Archimedes counts the sand

Posted by Graham Wheeler on Tuesday, January 5, 2010

In a previous post I described the Babylonian/Sumerian sexagesimal (base 60) counting system. Unlike this system, most cultures adopted a base 10 counting system due to the natural inclination to count with the fingers (leading to the term digits). Less common were quinary (base 5) systems, but vigesimal (base 20) were not unusual - for example, this was widespread in native American culture, including (with a novel variation) the Mayans. Duodecimal (base 12) has also played a role; we still have remnants in the groupings of dozens and gross.

Once the numbers in the base group are exceeded, we need to group to represent larger numbers - for example in decimal we have units, tens, tens of tens (hundreds), and so on. The Mayans used a novel variation where the second grouping only went up to 18, although after this the groupings reverted to base 20 again. This means the second order group represents numbers up to 360 rather than 400, and may relate to the Mayan calendar of 18 months of 20 days plus 5 extra days.

The larger groupings have names that are often quite recent introductions to language; while the term hundred can be traced back to a root meaning ten times (ten), the word thousand has no clear relation to the roots of Indo-European languages and is likely quite a late construction, seemingly from an early Germanic term meaning great hundred. An exception was the Hindus who seemed attracted to large numbers; there is a story from the life of the Buddha which mentions numbers up to \(2^{153}\). The Greeks typically stopped at myriad (ten thousand), and for a long time the Roman number system stopped at 100,000; the term for a million appeared around 1400AD in Italy. Numbers larger than a million have appeared so recently in European language that there is no common agreement - a billion in Europe is \(10^{12}\) while in America it is \(10^9\) (called a milliard in Europe), and things get worse for trillion and quadrillion.

Archimedes Thoughtful by Fetti (1620)

One of the first known efforts to systematize the number system to encompass large numbers is Archimedes’ The Sand Reckoning, a treatise addressed to his relative King Gelo of Syracuse, in which Archimedes constructs a systematic method for representing arbitrarily large numbers while addressing the problem of estimating the number of grains of sand in the universe. In his preface, he stated “There are some, King Gelo, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited.” Archimedes went on to use myriads raised to the the power of myriads, successively, to represent the very large numbers he needed for his calculations. In the process of doing this Archimedes discovered and proved the law of exponents:

$$10^a 10^b = 10^{a+b}$$

Archimedes had to estimate the size of the universe. He assumed a heliocentric model, with the earth revolving around the sun, the sun at the center, and the stars at the periphery. By his estimates the universe was about two light years across. Given we now know the next closest star to our sun is about 4 lights years away this was quite impressive for the time!