Eleven twelve and counting systems

Our math is strongly linked with our fingers. If we ever meet another civilization, even on this earth, the numbers will probably be different.  How come? This discussion is an exercise in basic mathematical and historic intuition. Very few people see math the way I present it here.

How do you count fingers?

We have ten fingers, so our natural counting is decimal. Fold a finger per number. Roman generals and Indian gurus used a decimal system independently of each other. The Brahmi numerals at the basis of the system predated 4th century BC.

By the way, there are at least five different decimal finger counting systems. With more systems being described.

Ancient Babylonians and occasionally Chinese pointed with the thumb to each finger bone on one hand (12 positions), while folding finders on another (5 positions).  The resulting system, called sexagesimal had the base 60 and special signs and names for the relevant numbers.

Many European languages used 12 as the basis with specials words for 11 and 12, like we do in English. But this is not all.

You have two ways of folding each finger: partially or fully. This generates a basis of 20, called vigesimal. We have rudiments of this in French, Georgian, and many other languages. From Maya in America to Yoruba in Africa and Maori in New Zealand. This was probably the most wildly used ancient counting system.

Why do we use Arabic numerals?

Who needed numbers more than everybody else? Possibly accountants. For the army, for the administration, and for engineering it was crucial to count items and prevent theft.

The most powerful driver for rigorous, fully self-consistent use of sexagesimal has always been its mathematical advantages for writing and calculating fractions [60=3*4*5]. In ancient texts this shows up in the fact that sexagesimal is used most uniformly and consistently in mathematical tables of data. Another practical factor that helped expand the use of sexagesimal in the past even if less consistently than in mathematical tables, was its decided advantages to merchants and buyers for making everyday financial transactions easier when they involved bargaining for and dividing up larger quantities of goods. The early shekel in particular was one-sixtieth of a mana.

Then the trade routes shifted. In the middle ages, spices, precious stones, and Damascus steel were often more valuable than gold. The medieval trade often passed through Arabs, who went to India, Byzant, and Europe. The first dated and undisputed inscription showing the use of a symbol for zero appears on a stone inscription found at the Chaturbhuja Temple at Gwalior in India, dated 876. The system was extended to include fractions, as recorded in a treatise y Syrian mathematician Abu’l-Hasan al-Uqlidisi in 952–953. Indian religion and science also spread to China. In China, Gautama Siddha introduced Hindu numerals with zero in 718, but Chinese mathematicians did not find them useful, as they had already had the decimal positional counting rods. Leonardo Fibonacci brought this system to Europe in the 12th century.

So while the Indian decimal system was initially inferior to the Babylonian system, the commercial and cultural power of India pushed the system into every medieval superpower. And the Indians also fixed all the shortcomings of their system.

Where do we still see base 60?

All astrology books were derived from Babylonian texts. So the astrologists, astronomers, and physicists were using Babylonian system of counting the time. Even Tamil astronomers used Hellenistic tables, which in turn used Babylonian tables. As a result, our hour is divided into 60 min, and they are divided into 60 seconds. And we also use six sixties of degrees to measure a circle.

Learning to count up to 60, especially in a different language, can be hard. But the more complex operations suddenly get easy. The mental math was superseded by slide rule only in the 17th century based on the emerging work on logarithms by John Napier. Before the advent of the electronic calculator, it was the most commonly used calculation tool in science and engineering.

The revolutionary metric system

The medieval measurement systems were complex, natural, and irrational. They were using feet, steps (yards), inches (1/12th), and other strange notations with non-decimal backbone. The French revolution was trying to promote a more rational way of life.

Charles Maurice de Talleyrand championed a new system based on natural units, proposing to the French National Assembly in 1790 that such a system be developed. They defined that length to be the ‘metre’ and its length as one ten-millionth of the length of a quadrant on the Earth’s surface from the equator to the north pole. In 1799, after the length of that quadrant had been surveyed, the new system was launched in France.

The new units were developed based on this meter for everything except time and temperature. For example, The kilogram was originally defined in 1795 as the mass of one litre of water. The litre The word litre is derived from an older French unit, the litron, whose name came from Greek and it was not the modern litre. To fix the issue the litre was redefined as 0.001 of a cubic meter.

The new system made engineering computations easier, and immediately after its introduction, we see an industrial revolution.

Modern systems

With new shiny computers, our math is changing. I think that I use hexadecimal numbers as much as I use regular decimal numbers. The decimal system appears just as strange as the (medieval) American system of weights and measures. And I constantly need to convert between the two.

The old systems are both very arbitrary and not accurate enough. Modern physics works with strange numbers and often uses the Planck units.

Originally proposed in 1899 by German physicist Max Planck, these units are a system of natural units because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are only one of several systems of natural units, but Planck units are not based on properties of any prototype object or particle (the choice of which is inherently arbitrary), but rather on only the properties of free space. They are relevant in research on unified theories such as quantum gravity.

These units are not very useful in everyday life. To be honest we even measure deferent objects, like gigabytes of data.

The percent is also something very natural, meaning “a proportion for every hundred” and it comes from the 16th-century trade. However, we supplement it with statistical measures like six sigma that come from Gaussian distribution from 19th century.

We still combine very conventional old numbers with floating numbers of engineering conventions, and with the naturally occurring numbers and measures. This is uncomfortable and confusing. It is probably not going to work forever. The singularity is coming and everything will possibly change.


Some of the old measures were made from very durable materials like iridium, and placed in large museums. We have some nostalgic feeling to the times when human fingers or the size of our planet were the measures for everything.

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