*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.

*A comical illustration of the dual function of the number 0 in our positional numeral system! Whoooo*

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.

*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.