The logical study of numerical and spatial relationships. It is usually divided into pure and applied mathematics. In pure mathematics the general theoretical principles are studied, often in abstract. Its branches are arithmetic, algebra, calculus, geometry, and trigonometry. Some form of mathematical calculation is an indispensable part of all financial transactions and all measurements. The ancient Egyptians, Sumerians, and Chinese were all using a form of abacus to carry out these calculations for thousands of years before the Christian era. But it was not until the 9th century AD that al-Khwarizmi introduced the idea of writing down calculations instead of carrying them out on an abacus.

The Venetian mathematicians of the 11th and 12th centuries were largely responsible for the introduction of these methods to the West; indeed it was they who showed that commercial calculations based on algorisms (a word derived from al-Khwarizmi's name) were superior to those performed on an abacus. However, the application of mathematics to the physical sciences (including astronomy) was largely a 16th-century development inspired by Galileo. It was from this development that applied mathematics grew. It is now largely concerned with mechanics and statistics. In the 20th century set theory and chaos theory were developed, while the introduction of electronic calculators and computers made calculations extremely easy.

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