Counting started at least forty thousand years ago, with tally sticks or bones carved with notches to mark a number of warriors, animals or the passage of time such as the lunar cycle.
Fingers and thumbs were a quick way of counting the number of animals or enemy warriors and this determined our use of base ten arithmetic. (Imaging using a different base if we had evolved with four fingers on each hand (1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 20. Computers use base two (binary) arithmetic 00, 01, 10, 11, 100, 101, 110, 111, 1000).
About 5000 years ago people needed a way to remember numbers larger than ten so they started counting the knuckles on each hand. This developed into the base 60 sexagesimal system which we still use for time and angles, including geographic coordinates.
(Twelve knuckles on the four fingers of one hand multiplied by 5 digits on other = 60. We still use 60 minutes in one hour, 60 seconds in a minute and 24 hours in a day (2 x 12). Early estimates of 360 days in year (6 x 60) were later corrected to 365.25 days. But there are still 360 degrees in a full circle and 360 degrees of longitude and latitude around the Earth).
Until the last century, British coinage included 12 pence in one shilling and 20 shillings or 240 pence (= 4 crowns of 60 pence) in one pound sterling. And we still buy donuts in boxes of six or twelve.
The first recorded use of numbers was 5400 years ago in Mesopotamia (Iraq). Farmers who were trading sheep or goats needed to know how many were for sale. This was the base 60 or sexagesimal system. The earliest known base 10 system dates from 5100 years ago in Egypt.Baked clay tokens were used in Iran about 6,000 years ago to record the number of sheep. Some tokens were marked to represent one sheep and others for ten sheep and they were strung like beads on a string. Tokens were marked with a different symbol for goats. These were placed in a sealed, hollow clay ball envelope and baked to ensure that nobody could alter the number and type of tokens.
To avoid damage to the tokens, number signs, symbol marks (+ indicated sheep) and witness seals were pressed into the outside of the balls before they were baked. Since there was seldom a reason to open the balls to verify the number, the markings on the outside became the first written language.
Beginning about 5500 years ago the tokens and balls were replaced by numerals impressed in flat clay tablets with a round stylus. A sharp stylus was then used to carve a pictograph representing the commodity being counted before the tablet was baked.
By 5100 years ago in the city of Uruk (in Southern Iraq), the Sumerians had invented more than a dozen incompatible numeric systems. There were separate number for animals, tools, and containers, cheese and grain products, volumes of grain (including fractions), beer ingredients, weights, land areas and time and calendar units. Adding to the confusion, these systems changed. For example, the size of the baskets used to measure volumes of grain changed over time.
The Sumerians also invented arithmetic. People who added and subtracted volumes of grain every day used their skills to count other things; areas and dimensions of land property in particular. Multiplication tables were baked into clay tablets to help with multiplication and division.
About 4000 years ago in Babylonia (Iraq), these various sign-number systems were gradually standardized into a sexagesimal system consisting of only two impressed marks. Sexagesimal numerals combined base 10 and base 60 in a sequence of cuneiform vertical wedges and chevrons. They were also used to represent fractions.
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.