Language Pulsations: Hindu-Arabic numerals Part I

The Indian place-value numeral system “has been propagated even more widely than the alphabet of Phoenician origin, and it has now become the only real universal language.” — Georges Ifrah

0, 1,2, 3, 4, 5, 6, 7, 8, 9, 10… We usually call these “Arabic numbers.” Like most conventional understandings of complicated things, this is misleading. Actually, our numbering system is an intricate and complex amalgam of various cultural and intellectual influences, from Sumer to Phoenicia to India to the Arabic world.

First, a caveat. When we talk about the history of a numbering system, there are at least three aspects we can discuss: the mathematical attributes of the system (such as the base), the evolution of the digit symbols (1, 2, 3,..), and naming conventions in a given language (one, two, three,…). In this post, I am going to discuss the mathematical attributes, and the other two I will save for a second post.

What do we know about our “Arabic” numbering system? First, it utilizes a base; second,  it is a decimal numeral system; third, it is place notational; and fourth, it uses zero to serve two different functions.

The most obvious means of inventing a counting system would be give each value a different name. This seems like a fine idea until you find yourself having to invent and memorize an infinite variety of different digits arranged in an arbitrary order. Learning such a numbering system would be akin to memorizing pi, but exceedingly more torturous, and doing mathematical operations is out of the question. Instead, we need to somehow limit the number of unique digits in our numbering system while maintaining the ability to count to higher values.

The mathematical base (or radix) does just this. The base goes back to our pebble-counting days in the fertile Tigris-Euphrates valley. Archaeological evidence from Sumer definitively proves that the number-system base was determined during the pre-writing era, when Sumerian farmers and traders had to keep records of their goods and transactions. At first, Sumerian used pebbles amongst themselves as tokens. The shape of the token indicated its value, and markings inscribed on the tokens would indicate the object being counted. Eventually, Sumerians invented an archiving method based on this token system:

They put the tokens in a clay sphere and baked them. Markings on the surface of the sphere indicated what was inside (i.e., the nature of the transaction). The next natural evolution was to convert the sphere into a tablet that described the transaction with no tokens actually involved. (Source.)

Spherical envelope and accountancy tokens, Louvre Museum

This process of token to globular envelope to tablet is crucial in the history of writing. According to a widely accepted theory by archaeologist Denise Schmandt-Besserat, these bookkeeping tokens are the precursor to Sumerian cuneiform and therefore the origin of all writing.

Now that we have decided on a base, what number will we choose for that base? Sumerians ended up with a base-60 numbering system because they had to reconcile preexisting counting and measurement systems. Their sexagesimal system lives on in a modified form in our measuring systems for time and angles. However, doing complex arithmetic is cumbersome with such a large base.

As it turns out, we find that a base-10 or “decimal” base is applied in most Mongolian, Semitic, and Indo-European languages. Why 10? Note that 10 is not a mathematically convenient choice — it neither has many divisors nor is a prime number. Twelve would be much more appropriate from a mathematical point of view.
Ten has huge benefits, though. Remembering ten things does not tax the human memory, so we can learn the number words and multiplication table with relative ease. Most importantly, the decimal base originates from our anatomy. Humans began counting on their fingers and simply abstracted that anatomical intuition (which means that if we had had 12 fingers, we would have a base-12 system. Weird right?).
Next, our numbering system works because it is positional — the position of the digits matters in defining the value of the number. 52 is not the same as 25 just as 3287 is not equal to 7382. Not all numbering systems must be positional. Rather, the Ancient Egyptian and Roman numbering systems were additive, utilizing sign-value notation. This kind of numbering system represents numbers by a series of numeric signs that add together and equal the number represented.
With sign-value notation, you can arrange the digits in any order you like and they still sum up into the same value (although in practice Roman and Egyptian had writing conventions to facilitate comprehension). Within the additive system, counting rods and abacuses easily facilitated arithmetic and you did not need to memorize multiplication tables. For centuries there was a standoff between the positional and additive-plus-abacus systems. The fact that we ended up with a positional system seems to be due to Indian (Brahmi) influence.
One thing that an additive system does not need is zero, but in our numbering system zero at least two roles. It is used to depict empty powers of ten, and it is also used to represent, say, the null value of 5 minus 5.
A comical illustration of the dual function of the number 0 in our positional numeral system! Whoooo
Babylonians had a positional system but did not have a zero sign, so they at first depended on context to distinguish between, say, 216 and 2016. Later, a punctuation sign developed to serve as an empty placeholder. Romans denoted zero by the word nulla “nothing”; zero was not considered a number at all.
In Indian mathematics, “zero” was first represented merely by empty columns in Indian counting tablets. Once the columns were done away with, the empty spaces had to be filled with something, and so either a small dot or a circle began to be added. This sign had the meaning of  “void.” As the idea of “number” became more abstract, zero could be conceived of as being a number in its own right. By around 650AD, the use of zero as a number came into Indian mathematics.
This discovery represented a pivotal “miracle moment” in the history of mathematics and ideas. As Douglas Harper at Etymology Online writes, the insight that “empty” and “null” are one and the same has, as far as we know,
only happened once in human history, somewhere in India, in the intellectual flowering under the Gupta Dynasty, about the 6th century C.E. There was no “miracle moment,” of course. It was a long, slow process.
No one has convincingly explained how the Indians managed to come up with this rich concept of zero, but thankfully they did. This zero made our numbers into a coherent, intellectually rigorous numbering system and made possible all of modern science and mathematics.
Where did our number symbols come from? Why are our numbers associated with the Arab world? What is the origin of the naming conventions for numbers in Indo-European languages? All of this and more will be answered in Part II (or 2, if you prefer positional numerals).
Postscript/More Resources

Mayan mathematics was awesome and amazing, but for obvious geographical reasons it was not influential in the development of the Hindu-Arabic numerals. Follow the link to learn more about it anyway.

See John Halloran for more work on Sumerian language and numbering. Also check out anything by  Schmandt-Besserat, such as How Writing Came About. Also Sumerian Calculation, and a nifty presentation on the Sumerian clay tokens.

The history of zero and it’s significance in the history of mathematics. A much better summary of the origin of Hindu-Arabic numerals, but my post has better pictures.

Another history of zero. Because it’s that important.

Georges Ifrah, From One to Zero: A Universal History of Numbers.

Karl Menninger, Number Words and Number Symbols.

Peter Rudman, How Mathematics Happened: The First 50,000 years.

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