The Sand Reckoner

In his short work The Sand Reckoner, Archimedes (c. 287 BC – c. 212 BC) identifies a number larger than what he believed was the number of grains of sand which would fit into the Universe. He was hampered by the fact that the largest number-word he knew was myriad (10,000), so that he had to invent his own notation for large numbers (I will use modern scientific notation instead).

Archimedes’ began with poppyseeds, which he estimated were at least 0.5 mm in diameter (using modern terminology), and which would contain at most 10,000 grains of sand. This makes the volume of a sand-grain at least 6.5×10⁻¹⁵ cubic metres (in fact, even fine sand-grains have a volume at least 10 times that).

Archimedes estimated the diameter of the sphere containing the fixed stars (yellow in the diagram below) as about 2 light-years or 2×10¹⁶ metres (we now know that even the closest star is about 4 light-years away). This makes the volume of the sphere 4×10⁴⁸ cubic metres which means, as Archimedes shows, that less than 10⁶³ grains of sand will fit.

A more modern figure for the diameter of the observable universe is 93 billion light-years, which means that less than 10⁹⁵ grains of sand would fit. For atoms packed closely together (as in ordinary matter), less than 10¹¹⁰ atoms would fit. For neutrons packed closely together (as in a neutron star), less than 10¹²⁶ neutrons would fit. But these are still puny numbers compared to, say, 2^77,232,917 − 1, the largest known prime!

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