Numeral system

A numbering system is used to represent numbers. A number is represented according to the rules of the respective number system as a sequence of numbers or numerals.

Modern research distinguishes between additive, hybrid and positional ( status ) number systems.

Addition systems

In addition system, a number is represented as the sum of its digits. The position of each item does not matter.

An example is the line system ( Unärsystem ), which lends itself, if something is to be counted in writing ( such as the drinks on a beer mat ). Here, the number is represented by dashes. This is probably one of the oldest counting systems at all. The Unärsystem is very hard to handle the preparation of relatively large numbers. Therefore, it is customary to combine the numbers in blocks, for example by placing every fifth line across the previous four single lines. Although it is not suitable for this reason, represent large numbers, it is still used in everyday life in some situations. An addition to a numerical value is easily done by adding a stroke. Conventional systems do not allow for such a simple and fast expansion in general.

Hybrid systems

Here, a basic figure is preceded by a character, which represents a power of the base; the values ​​of both are multiplied with each other. In the European number systems such hybrid systems were not as good as before, but rather, since the beginning of the second millennium BC in Mesopotamia, and later in China and the Middle East generally. Both of Ethiopia, as well as from southern India and Sri Lanka, as well as the Mayan culture are known such hybrid payment systems.

Examples in Japanese / Chinese number system

23:二十 三(2 x 10 3 ) 30,000 :三万(3 x 10,000 ) Value systems

Location

In the place value system ( positional system ) used in Europe in everyday life and science, a number is represented by a sequence of digits, each digit must be evaluated differently depending on the position in the network. Each place in this arrangement to be taken by a digit or to occupy, is a place ( " one's place ", " tens ", ...). The " lowest " point is always right.

A priority system has a base ( also referred to as a p-adic number system ). Each digit or digit position has a value corresponding to a power of the base. Numbered to the units digit to 0, then the - th digit has the value.

The calculation of the numerical value by multiplying the single digit values ​​together with the respective values ​​and the addition of these products:

For the presentation of this numerical value points are needed and assigned.

Digit

Digits are characters ( digit symbols ) for the first natural numbers from zero.

Any integer can be used as the basis of a value system. The most common bases are two (dual system ), 8 ( octal ), 10 (our familiar decimal ) or 16 ( which is important in computer science Sedezimalsystem ).

For the presentation in such a system digits are required, ranging from to.

Properties

• Two adjacent point values ​​differ by a factor.
• With the restriction of the lowest exponent to 0 can only represent integers. Leaving negative exponent, you can write any rational numbers in a place value system, the transition from non-negative to negative exponents is marked by a delimiter such as a comma:

• The digits of a rational number is obtained by the method of long division. In numeric system is also called a decimal fraction development. In many rational numbers and irrational numbers in the number of jobs is unlimited. The necessary limitation in the number of points to be specified based by how far the points are significant digits.
• The base need not necessarily be an integer. It has been demonstrated that all the complex numbers can be used as the basis of an amount greater than 1 value system. Similarly, number systems with mixed bases possible. Examples of this can be found in Knuth: The Art of Computer Programming.