks on a beer mat). The number is represented by dashes. This is probably one of the oldest counting systems around. The unary system quickly becomes confusing when displaying larger numbers. That is why it is usually common to summarize the numbers in blocks by placing about every fifth line across the previous four single lines. Although it is not suitable for displaying large numbers for this reason, it is still used in everyday situations in some situations. Adding a numerical value is possible simply by adding a dash. Conventional systems generally do not allow such a simple and quick expansion. ${\ displaystyle n}$${\ displaystyle n}$

## Hybrid systems

A basic digit is placed in front of a character that represents a power of the base; the values ​​of both are multiplied together. Such hybrid systems hardly ever appeared in European number systems, but they did, since the beginning of the second millennium BC. BC, in Mesopotamia, later also in China and in the Middle East in general. Such hybrid number systems are known from Ethiopia as well as from South India and Sri Lanka as well as the Maya culture.

Examples in the Japanese - Chinese number system :

```    23:  二十三  (2 × 10 + 3)
30.000:  三万    (3 × 10.000)
```

## Place value systems

### construction

In everyday life and in science, a number is usually represented by digits (0, 1, 2, ..., 9, which only represent the first ten of the natural numbers , and letters) and other numerical signs such as signs (plus, minus) and separators (comma , Spaces). The number of digits used is called the "basis of the place value system". The most common bases are 2 (for the dual system ), 8 (for the octal system ), 10 (for the decimal system used in everyday life ) or 16 (for the hexadecimal system, which is important in data processing ).

The digits have an order of their value determined by convention. When counting up (this corresponds to the addition of a one), the sequence proceeds to the next digit. When adding a one to the most significant digit, a move is made to the least significant digit, and a one is added to the next higher digit.

For this purpose, the digits are valued differently depending on their position, the place value being a power of the base (for example "units", "tens", "hundreds", ...). The position with the lowest rating is on the far right. The numerical value is then calculated by multiplying the individual numerical values ​​with the associated place values ​​and adding these products.

In this way, any natural number can be represented in a place value system. For the extension to negative numbers, a sign is placed to the left of the sequence of digits, which indicates whether a number is positive or negative . By using negative exponents, rational numbers can also be written in a place value system , with the transition from non-negative to negative exponents being marked by a separator in the number representation, for example a comma or a point.

### Display area

Number line
Number circle

The number of numbers that can be represented can be illustrated with an unlimited number of digits on a number line . If only a limited number of positions is available, this is illustrated by a circle of numbers . With this restriction, adding or subtracting numbers can lead out of the range of the representable numbers.

## literature

• Georges Ifrah: Universal History of Numbers . 2nd Edition. Campus-Verlag, Frankfurt / Main 1987, ISBN 3-593-33666-9 .
• John D. Barrow: Why the world is mathematical . Campus-Verlag, Frankfurt / Main 1993, ISBN 3-593-34956-6 .
• Guido Walz (Ed.): Lexicon of Mathematics. Volume 5: Sed to Zyl. 2nd Edition. Springer, Mannheim 2017, p. 442 f. (Payment system).