Namen großer Zahlen

For several centuries, "billion" meant two different quantities depending on which country you were in, and the confusion was nobody's fault in particular — it came from two reasonable readings of the same naming scheme.

The French mathematician Nicolas Chuquet set out the -illion names in a manuscript of 1484, grouping digits in sixes: a million was a million, a billion was a million millions, and so on. Jacques Peletier du Mans discussed the system in the mid-sixteenth century and is associated with milliard for the intermediate thousand-million, which the six-digit grouping otherwise leaves unnamed. That is the long scale.

French practice later shifted to grouping in threes, making a billion a thousand millions, and this short scale travelled to the United States, where it became standard. Britain kept the long scale for much longer, which meant a British billion was a thousand times a American one. The British position changed officially in December 1974, when Prime Minister Harold Wilson confirmed in a written parliamentary answer that government statistics would thereafter use billion in the American sense. Most of continental Europe still uses the long scale, so the ambiguity persists across languages even though it has largely gone within English.

The names beyond trillion are mostly a curiosity; scientific writing uses powers of ten or SI prefixes instead, precisely because they cannot be misread. Googol is the exception that escaped into ordinary speech. The mathematician Edward Kasner asked his nine-year-old nephew Milton Sirotta for a name for 10^100, got "googol", and published it in Mathematics and the Imagination in 1940 along with googolplex, 10 to the power of a googol.

Past that point, names stop describing digit strings and start describing constructions. Graham's number, an upper bound from Ramsey theory, cannot be written as a power tower at all — it needs Knuth's up-arrow notation applied in 64 successive layers. TREE(3) is larger still. Both are finite, and in Graham's case the final digits are known even though the number itself can never be written down.

Wichtige Eigenschaften

• Short scale: each new name is 1,000 times the previous, so billion is 10⁹ and trillion is 10¹².
• Long scale: each -illion is 1,000,000 times the previous, so billion is 10¹² and trillion is 10¹⁸.
• The long scale fills the gaps with -illiard names: milliard is 10⁹ and billiard is 10¹⁵.
• The UK formally adopted the short scale for government statistics in 1974; much of continental Europe still uses the long scale.
• A googol is 10¹⁰⁰, which is larger than the commonly cited estimate of the number of atoms in the observable universe (around 10⁸⁰).
• A googolplex is 10 raised to a googol. It cannot be written out in decimal anywhere, because the universe does not contain enough matter to record the digits.
• The SI prefixes ronna (10²⁷) and quetta (10³⁰) were adopted by the General Conference on Weights and Measures in November 2022, the first additions since 1991.
• Graham’s number is finite and its last digits are computable, even though the number itself cannot be written in any positional notation.

Wo sie auftauchen

• Financial and government reporting, where the short-versus-long scale ambiguity has caused real translation errors in international documents.
• Computing storage and bandwidth, which use SI prefixes — and the separate binary prefixes kibi, mebi and gibi, introduced because 1 kB and 1 KiB differ by 2.4%.
• Cosmology and particle physics, where quantities like the estimated atom count of the observable universe (~10⁸⁰) are routinely compared against a googol.
• The name of the search company Google, a respelling of googol chosen to suggest indexing a very large number of pages.
• Combinatorics, where Graham’s number and TREE(3) arise as genuine upper bounds in proofs rather than as novelties.