## Names of Large Numbers

This article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions. There are two main ways of naming a number: scientific notation and naming by grouping. For example, the number 500,000,000,000,000,000,000 can be called 5 x 1020 in scientific notation since there are 20 zeros behind the 5. If the number is named by grouping, it is five hundred quintillion (American) or 500 trillion (European).

At times, the names of large numbers have been forced into common usage as a result of excessive inflation. The highest numerical value banknote ever printed was a note for 1 sextillion pengő (1021 or 1 milliard bilpengő as printed) printed in Hungary in 1946. In 2009, Zimbabwe printed a 100 trillion (1014) Zimbabwean dollar note, which at the time of printing was only worth about US\$30.

Names of larger numbers, however, have a tenuous, artificial existence, rarely found outside definitions, lists, and discussions of the ways in which large numbers are named. Even well-established names like sextillion are rarely used, since in the contexts of science, astronomy, and engineering, where large numbers often occur, numbers are usually written using scientific notation. In this notation, powers of ten are expressed as 10 with a numeric superscript, e.g., ‘The X-ray emission of the radio galaxy is 1.3×1045 ergs.’ When a number such as 1045 needs to be referred to in words, it is simply read out:’ten to the forty-fifth.’ This is just as easy to say, easier to understand, and less ambiguous than ‘quattuordecillion,’ which means something different in the long scale and the short scale.

Take 642,500,000,000 which in scientific notation is 6.425 x 1011 or 642 billion, 500 million (US) or 642 milliard, 500 million (Eur). The American way or ‘hort form’ for naming large numbers is different from the European way or ‘Long form’ of naming large numbers. This is mainly because of American finance. Short form numbering is based on thousands and Long form is based on millions. Because of this, in short form a billion is one thousand millions (109) while in Long form it is one million millions (1012). The change in the United Kingdom to short form numbering happened in 1974. Today, Short form is most commonly used in most English speaking countries.

Apart from million, the numbers ending with -illion are all derived by adding prefixes (bi-, tri-, etc.) to the stem -illion. Centillion (10303) appears to be the highest number ending in -‘illion.’

When a number represents a quantity rather than a count, SI prefixes can be used—thus ‘femtosecond,’ not ‘one quadrillionth of a second’—although often powers of ten are used instead of some of the very high and very low prefixes. In some cases, specialized units are used, such as the astronomer’s ‘parsec’ and ‘light year’ or the particle physicist’s ‘barn.’ Nevertheless, large numbers have an intellectual fascination and are of mathematical interest, and giving them names is one of the ways in which people try to conceptualize and understand them.

One of the first examples of this is ‘The Sand Reckoner,’ in which Archimedes gave a system for naming large numbers. To do this, he called the numbers up to a ‘myriad’ (108) ‘first numbers’ and called 108 itself the ‘unit of the second numbers.’ Multiples of this unit then became the second numbers, up to this unit taken a myriad myriad times, 1016. This became the ‘unit of the third numbers,’ whose multiples were the third numbers, and so on. Archimedes continued naming numbers in this way up to a myriad myriad times the unit of the 108-th numbers. Archimedes then estimated the number of grains of sand that would be required to fill the known Universe, and found that it was no more than ‘one thousand myriad of the eighth numbers’ (1063).

Since then, many others have engaged in the pursuit of conceptualizing and naming numbers that really have no existence outside of the imagination. One motivation for such a pursuit is that attributed to the inventor of the word googol, who was certain that any finite number ‘had to have a name.’ Another possible motivation is competition between students in computer programming courses, where a common exercise is that of writing a program to output numbers in the form of English words.

Most names proposed for large numbers belong to systematic schemes which are extensible. Thus, many names for large numbers are simply the result of following a naming system to its logical conclusion—or extending it further.

The words bymillion and trimillion were first recorded in 1475 in a manuscript of Jehan Adam. Subsequently, Nicolas Chuquet wrote a book ‘Triparty en la science des nombres’ which was not published during Chuquet’s lifetime. However, most of it was copied by Estienne de La Roche for a portion of his 1520 book, ‘L’arismetique.’ Chuquet is often mistakenly credited with inventing the names million, billion, trillion, quadrillion, and so forth. Million was certainly not invented by Adam or Chuquet, rather it is an Old French word thought to derive from Italian milione, an intensification of mille, a thousand. That is, a million is a big thousand.

From the way in which Adam and Chuquet use the words, it can be inferred that they were recording usage rather than inventing it. One obvious possibility is that words similar to billion and trillion were already in use and well-known, but that Chuquet, an expert in exponentiation, extended the naming scheme and invented the names for the higher powers.

The names googol and googolplex were invented by Edward Kasner’s nephew, Milton Sirotta, and introduced in Kasner and Newman’s 1940 book, ‘Mathematics and the Imagination,’ in the following passage:

‘The name ‘googol’ was invented by a child (Dr. Kasner’s nine-year-old nephew) who was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested ‘googol’ he gave a name for a still larger number: ‘Googolplex.’ A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would actually happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, equal to 1 with a googol zeros after it.’