Arithmetic
The historical development of arithmetic may have its origins in ancient Mesopotamia around 3000 b.c.; however, the civilizations of Egypt, India, and China have also been credited with its discovery.
The oldest known artifact of mathematical significance is a bone tool handle, bearing notches arranged in numerical patterns, with a piece of quartz at the head of the handle. Known as the Ishango bone, it was discovered in 1962 on the shore of Lake Edward in the Congo (now Zaire) and dates back to the period between 9000 and 6500 b.c. It is believed to be an early counting device, however, its exact purpose is unknown.
The discovery of Sumerian tablets show that Mesopotamian merchants in 3000 b.c. were already familiar with the use of bills, receipts, notes, accounts and other systems of measure. An extensive collection of mathematical artifacts unearthed in the region of Babylon show that arithmetic processes had already been clearly established by 2150 b.c. Multiplication and division tables, tables of squares and square roots, as well as geometric progressions have been discovered dating back to that time.
The abacus, believed to have originated in the Middle East or India around 3000 b.c., marked an important advance in the calculation of numbers. The abacus was the most widely used arithmetical device in existence until algorithms and Hindu-Arabic numerals began to dominate European mathematics in the sixteenth century.
The discovery of Greek multiplication tables as well as addition and subtraction artifacts traceable to the beginning of the Christian era indicate arithmetic was being widely used in Greece as well. However, the ancient Greeks used the term logistics to refer to arithmetic; their use of the word arithmetic alluded to the search for relationships among numbers--a science now known as number theory.
While arithmetic today refers to the addition, subtraction, multiplication and division operations, up until a.d. 1250 there were nine such logistic operations--numeration, addition, subtraction, duplation (doubling), mediation (halving), multiplication, division, progression and extraction of roots. By 1543, arithmetic was reduced to five processes, later four.
Arithmetic has undergone many changes to reach its present state. The term addition was known as aggregation up until the thirteenth century. The operation of subtraction has never been completely standardized. There are roughly four known subtraction methods including: the complementary method, dating back to 1150-1300; the borrowing and repaying method, used by Leonardo Fibonacci and possibly inherited from Eastern Arabs; simple borrowing, dating back to 1140; and the addition method, or Austrian method, first used around 1159.
Likewise, multiplication has evolved from several techniques, including what is called the duplation method, which was used in ancient Egypt. The Greek and Romans employed the abacus and wax tablets for multiplication, while an additional eight methods were discovered and published in 1494 by Italian mathematician Luca Pacioli (1445?-1514?).
Throughout its history, division has been described as the most difficult operation; however, little of its historical development is known. Today, there are about six known methods including: short division; Gerbert's method, dating back to 980 AD ; division by factors, from the late Middle Ages; division by parts, used in the sixteenth century; the galley method, believed to be of Indian origin, in use before 1600; and long division, developed in the thirteenth or fourteenth century.
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