Names of (very) large numbers

Introduction

There are about 17 sextillion protons in a single drop of water. How many are 17 sextillion protons?

In finance, in popular science, in news articles aimed at a general audience, large numbers are often expressed using names such as "million", "billion", "trillion", "quadrillion". Large numbers are named according to a particular system. There are different systems, called "scales", and the scale you use depends on what part of the world the text focuses on (and when the text was written).

In this article three often used mutually related systems are described. These scales are common in areas where a European language is spoken. The 17 sextillion mentioned above should be interpreted using the scale used in most English-speaking countries. It means 17 000 000 000 000 000 000 000 (21 zeros). However, a Dutch reader might misinterpret 17 sextillion as 17 followed by 36 zeros, because that would be in line with the scale used in the European Union.

Be aware that mathematicians, scientists and technicians usually do not use these names for large or small numbers in scientific or technical texts. They use scientific notation or they use appropriate units, often with SI prefixes (milli-, kilo-, giga-, etc.), when dealing with quantities rather than counts.

Different scales

In European languages the digits in a large number are generally separated in groups of thousands, corresponding with the pronunciation of the number. Every three digits from the right, a "thousands separator" occurs. The commonly used type of separator varies among countries (e.g. dot or comma), but the international standard (SI/ISO 31-0) is a (thin) space.

For instance, the number 12 345 678 is pronounced as "twelve million, three hundred (and) forty five thousand, six hundred (and) seventy eight". So the groups of three digits are pronounced as numbers up to 999, and at the positions of the thousands separator a specific word or name is pronounced; each separator with its own name. These separator names start off as (from right to left): thousand, million, etc. So, 23 070 000 is pronounced as "twenty three million, seventy thousand".

This is just viewed from a perspective of pronunciation. The separators do not really have a name. The names belong to digit positions. The fourth digit position from the right is the thousands (10³) position, the seventh digit position from the right is the millions (10⁶) position, etc. Just like the 6 in 600 is in the hundreds position and pronounced as "six hundred", the 2 in 2 000 000 is in the millions position and pronounced as "2 million". So, we can say that 45 in 45 000 is in the thousands position, pronounced as "forty five thousand". This is sometimes also applied to digits forming a number from 10 to 99 in the hundreds position: 1973 pronounced as "one thousand nine hundred seventy three" or as "nineteen hundred seventy three".

So the names "thousand", "million", etc. are names for digit positions 10ⁿ, with n being a multiple of 3. These names follow a certain system or "scale". There are different scales used in different parts of the world. Up to "thousand million" the scales are equal, but a "thousand million" and larger numbers are named different in different scales. A billion means something different in the US than in the EU. Moreover, the three scales discussed here predominantly apply to European languages. Other languages may use (very) different naming procedures.

The three scales discussed in this chapter are the short scale, the long scale and the Peletier long scale.

Short scale

In the short scale naming system, a million, a billion, a trillion, a quadrillion, etc. each represents a number a factor 1000 larger than the previous "-illion" named number. For instance, a trillion is a thousand billion and a billion is a thousand million. This corresponds with the pronunciation of numbers: at each position of a thousands separator a specific word or name is pronounced. However, the relation between the prefixes bi-, tri-, quadri-, etc. and the basis, a million, is not immediately apparent. Later more about this. The short scale is used in most English-speaking countries, in Brazil, and in several other countries.

Long scale

In the long scale naming system, a million, a billion, a trillion, a quadrillion, each represents a number a factor a million larger than the previous "-illion" named number. For instance, a billion is a million million. A thousand million in the long scale is a billion in the short scale. The relation between the prefixes (bi = 2, tri = 3, quadri = 4, etc.) and a million is clear: a billion is a million squared, a trillion is a million cubed, a quadrillion is a million raised to the power of 4, etc. Later more about this. The long scale was used in the United Kingdom until 1974 when they officially adopted the short scale, probably under the influence of American hegemony.

Peletier long scale

The Peletier long scale equals the long scale except now the intermediate "thousand million", "thousand billion" etc. are named "-illiard" with the same prefix as the immediately preceding "-illion". Thus, a "thousand million" becomes a "milliard", a "thousand billion" becomes a "billiard", etc. The Peletier long scale is used in non-English-speaking areas in Europe and in Spanish-speaking countries in Latin America. Of course the "-illion" and "-illiard" suffixes and the Latin prefixes have their own equivalents in languages other then English. For example: "Oktillion" (10⁴⁸) and "Dezilliarde" (10⁶³) in German, "octiljoen" and "deciljard" in Dutch or "octillón" and "decillardo" in Spanish.

The naming system

How to pronounce the number 12 345 678 912? The 12 is followed by 9 digits (12.345678912 × 10⁹). So, we start off with 12 billion (short scale), 12 thousand million (long scale) or 12 milliard (Peletier long scale). Then 345 with 6 trailing digits follows, thus 345 million, followed by 678 thousand and finally 912. Therefore this number is pronounced as:

• Short scale: "twelve billion, three hundred forty five million, six hundred seventy eight thousand, nine hundred twelve".
• Long scale: "twelve thousand three hundred forty five million, six hundred seventy eight thousand, nine hundred twelve".
• Peletier long scale: "twelve milliard, three hundred forty five million, six hundred seventy eight thousand, nine hundred twelve".

Numbers (decimal fractions) between zero and one are named following the same principle: a tenth (10⁻¹), a hundredth (10⁻²), a millionth (10⁻⁶), a billionth (10⁻⁹, short scale or 10⁻¹², long scale), etc.

Similar to scientific notation, large numbers are often rounded because the ones, tens, hundreds may not be very significant compared to the entire number. So, for instances, 12 345 678 912 may become "12.3 billion" (short scale) or "12.3 milliard" (Peletier long scale). In the long scale this would be something like "twelve thousand three hundred million"?

The names in the above table are formed by concatenating a prefix representing an integer to a suffix "-illion" or "-illiard". The prefix refers to integer n occurring in 10³ⁿ⁺³ (short scale) or 10⁶ⁿ (long scale). The numeral prefixes are derived from Latin numerals. The smallest "n-illion", a million, is not such a concatenating (although sometimes the "m" is mistakenly attributed to the Greek numeral prefix "mono"). A "million" translates to something like "super thousand", where "milli-" stands for "thousand" in Latin. Latin (and Greek) numeral prefixes are often used in European languages like "monologue" (one person speech), "bicycle" (two-wheeler) or "November" (the ninth month in the old calendar of Romulus). How many limbs does an octopus have?

Other names: a googol

Apart from names following the above discussed name convention you may occasionally find other names used to denote (extremely) large numbers. One of them is the googol. A googol is 10¹⁰⁰, i.e. a digit 1 with one hundred trailing zeros. How to name a googol using the name conventions discussed previously?

For the long scale we find the largest multiple of 6 smaller than 100. This is 96 = 6 × 16. The latin prefix for 16 is "sedeci" or "sexdeci", so 10⁹⁶ is named "sedecillion" or "sexdecillion". Then, 10⁹⁹ is named "sedecilliard" or "sexdecilliard" (a factor 1000 more). And finally we need to multiply by 10 to get 10¹⁰⁰: "ten sedecilliard" or "ten sexdecilliard". Finding the name for the short scale is not as obvious. A googol is:

• Short scale: "ten duotrigintillion".
• Long scale: "ten thousand sedecillion" (or "ten thousand sexdecillion").
• Peletier long scale: "ten sedecilliard" (or "ten sexdecilliard").

The name "googol" was popularized in 1940 by the best-selling mathematics book Mathematics and the Imagination by Edward Kasner. The technology company Google was named after a "googol", as it was actually an accidental misspelling of "googol".

A googol may be the most famous non-conventional name for a specific large number, but there are many more of them, such as a "Lcillion" (10⁵⁰ i.e. the square root of a googol) or a "faxul" (factorial of 200) or names for really obscure numbers such as a "myrillion", a "maximusmillion", a "googolgong" or a "googolplex". And of course there are the often used fantasy names such as "zillion", "bazillion" or "bajillion" for unspecified "very large" numbers.

How many?

Can we imagine huge quantities like a sextillion items or a quindecillion pieces? What about the number of grains of sand in the Sahara? We may be able to somewhat imagine the humongous amount of sand in this iconic desert. So let's try to make a very rough estimate of the number of grains it contains.

Let's assume that on average 8000 grains fit into one cubic centimeter. That makes 8 × 10⁹ grains per cubic meter. That is eight milliard per m³ in the Peletier long scale and eight billion per m³ in the short scale.

Now let's assume that 2 million km² of the Sahara's surface are sand dunes with an average depth of 100 meters. That results in 2 × 10¹⁴ m³ sand. That is 200 billion m³ sand in the long scale and 200 trillion m³ sand in the short scale.

That makes 2 × 10¹⁴ × 8 × 10⁹ = 1.6 × 10²⁴ grains of sand in the Sahara. That is 1.6 quadrillion grains in the long scale and 1.6 septillion grains in the short scale. Or in other words: one quadrillion, six hundred trilliard in the Peletier long scale and one septillion, six hundred sextillion in the short scale.

As mentioned in the introduction: there are about 17 × 10²¹ (17 sextillion or 17 trilliard) protons in a drop of water. It is estimated that there are roughly 10⁸⁰ (100 quinvigintillion or 100 tredecillion) protons in the known, observable universe.

Speaking about the observable universe, astronomer Carl Sagan estimated that it would take roughly 10⁹⁵ grains of sand to completely fill the observable universe with sand. This means one googol (10¹⁰⁰) of sand grains fills a hundred thousand observable universes with sand 😵