Googol and Googolplex

Googol and Googolplex

The American mathematician Edward Kasner once asked his nine-year-old nephew to invent a name for an extremely large number – ten to the power one hundred (10 in 100), a one followed by a hundred zeroes. The boy called it googol.

You, too, can make up your own very large numbers and five them strange names.

If a googol seems large, consider a googolplex. It is ten to the power of a googol – that is, a one followed by a googol zeros. By comparison, the total number of atoms in your body is about 10 (28), and the total number of elementary particles – protons and neutrons and electrons – in the observable universe is about 10 (80). If the universe were packed solid with neutrons, say, so there was no empty space anywhere, there would still be only about 19 (128) particles in it, quite a bit more than a googol but trivially small compared to a googolplex, do not approach, they come nowhere near, the idea of infinity.

The spirit of this calculation is very old. The opening sentences of Archimedes’ The Sand Reckoner are: “There are some, King Gelon, 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. And again, there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude”. Archimedes then went on not only to name the number but to calculate it. Later he asked how many grains of sand would fit, side by side, into the universe that he knew. His estimate: 10(65), which corresponds, by a curious coincidence, to 10(83) or so atoms.

Transmute the elements: cut the atom! Image: © Megan Jorgensen (Elena)

A googolplex is precisely as far from infinity as is the number one. We could try to write out a googolplex, but it is a forlorn ambition. A piece of paper large enough to have all the zeroes in a googolplex written out explicitly could not be stuffed into the known universe. Happily, there is a simpler and very concise way of writing a googolplex: 10 (10(100)), and even infinity (pronounced infinity).

In a burnt apple pie, the char is mostly carbon. Ninety cuts and you come to a carbon atom, with six protons and six neutrons in its nucleus and six electrons in the exterior cloud. If we were to pull a chunk out of the nucleus – say, one with two protons and two neutrons – it would be not the nucleus of a carbon atom, but the nucleus of a helium atom. Such a cutting or fission of atomic nuclei occurs in nuclear weapons and conventional nuclear power plants, although it is not carbon that is split. If you make the ninety-first cut of the apple pie, if you slice a carbon nucleus, you make not a smaller piece of carbon, but something else – an atom with completely different chemical properties. If you cut an atom, you transmute the elements.