Neolithic

Part 6 - Numbers

The system was widely used in commerce and also used in astronomical and for other calculations. The sexagesimal system was exported from Babylonia and used throughout Mesopotamia (Iraq), and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Syrians (Iran).

The earliest abacus probably dated from the practice of using pebbles on a board to count. Shown below is a later Chinese abacus.

Ancient Greeks used a number system based on the Greek alphabet where alpha was 1, beta 2, gamma 3, delta 4 and so on. This permitted them count up to 999. Larger numbers employed the same numbers to represent thousands, tens of thousands, and hundreds of thousands.

Roman numerals evolved from a primitive system of cutting notches or as a representation of a hand (four finger and the thumb held out looked like a V = 5). Despite its cumbersome appearance it was still used throughout Europe well after the fall of the Roman Empire.

The modern decimal Hindu–Arabic numeral system with zero (0 1 2 3 4 5 6 7 8 9) was developed in India by around 700 CE (current era). (Before that time the concept of zero was not accepted. 'How can nothing be something,' according to one Greek philosopher. Which was the reason why the calendar jumped from 1 BCE to 1 CE. (Jesus was not born in the year zero). Hindu–Arabic numerals slowly spread throughout the middle east and Europe and was in general use by 1500 CE although Roman numerals lingered on clock faces and some documents.

The late Olmec people of south-central Mexico began to use a symbol for zero, a shell glyph, in the New World, possibly 2400 years ago but certainly by 2000 years ago, which became an integral part of Maya numerals and the Maya calendar. Mayan arithmetic used base 4 and base 5 written as base 20.

The concept of negative numbers was recognized about 2000 years ago in China. The first reference in a Western work was 2300 years ago in Greece when Diophantus referred to the equation equivalent to 4x + 20 = 0 (the solution, x, is negative) in Arithmetica, saying that the equation gave an absurd result.

In the 7th century CE, negative numbers were used in India to represent debts. Diophantus' comment was discussed more explicitly by Indian mathematician Brahmagupta, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century CE, in India, Bhaskara found negative roots for some quadratic equations but commented that the negative value, " in this case not to be taken, for it is inadequate; people do not approve of negative roots."

Al-Khwarizmi's published The Compendious Book on Calculation by Completion and Balancing between 813–833 CE. It was first systematic method of solving linear and quadratic equations by "reduction" and "balancing" (the cancellation of terms on opposite sides of the equation), The term algebra came from the word al-jabr meaning "completion" or "rejoining". The term algorithm is derived from his name.

In the 10th century CE, Middle-Eastern mathematicians extended the decimal numeral system to include fractions. The decimal point notation was introduced by Sind ibn Ali, who also wrote the earliest treatise on Arabic numerals.

European mathematicians generally resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts and later as losses. At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral.