Douglas Crockford

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Sexagesimal

Long ago in Mesopotamia, agriculture took root. Many farming settlements were established along the rivers, growing into villages and cities. Accounting was invented to deal with the consequences of growth. It began with notches and marks, and eventually evolved into columns of numbers. Writing was invented to annotate the numbers. Some cities based their numbers on ten, a factor that was obtained by counting their fingers. Some cities based their numbers on twelve, a factor that was obtained by counting their fingers and feet.

King Alulim began the process of unifying the cities into a single nation. He immediately encountered a problem with the various number bases. Having incompatible number systems made administration and budgeting difficult. It also impacted his ability to levy taxes. Financial reports were significantly delayed, and the error rate was high. Something had to be done.

Alulim appointed the world's first standards committee composed of the top accountants and mathematicians of the region. He ordered them to standardize the number system.

The committee started by surveying all of the systems in use and decided that the two most popular, ten and twelve, were the only viable candidates. About half of the committee favored ten. The other half favored twelve. They made long impassioned presentations to each other, trying unsuccessfully to convince their opponents to adopt the better base. After many exhausting debates, it was observed that if a winner were chosen, the losers would be forced to become students of the new system, which would be a huge loss of professional status. They would also appear to be foolish for having advocated the losing system, which was not only humiliating, it was also unfair because neither system could be proven to be inferior to the other. In the end, the committee was unanimous in determining that the problem was unsolvable, and they delivered their consensus to King Alulim.

Alulim was furious at the committee's failure. The future of the nation depended on this. It seemed to him to be a trivial decision. All they had to do was to pick a number. He ordered the committee to reconvene, and he warned them that if they failed again, he would toss their heads into the river.

The committee resumed its debating, but it was clear that even the threat of beheading and/or drowning was not sufficient to motivate the membership to change their positions. Still at an impasse, they were unanimous in adopting a principle that would allow them to move forward:

All members should be equally inconvenienced.

That suggested that neither ten nor twelve be the standard, that some other number be chosen. But which number?

The first suggestion was eleven, being an obvious compromise. Unfortunately, this ran into another problem. The ten system wrote the numbers with vertical marks. The twelve system wrote the numbers with horizontal marks. How should the eleven system be written? They were at another impasse until an astrologer suggested that they consider the least common multiple, which is sixty, containing both 10 and 12 as factors. Both sides can declare victory.

But there was an obvious problem. In the twelve system, the symbol for 11 was 11 horizontal marks. In the ten system, the symbol for 9 was 9 vertical marks. The sixty system can not require that 59 be written with 59 marks. That would be much too inefficient and error prone, and it revived the vertical/horizontal mark issue.

After hours of arguing, the committee came up with an elegant solution. A number could be made up of two symbols, a higher symbol and a lower symbol. The higher symbol was written horizontally in the twelve system style, representing 10, 20, 30, 40, or 50. The lower symbol was written vertically in the ten system style, representing 1 thru 9. One or two symbols could represent 1 thru 59. Larger numbers could be represented by using more symbols.

Satisfied with their mutually disagreeable compromise, they presented their solution to Alulim, who thought it was unnecessarily complicated, but accepted the committee's assertion that it would work. And indeed it did work for many centuries, for many civilizations, including Sumeria, Akkadia, and Babylon.

Meanwhile, Egypt and China independently chose ten which has become the entire planet's base. The sixty system still survives, however, in timekeeping. There are 60 (not 100) seconds in a minute, and 60 (not 100) minutes in an hour. We will forever be trapped in this ridiculous system because a committee that met five thousand years ago could not figure out how to flip a coin.