What are the uses of number systems
Excursus: number systems¶
Numbers can be represented in different "number systems" in different forms without changing their mathematical meaning.
Historical number systems
The art of counting probably began with the use of bars to represent numbers:
Obviously, this type of representation is very complex and confusing for larger numbers. One of the goals of the various number systems that have developed over time was therefore to connect the respective digits with one another in such a way that the numbers can be represented as simply, clearly and / or appropriately as possible. There are two options:
- Addition systems:
In addition systems, the values of the consecutive digits are linked by addition (and, if necessary, by subtraction). The choice of certain symbols as basic digits is of fundamental importance here,
Addition systems generally have the disadvantage that they quickly become illegible with large numbers.
At Position systems the values of the digits multiplied by a place value resulting from the position are added. The choice of the base and thus the number of digits is of fundamental importance here,
In practice, position systems are used almost exclusively nowadays, as they can also be used to represent large numbers and fractions with great accuracy.
The decimal system
The decimal system is a position system with the base . This results in the following values as position factors:
With these position factors ("place values") the individual digits of a number are multiplied starting from the right.
In the opposite case, a number can be represented by repeated division by the base in the form of its residual division:
The decimal digits of the number to be displayed correspond to the division remainders, provided that these are read from bottom to top. This method of representation is used in particular for conversion, i.e. the transfer of a number from one number system to another.
The binary system
The binary system (also called "dual system") is a position system with the basis . This results in the following values as position factors:
To convert a decimal number to a binary number, the division by remainder method described in the previous section is used. The binary number results from the decimal number by repeated division with , whereby the result can be read from the division remainder from bottom to top.
In order to make the change of the number system clearly recognizable, the respective number base is often used indicated by a corresponding index.
If, in the opposite case, a binary number is to be converted into a decimal number, the digits that occur must be multiplied by their respective position factors and the results then added up.
Even if the long sequences of ones and zeros seem unusual at first, they have proven to be of fundamental importance, especially in the development of computer systems. Even with the current state of the art, binary numbers make the storage and transmission of data considerably easier and make their processing possible in the first place with the help of microcontrollers.
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