Tallies by carving notches in wood, bone, and stone were used for at least forty thousand years. Stone age cultures, including ancient American Indian groups, used tallies for gambling with horses, slaves, personal services and trade-goods.
Roman Numerals evolved from this primitive system of cutting notches. The V for five was cut as two notches to represent a person's hand of five fingers (four fingers separated from the thumb by a V shaped gap). The X for ten was cut as two crossed notches to represent two hands.
Beginning about 3500 BC the tokens and envelopes were replaced by numerals impressed with a round stylus at different angles in flat clay tablets which were then baked. A sharp stylus was used to carve pictographs representing various tokens. Each sign represented both the commodity being counted and the quantity or volume of that commodity.
About 3100 BC written numbers were dissociated from the things being counted and abstract numerals were invented. The things being counted were indicated by pictographs carved with a sharp stylus next to round-stylus numerals.
The Sumerians had a complex assortment of incompatible number systems and each city had their own local way of writing numerals. In the city of Uruk about 3100 BC, there were more than a dozen different numeric systems. One number system was used for counting discrete objects such as animals, tools, and containers. A different system was for counting cheese and grain products. Another system was used to count volumes of grain and included fractions. Another system counted beer ingredients. Another system counted weights. Another system counted land areas. Another system counted time units and calendar units. And these systems changed over the years. Numbers for counting volumes of grain changed whenever the size of the baskets changed. People who added and subtracted volumes of grain every day used their arithmetic skills to count other things that were unrelated to volume measurements.
The Sumerians invented the wheel and also invented arithmetic. Multiplication and division were done with multiplication tables baked in clay tablets.
Between 2700 BC and 2000 BC, the round stylus was gradually replaced by a reed stylus that had been used to press wedge shaped cuneiform signs in clay. To represent numbers that previously had been pressed with a round stylus, these cuneiform number signs were pressed in a circular pattern and they retained the additive sign-value notation that originated with tokens on a string. Cuneiform numerals and archaic numerals were ambiguous because they represented various numeric systems that differed depending on what was being counted. About 2100 BC in Sumer, these proto-sexagesimal sign-value systems gradually converged on a common sexagesimal number system that was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions. This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.
Sexagesimal numerals were a Mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. Sexagesimal numerals became widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. In Arabic numerals, we still use sexagesimal to count time (minutes per hour), and angles (degrees).