Large Numbers in the Universe (Math Lair)

"Many people believe, King Gelon, that the grains of sand are infinite in multitude; and I mean by the sand not only that which exists around Syracuse and the rest of Sicily, but also that which is found in every region, whether inhabited or uninhabited. Others think that although their number is not without limit, no number can ever be named which will be greater than the number of grains of sand. But I shall try to prove to you that among the numbers which I have named there are those which exceed the number of grains in a heap of sand the size not only of the earth, but even of the universe."
—Archimedes, The Sand Reckoner (opening sentences)

In Archimedes' The Sand Reckoner, addressed to Gelon, King of Syracuse, he attempted to count the number of grains of sand required to fill the entire universe. Assuming that one poppy-head would not contain more than 10,000 grains of sand, and that its diameter is not less than ¹/₄₀^th of a finger's breadth, and assuming that the sphere of the fixed stars, which was to Archimedes the boundary of the universe, was less than 10⁷ times the sphere exactly containing the orbit of the sun as a great circle, he found that the number of grains of sand required to fill the universe turns out to be less than 10⁶³. In order to perform this feat, Archimedes had to invent a notation for expressing these large numbers.

This is a unique and extraordinary achievement for Archimedes' time. The ancient Greeks had little interest in numbers outside of geometry.

Nowadays, the total number of particles in the universe has been variously estimated at numbers from 10⁷² up to 10⁸⁷. The total number of atoms in your body is about 10²⁸. If the universe were packed solid with neutrons, there would still be only 10¹²⁸ particles, a number larger than a googol but much smaller than a googolplex.

You don't need huge volumes of particles to get a glimpse of large numbers, however. All you need is combinatorics. How many ways are there to arrange 30 books on a bookshelf? The answer is 30 factorial (which is abbreviated 30!), or 265,252,859,812,191,058,636,308,480,000,000.

Mathematicians often have to use really big numbers in their work. There is much interest in finding huge numbers with certain properties. Examples are odd perfect numbers (none of which has yet been found) and Mersenne primes,