The Sumerian civilization also provided many inventions and innovations to the path of human progress, such as cuneiform writing, wheel,  saw,  spear,  plough, irrigation, money, and others.

In addition to all of this, a number of archaeologists and arithmetic scholars were interested in the emergence of mathematics and arithmetic at the hands of the Sumerians and Babylonians. Mathematics flourished in Mesopotamia in the period extending from the beginning of the appearance of the Sumerians until the fall of Babylon in the year 539 AD, where numbers and manual computing were born at the hands of the ancient Iraqis who created the first human civilizations in the valley of Mesopotamia.

 The inhabitants of the first known civilization of Sumer (ca. 5000 BC) were the first to keep records of commercial transactions on baked clay tablets.

the beginning

Basic concepts of numbers began to emerge when the world's first civilization, the Sumerian civilization, established its roots in Mesopotamia more than 5,000 years ago. This digital system used the positional feature for the first time (that is, the value of the symbol used depends on its position in the number).

In fact, the Sumerian accountant who lived in the lower part of the Mesopotamian Valley around 3200 BC may have been the first person to record numbers as a means of storage, as he used the sexagesimal numeric system, which is based on bases 6 and 10, which can be done using the 12 joints in the fingers of one hand along with 5 fingers of the other hand.

Overall, the Sumerians laid crucial foundations for the development of mathematics that would influence later civilizations in Mesopotamia, Egypt, Greece, and beyond. Their number systems and computational abilities were remarkably advanced for their time.

The discovery of arithmetic brought tangible benefits to the Sumerian civilization, including the ability to numerically determine the products of their economy, and contributed to the growth of Mesopotamian trade, and other civilizations later benefited from it.

Why did the Sumerians advance in mathematics?

The Sumerians initially developed mathematics as a response to bureaucratic needs as their civilization settled and developed agriculture (possibly as far back as the 6th millennium BC) to measure agricultural land, calculate individual taxes, and the like.

In addition, the Sumerians and Babylonians needed to write relatively large numbers as they attempted to chart the night sky and construct their complex lunar calendar.

So, a small clay cone began to be used to represent the number 1, a clay ball to represent the number 10, and a large cone to represent the number 60. This was during the fourth millennium BC.

During the third millennium, these shapes were replaced by other symbols in cuneiform writing, so that they were written using the same pen used to write words.

Digital systems

Basic concepts of numbers began to emerge when the world's first civilization, the Sumerian civilization, established its roots in Mesopotamia more than 5,000 years ago, this digital system used the principle of position for the first time (that is, the value of the symbol used depends on its position in the number).

The Sumerians had a complex variety of numerical systems, and each city had its own local way of writing numbers.

In the city of Uruk, around 3100 BC, there were more than ten different numerical systems. For example, a number system was used to count various things such as animals, tools, and containers.

There was a different system for counting cheese and grain products. Another system was used to calculate quantities of grains, another for beer ingredients, another for weights, another for land areas, time units, and calendar units, and these systems changed over the years.

The numbers changed to calculate the quantities of grains as the size of the baskets changed, people handled quantities of grains every day and used their arithmetic skills to calculate other matters that were unrelated to volume measurements.

The Sumerians invented arithmetic, including multiplication and division, and multiplication tables were written on clay tablets with a round pen.

The ancient Sumerians of Mesopotamia developed a complex system of measurement as early as 3000 BC. Beginning in 2600 BC, the Sumerians wrote multiplication tables on clay tablets and dealt with geometry exercises and division problems, and the earliest traces of the beginning of Babylonian numerals also date back to the same date.

These symbols developed into complex systems of numbers around 3000 BC, capable of registering very large quantities of goods, the writer uses different sets of symbols to record quantities of sheep or grain, more than a dozen different such systems are known to have existed. 

There was no unified rule, just as we do not have any unified rule for all the different units of weights and measures outside the metric system.

The sexagesimal system

At that time, the Sumerians developed unique number systems, using the base of sixty in scientific terms, and this system is called sexagesimal, the base of the system was 60, while the base of ten is what we use today.

The Sumerians counted things with sixty as one unit. They had the same symbol for the numbers 1 and 60. To express 70, they literally expressed it as the sum of 60 and 10. Likewise, they expressed 125 as the sum of two units of 60 and one unit of 5. The Sumerians used sexagesimals not only because the number 60 has many divisors or is countable on the fingers of both hands but because 60 is the least common multiple of the number of fingers on both hands and the number of months in the year.

The Sumerians used their knuckles to count the duodenal system (12), and divided the day, from sunrise to sunset, into 12 parts, so that night and day together were divided into 24 parts.

The Babylonians developed cuneiform based on the cuneiform or “wedge form”. They wrote these symbols on wet clay tablets that were baked in the hot sun, several thousand of these discs are still present today in museums inside and outside Iraq. The Babylonians used the wedge to imprint symbols on clay because curved lines could not be used easily.

Between 2700 BC and 2000 BC, the round pen was gradually replaced by the reed stylus, which was used to press wedges that formed cuneiform marks in clay, to represent numbers whose symbols had previously been represented by the round pen.

Ancient cuneiform numbers were ambiguous because they represented various numerical systems that varied depending on what was being sorted.

In about 2100 BC in Sumer, these systems gradually converged on a common sexagesimal number system which was a place-value system consisting of only two signs, the vertical and the horizontal wedges, which could also represent fractions.

This sexagesimal system was fully developed at the beginning of the ancient Babylonian period (circa 1950 BC), and became the standard in Babylon. 

The sexagesimal numbers, which retained base 10 and base 6, alternated in a series of vertical and horizontal cuneiform symbols (wedges), one of the oldest numbering systems.

The first mathematics can also be traced back to ancient Babylon, during the third millennium BC, where the Babylonians were most familiar with tables that helped them solve problems. The sexagesimal system became the system widely used in trade, but it was also used in astronomical and other calculations.

The Sumerian System, called "sexagesimal", combined a mundane 10... with a "celestial" 6, to obtain the base figure 60. This system is in some ways superior to our present one, and much superior to later Greek and Roman systems. It enabled Sumerians to divide into fractions and multiply into the million, to calculate roots or raise numbers several powers.

 

Also, to represent the numbers 1 - 59 within each place value, two distinct symbols were used, a unit symbol (Description: 1) and a ten symbol (Description: 10) which were combined in a similar way to the familiar system of Roman< numerals (e.g. 23 would be shown as Description: 23). Thus, Description: 1Description: 23represents 60 plus 23, or 83. However, the number 60 was represented by the same symbol as the number 1 and, because they lacked an equivalent of the decimal point, the actual place value of a symbol often had to be inferred from the context.

 

Over the next five hundred years, writing gradually developed into cuneiform and the advent of arithmetic came hand in hand with the advent of cuneiform. Clay tablets are no longer used merely to record the numbers of goods, but we find them in recording arithmetic totals and accounts.

Even before the Old Babylonian period, around 2000 BC, there was fully developed mathematics. Thousands of mathematical and economic tablets were discovered, showing a remarkable knowledge of arithmetic, linear and quadratic equations, and many geometric and arithmetic constructions.

There are multiplication tables, tables of areas, square roots, and common constants. There are lists of arithmetic problems created by teachers, and the solutions they provided to students. At this time, the number system for counting had settled on the sexagesimal system, or base of sixty, and it was easier for them to calculate fractions.

How has Sumerian mathematics affected current mathematical applications?

Although it is no longer used for general computation, the sexagesimal system is still used to measure angles, geographic coordinates and time.

Today, our decimal number system uses base ten, not sixty, as the basic unit, but this does not mean that the Sumerians' invention has become obsolete.

 As a matter of fact, it still plays a crucial role in our daily lives. For example, have you ever wondered why an hour has 60 minutes and a minute has 60 seconds? Have you ever thought why a full circle has 360 degrees? As it turns out, this was what kept the Sumerians on track of their time, and it was how they came full circle. The Babylonians divided the day into twenty-four hours, every hour into sixty minutes, and every minute into sixty seconds. 

This form of counting time has survived for four thousand years to this day