History of Numeral Systems - 2

   The spread of the Hindu-Arabic numeral system was to a large extent facilitated by the translation into Latin in the XII century of "Arithmetic Treatise" by the prominent Arab scholar al-Khwarizmi (circa 780 – circa 850). In this treatise he justified the strict rules of arithmetic operations in a system, which was new for Europe. Subsequently, these rules were called algorithms, from the Latinized form of the name al-Khwarizmi, who was known in Europe as Algoritmi. Now the concept of the algorithm is one of the most common in science, especially in mathematics. An algorithm is any rule for performing certain actions in order to achieve a specific result, if this rule is reduced to a finite number of clearly defined and described basic operations – algorithm steps. Incidentally, the well-known rules of columnar addition, long multiplication and division are algorithms. Solving any geometric construction problem boils down to the elaboration of an appropriate algorithm.

   The first "European" treatise on justification of algorithms for performing operations in the Arab-Indian numeral system was known starting from 1202. It was created by the Italian mathematician Leonardo of Pisa (circa 1170 - after 1228). However, only in the XVI century did the Hindu-Arabic numeral system gain universal acceptance in Europe.

   Basic arithmetic operations with numbers – addition, subtraction, multiplication and division – as we know, rest upon addition and multiplication tables, which state the results of operations for few-digit numbers. For example, in order to learn the multiplication table in the decimal system you only need to remember (taking into account the commutative property and elementary multiplication by zero and one) 8 + 7 + 6 + ... + 2 + 1 = 36 small numerical results. However, with increasing the base of the numeral system, such tables rapidly grow in size. So, in the Ancient Babylonian sexagesimal system, the multiplication table is composed of 58 + 57 + ... + 2 + 1 = 1711 main results. That's why this positional number system – the first one in history – did not spread.

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