# Translations of Encyclopedia about Mathematics

## Numbers and Symbols

Our oldest ancestors already knew how to count, such as the number of items, people or animals. They were either able to count objects and visualise numbers or were able to calculate the number of objects in a set. For simple counting, we use basic numbers or cardinal numbers, such as 1, 2, 3, 4 and so on. Serial or ordinal numbers determine order, such as 1st, 2nd or 3rd.

Numbers and symbols are what mathematics is actually made of. There are many numerical and symbol systems in the world.

We most often use the decimal system, which is most common. A number in the decimal system (although generally also in other numerical systems) is, by rule, made up of several numbers or digits, where the value of each digit depends on its position in the number. Digits within the decimal system are the Indian symbols for 1 to 0, which were adopted by al-Choresmi for the Arabian mathematical system.

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## Numbering Systems

Numbers can look quite differently. Some numbers were in wedge shaped form, and Arabian numbers, which originally came from India, also look different. The Romans and the Chinese also used totally different symbols.

The Roman numbering system is based on the adding system, where there are a certain number of basic numbers and aiding symbols, combined according to some rules to form a number with the aim of using as few amount of symbols as possible. Even so, larger numbers under this system are not easily readable and, with very large numbers, new symbols must be invented.

Under this system, the capital letter of I is used to represent one unit, V to represent five units, X ten and C one hundred.

Therefore, seven would be written as VII (V+I+I). If a smaller unit symbol is to the left of a larger one, then it is subtracted, such that the number 9 is written as IX (X-I).

Thirty is written as XXX and 321 as CCCXXXI (100+100+100+10+10+1).

The present numbering system, where a number’s value is described by the number of the digits with their individual positions, has its origin in India. According to this system, the position of a number determines its value. Therefore, for the number 321, the number 1 has a value of 1, the number 2 twenty and the number 3 three hundred.

This system was brought to Europe by the Brit Aderland von Bath in the year 1100.

The present numbering system is based on multiples of ten and use the digits from 1 to 0.

Another numbering system is the binary system. In this system, there are only two value numbers. For example, the number 10101 is read as follows: 1 is one, 0 is two, 1 is four, 0 is eight and 1 sixteen. When we add all these together, we get the number 21 (according to our decimal numbering system).

The symbols 0 and 1 are used to represent the numbers 0 to 9 as well. Under the binary system, the number 9 is represented as 1001. The binary numbering system is read from right to left, starting with the exponent 0. Each digit to the left has one higher exponent number, where the binary symbol representing each digit is multiplied by the number 2, which has a power of the exponent associated with that digit, such that: .

The binary numbering system is used mostly in semi-conductor electronics when describing on-off chain states.

Other numbering system use the numbers 5 or 16 as their base.

Below are shown several numbering systems:

Decimal

Symbols: 0,1,2,3,4,5,6,7,8,9
Base:10

Binary

Symbols:0.1
Base: 2

Octal

Symbols: 0,1,2,3,4,5,6,7
Base: 8

Symbols: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Base: 16

Threescore

Symbols: 0…59
Base: 60

The most varied of ways to describe values have been invented, such as the symbols used by the Romans or the Arabs (Indian).

Numbers can then be classified differently, such as "natural numbers", "rational numbers" or "whole numbers". Each category has its own characteristic which is revealed through its mathematical use.

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