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Tuesday, February 01, 2011

History of Mathematics - Arithmetic

Numbers and counting are a part of everyone’s life and understanding the numbers and their structure is essential to progress the in mathematics, mostly in arithmetic and algebra (Nataraj & Thomas, 2009). Arithmetic come from the Greek word (αριθμός) arithmos meaning number is the oldest and most basic branch of mathematics. They came about because human beings wanted to solve problems and created numbers to solve these problems. It is used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. It is also known as "science of numbers."

The number systems we have today have come through a long route, and mostly from some faraway lands, outside of Europe. The prehistory of arithmetic is limited to a very small number of small artifacts which may indicate conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC although its interpretation is disputed. 

The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. Egyptians were one of the first civilizations to use mathematics in an extensive setting. Their system was derived from base ten and this was probably so because of the number of fingers and toes. These artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. 

Their number system worked very well when doing addition or subtraction. The numbers were grouped together in no particular order and the operation was performed. In one example, from the Rhind Papyrus, addition and subtraction signs were represented through figures which resemble the legs of a person advancing for addition, and departing for subtraction. 

The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base but did not include positional notation. Although addition was generally straightforward, multiplication in Roman arithmetic required the assistance of a counting board to obtain the results. The counting board is ancestor of the abacus, and the earliest known form of a counting device excluding fingers and other very simple methods. Counting boards were made of stone or wood, and the counting was done on the board with beads or pebbles. The oldest known counting board in 300 BC was discovered on the Greek Island of Salamis in 1899. It is thought to have been used by the Babylonians in about 300 BC and is more of a gaming board than a calculating device. 

Early number systems that included positional notation were not decimal, including the sexagesimal (base 12) system for Babylonian numerals and the vigesimal (base 20) system that defined Maya numerals. For this place value concept, the ability to reuse the same digits for different values contributed to more simple and more efficient methods of calculation.

The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. The book entitled Introduction to Arithmetic was written by Nicomachus almost 2,000 years ago and contains both philosophical prose and very basic mathematical ideas. It covers Pythagorean number theory and contains the multiplication table of Greek origin. His book was different with Euclid's book, which represents numbers by lines where Nicomachus used arithmetical notation expressed in ordinary language. Nicomachus referred to Plato (429 - 347 BC) quite often, and wrote about how philosophy can be possible only if one knows enough math. This is his only complete book that has survived to our day. Nicomachus describes how natural numbers and basic mathematical ideas are eternal and unchanging, and in an incorporeal realm. 

The derivation of the Greek numerals of hieratic Egyptian system lacked positional notation, and therefore imposed the same complexity on the basic operations of arithmetic. For example, the ancient mathematician Archimedes (287 - 212 BC) devoted an entire work The Sand Reckoner merely to devising a notation for a certain large integer where he computed the number of grains of sand to fill the universe.

Although the Codex Vigilanus described an early form of Arabic numerals (omitting zero) by 976 AD, Fibonacci was primarily responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202. He considered the significance of this "new" representation of numbers, which he styled the "Method of the Indians" (Latin Modus Indorum), so fundamental that all related mathematical foundations, including the results of Pythagoras and the algorism describing the methods for performing actual calculations, were "almost a mistake" in comparison.

In the middle ages, arithmetic was one of the seven liberal arts taught in universities. The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the huge simplification of computation through decimal notation. Various types of tools exist to assist in numeric calculations. Examples include slide rules (for multiplication, division, and trigonometry) and nomographs in addition to the electrical calculator.

Decimal Number
The gradual development of Hindu-Arabic numerals independently devised the place value principle and positional notation, which combined the simpler methods for computations within decimal base and the use of a digit representing zero to nine. This allowed the system to consistently represent both large and small integers. This approach eventually replaced all other systems. In the early 6th century AD, the Indian mathematician Aryabhata incorporated an existing version of this system in his work, and experimented with different notations.

Negative Number
Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art (Jiu zhang suan-shu), which in its present form dates from the period of the Han Dynasty (202 BC. – AD 220). The Nine Chapters used red counting rods to denote positive coefficients and black rods for negative. These were used for commercial and tax calculations where the black cancelled out the red. 

The use of negative numbers was known early in India, and their role in situations like mathematical problems of debt was understood. During the 7th century AD, negative numbers were used in India to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written in A.D. 628), discussed the use of negative numbers to produce the general form quadratic formula that remains in use today and determined the results for multiplication, division, addition and subtraction of zero and all other numbers, except for the result of division by zero. His contemporary, the Syriac bishop Severus Sebokht described the excellence of this system as "...valuable methods of calculation which surpass description". He also found negative solutions of quadratic equations and gave rules regarding operations involving negative numbers and zero, such as "A debt cut off from nothingness becomes a credit; a credit cut off from nothingness becomes a debt.” He called positive numbers as "fortunes”, zero as “cipher” and negative numbers as "debts." The Arabs also learned this new method and called it as hesab.

Odd and even number
The distinction between odd and even number is one of the most ancient features in the science of arithmetic. The Pythagoreans knew it and their founder may well have learned it in Egypt or in Babylon. The Pythagoreans used the term gnomon for the odd number. A fragment of Philolaus (425 BC) says that "numbers are of two special kinds, odd and even, with a third, even-odd, arising from a mixture of the two." Euclid, Book 7, definition 6 is "An even number is that which is divisible into two parts.”

In English, gnomon is found in 1660 in Stanley, Hist. Philos. (1701): "Odd Numbers they called Gnomons, because being added to Squares, they keep the same Figures; so Gnomons do in Geometry" (OED2). So the ancient Greeks had a word for "odd" that was the word they used for this kind of shape:

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