From body cells to chess games, real numbers get astonishingly big
Your body holds an estimated 37.2 trillion cells, and your brain around 100 trillion neural connections. Those figures are tiny next to the possible games of chess, estimated at a number far larger than the count of atoms in the observable universe. Large numbers turn up everywhere, and mathematicians keep inventing bigger ones.
Writing such numbers out is clumsy, so shortcuts evolved. Scientific notation writes a value as a small decimal times a power of ten, which makes a billion three characters instead of ten; computers often swap in the letter e. Comparing two numbers then means checking the exponents first and the leading digits only when exponents match. Named units help too, though they can mislead: Carl Sagan stressed the b in billions on Cosmos, but he never actually said billions and billions. That catchphrase came from Johnny Carson parodying him on the Tonight Show.
Physical reality supplies plenty of giants. A mole of carbon holds about 602.2 sextillion atoms. The observable universe is about 13.8 billion years old, spans 93 billion light years and contains roughly 125 billion galaxies, with perhaps 100 quinvigintillion atoms in all. The chess estimate is known as the Shannon number, and larger-board variants push it higher still. The physicist Don Page calculated what may be the longest finite time any physicist has worked out: roughly how long before a universe-sized black hole in a box would drift back to a state resembling today's.
Pure mathematics goes much further. Counting arrangements makes factorials explode faster than any exponential, and statistical mechanics deals in numbers so vast that people handle them through logarithms. Graham's number is too large for ordinary towers of powers and needs Knuth's up-arrow notation, while TREE(3), from a theorem about graphs, dwarfs even that.
Rayo's number, named after philosopher Agustín Rayo, emerged on 26 January 2007 from a contest at MIT to name the biggest number, and has been claimed as the largest named number. The study of such monsters has its own name: googology.
Source: Large numbers
