1.01. Numbers in the English Language
There's no more natural point to start a discussion of large numbers than the start, as in the first numbers. I originally kicked off this site with the -illions which are among the smallest numbers we generally think of as large, and good building blocks in googology. However, now I feel that some background on the smallest numbers would be somewhat more natural of a starting point.
Numbers 1 to 10
You should already know what numbers up to 1000 are, but you might not know the full history behind them or how they got their names. We won't really focus on negative numbers or non-integers, and instead start with 0.
Zero, obviously, is the number representing "nothingness", a lack of objects. Because it represents nothingness, most people in ancient times didn't think of 0 as a number! The earliest known usage of zero was around 250 BC by the Olmec civilization, and usage of zero came to Europe around the 1500s. The name "zero" didn't come from Indo-European roots like the other numbers, but rather from Arabic "sifr" (meaning "void" or "nothing") which became "zefiro" in Italian, later contracted to "zero". For our purposes we can define 0 as a number whose existence is an axiom. An axiom is a statement that is true by definition rather than by proof. That means the fact that 0 exists is not something to prove, but essentially given. It is also sometimes defined as the number a with the property a+x = x for any x, but we'll define addition a little later.
If you're thinking, "but doesn't everyone already know how addition works?", I think it's still helpful to pick it apart in a little more depth, serving as a warm-up for the complex notations used for making gigantic numbers.
Up next is the unique number one. Here's what one object looks like:
Unlike 0, most (but not all!) people in ancient times thought of one as a number. However, it was treated differently from numbers like seventy-five or whatever else, since it represents a single quantity instead of a multiple quantity. But anyway, for our purposes we can define one based on zero with a "fundamental function", S(x) (the successor function), which is equal to x+1. It's a fundamental function here because it forms the basis of all functions, and as with zero we'll treat its existence as an axiom. More specifically, we'll make it an axiom that S(x) exists for all x; in other words, S(x) is a total function. This is one of the most common axioms in mathematics. For example, one of the Peano axioms, a set of axioms that define the natural numbers, is that the successor function is a total function.
Treating some things as axioms in mathematics is necessary for forming its very basis, and for our purposes we can consider S(x) a kind of base function. In some forms of mathematics the number one's existence is treated as an axiom, but I don't find that usage as natural as basing it upon zero, or alternately defining it as the number a with the property that a*x = x. So for our purposes 0 is the only number with a "fundamental" definition.
Note of passing interest: S(x)'s existence can define the negative numbers: -1 is the number with the property that S(x) = 0, -2 is the number with the property that S(x) = -1, and so on and so forth. But for now let's move on to two.
Two is the number that is defined as S(1) with the successor function, or alternately as S(S(0)). Two objects look like this:
and it's represented with the glyph "2". That glyph evolved from a less arbitrary glyph for two, 二 (which is still used in Chinese and Japanese when writing "two" as a word). Like one, two has many unique properties, largely for trivial reasons. If you're interested in such properties look at my number list's entry for two.
Likewise, three is defined as S(2) and three objects looks like this:
o o o
Three is represented with the glyph "3", which evolved from a glyph for three 三 (three lines) which is still used in Chinese and Japanese like 二 for two. It's notable as the smallest number of repetitions or anything needed to recognize a pattern, and therefore holds a significant place in culture with many occurrences, such as the three little pigs, or Goldilocks and the three bears, or Newton's three laws of motion, and so much more (for more occurrences look at my number list's entry for 3).
Four, of course, is S(3), and four objects looks like this:
o o o o
Four is represented with the glyph "4" which evolved from a glyph that looked like a cross (like +) which was simplified from four lines. Four is the smallest composite number, the smallest non-trivial perfect power, the period of the powers of i (the imaginary unit, for information on those kinds of numbers look here), and some things that come in four are the four seasons, the four cardinal directions, the four legs of most animals, and the four human blood types (see my entry for 4).
After four comes five (5, o o o o o). The glyph "5" is the first number glyph that does not clearly originate from a depiction of that many of something. Five is a number that has gained some significance to humanity as half of our numeral base, 10. It's gained some fame in other places. For example, there are exactly five Platonic solids, the famous three-dimensional solids which were given a mystical cult significance in ancient times. See my number list's entry for five for more information.
Six (6, o o o o o o) is a number notable for several reasons: it's the smallest perfect number (its divisors, 1, 2, and 3, add up to itself), as well as a factorial, a primorial, a triangular number, the number of faces on a cube, and more. It's also notable for being about the limit of quantities we can immediately recognize by looking at them. I say "about" because depending on conditions and people, that limit may vary from four to about eight, and the limit is often considered six thanks to Robert Munafo and his idea of classes of numbers.
Once you get to seven (7, o o o o o o o), you can't look at that many objects in a row anymore and immediately tell how many there are; instead, you'll have to count them one by one. Four objects in a row can be immediately recognized as four, and five and six are in a weird middle ground before you reach numbers that you'll need to count to recognize. In that sense, seven can be seen as a respectably large number... well, at least by primitive standards.
Aside from this, seven is a number given much spiritual significance: for example, there are the seven deadly sins, the seven wonders of the ancient world, the seven dwarves in Snow White, the count of seven moving objects in the sky (sun, moon, Mercury, Venus, Mars, Jupiter, Saturn), and don't forget about 7's connotation of good luck. It was my favorite number as a kid, and it still is, because it's somehow appealing (not really sure why).
Eight (8, o o o o o o o o) is the number of vertices of a cube, the number of bits in a byte, and the number of directions most people choose from when specifying directions on a map (like east or northwest). Even though it's quite a divisible number for its size, it isn't anywhere near as common to group things into eight types as it is to group them in seven.
Nine (9, o o o o o o o o o) is the third square number and the largest number that is represented with a single glyph. It's part of the unique pair of numbers 8 and 9: eight (23) and nine (32) are the only pair of perfect powers that neighbor, and they're the only pair of integers xy and yx that satisfy both properties, xy > yx and x > y > 1. Sometimes, items or characters may come in a group of nine that is further divided into three groups of three.
After zero through nine, which are all represented with a single glyph, comes ten (o o o o o o o o o o). It's not represented with one glyph, but rather combining the glyphs "1" and "0" to make "10" to represent ten. This is because humanity generally has used ten as a base of their number system, which is because there are ten fingers on a human being. With the number ten, we are introduced to place value notation for making numbers, which we all learned in elementary school. Place value notation is just a sequence of the glyphs 0, 1, 2, 3, 4, 5, 6, 7, 8, and/or 9. The glyphs used in place value are known as "digits", and the sequence of digits anan-1 ... a1a0 is a compact way of expressing:
an*10n + an-1*10n-1 + ... a1*10 + a0
This tedious task is often taught in elementary school to emphasize this purpose of place value notation. Now that we have the numbers one through ten covered, we now will want to cover some of the basic operators under integer outputs, the ones we've been using lately.
The Basic Operators
For a kind of intermission, let's take a further look at how we can define the operators addition, multiplication, and exponentiation under whole number inputs. Even though you should already know how these operators work, listing formal definitions of them is useful because it serves as a warm-up to understanding how to define googological functions.
First off is the familiar operation of addition, always denoted with a plus sign (+). You should already know how it works, but for the purposes of this article, we'll define it using the successor function as follows:
a+b (pronounced a plus b) =:
b = 0: a+b = a
b > 0: a+b = S(a+(b-1)), where b-1 is the solution x to the equation S(x) = b
or for short:
a+b = S(S( ... S(a) ... )))
With that definition, we can prove that addition is commutative, i.e. a+b = b+a, and also that addition is associative, i.e. (a+b)+c = a+(b+c). But proving those facts is far more tedious and complex than you might think! LittlePeng9 of Googology Wiki made a nice graphic proving this property fully formally with showing every step, which you can look at here.
Here are some small examples of addition in action:
0+0 = 0
1+0 = 1
0+2 = S(0+1) = S(S(0)) = S(1) = 2
1+2 = S(1+1) = S(S(1)) = S(2) = 3
2+3 = S(2+2) = S(S(2+1)) = S(S(S(2))) = S(S(3)) = S(4) = 5
Of course, a faster way to add a and b is just to count up from b a times. For example, to calculate 2+3, count up from 2 three times (three, four, five) to get 5. For large numbers, it helps a lot to use place-value addition which you probably learned in elementary school. For example, to add 123+748 you use the methods like so:
The 1 on top of the 2 is just carried over as the first digit of 11 (3+8), of course. But you probably already know all this, so let's go to the next operator, multiplication.
Multiplication is another familiar operator, one that can make large numbers much more quickly than addition can. Just as addition can be defined using the successor function, multiplication can be defined using addition like so:
a*b (pronounced a times b) =:
b = 0: a*b = 0
b > 0: a*b = a*(b-1) + a
or for short:
a+a+a+a ... +a with b a's. The parentheses are left out because addition is associative, and therefore it doesn't matter where you place parentheses in a sum of numbers.
Like with addition, multiplication is both commutative (a*b = b*a) and associative ( (a*b)*c = a*(b*c) ).
There are several ways to note multiplication: in English-speaking countries, kids in elementary school are taught to use a × b, though in handwriting, that conflicts with the common usage of the letter "x" as an unknown. Because of this conflict, many people avoid that usage of "×" for multiplication. Therefore, in many European countries and in middle school in English-speaking countries, you are often taught to use a·b instead, using a middle dot instead of a cross. But since middle dots aren't easy to type on a keyboard, generally a*b is instead used for multiplication when typing (that's the standard usage in computing), which is the way I'll usually denote it on this site.
Additionally, absence of a symbol between two expressions (if they're not both numbers!) can indicate multiplication. For example, multiplying 2 by an unknown constant x is usually denoted 2x, and multiplying the unknowns x and y is denoted xy. For longer expressions parentheses are used—for example, if you multiply a+b by c+d you use (a+b)(c+d).
Here are some examples of multiplication in action, making use of the counting and similar ideas in addition:
0*0 = 0
1*0 = 0
0*2 = (0*1)+0 = 0*1 = 0*0 + 0 = 0*0 = 0
1*2 = (1*1)+1 = ((1*0)+1)+1 = (0+1)+1 = 1+1 = 2
2*3 = (2*2)+2 = ((2*1)+2)+2 = (((2*0)+2)+2)+2 = ((0+2)+2)+2 = (2+2)+2 = 4+2 = 6
Similarly, we can evaluate some larger values:
5*5 = 5+5+5+5+5 = 25
20*30 = 20+20+20 ... (30 20's) ... +20 = 600
You can see that multiplication has the properties, 0*x = 0 and 1*x = x. The property 1*x = x means that 1 is the multiplicative identity; similarly, 0+x = x means that 0 is the additive identity. Multiplication can produce large numbers somewhat more efficiently than addition, but if you want to quickly make bigger numbers you'll need our last standard operator for now, exponentiation.
The next step after multiplication, exponentiation is where things start to get fun. While we started with the boring, plain successor function, it only took a few steps of building up from it to get to exponentiation, which we can define as:
ab or a^b (pronounced a to the power of b or a to the bth power, also a squared when b is two or a cubed when b is 3) = :
b = 0: ab = 1
b > 0: ab = ab-1 * a
or a*a*a ... *a with b a's for short
As you can see, while multiplication is repeated addition, exponentiation is repeated multiplication. In exponent expressions a is the base and b is the exponent. Here are some small examples of evaluating exponents with our rules:
00 = 1 since a0 = 1; though 00 is actually an undefined expression in mathematics, this definition works fine for our purposes.
01 = 00*0 = 1*0 = 0
10 = 1 by definition
22 = 21*2 = (20*2)*2 = (1*2)*2 = 2*2 = 4
Some larger examples are:
33 = 3*3*3 = 27
65 = 6*6*6*6*6 = 7776
1010 = 10000000000, more often denoted 10,000,000,000 since as we will see later it became convention to denote 5-or-more-digit numbers (and also sometimes 4-digit numbers) with commas every three digits.
2030 = 1,073,741,824,000,000,000,000,000,000,000,000,000,000, which is far bigger than 20*30 = 600.
Exponentiation lacks many of the properties of addition or multiplication. For example, it is not commutative, i.e. generally ab ≠ ba . This is not always false; if a and b are equal that is true, and there are some other special cases—for example, if you set a = 2 and b = 4, then ab = 24 = ba = 42 = 16. 16 is special because it's the only integer case where ab = ba and a ≠ b. But generally ab won't be equal to ba: for example, if a = 2 and b = 10, ab = 1024 but ba = 100.
Neither is exponentiation associative, i.e. generally (ab)c ≠ a(b^c). For example, (33)3 = 273 = 19,683, but 3(33) = 327 = 7,625,597,484,987, which is much larger. This is important partly because it makes all the difference when defining an operator which is repeated exponentiation, but let's not get ahead of ourselves. Another important reason is something that is important to realize: a^b^c refers to a(b^c), not (ab)c. For example 3^3^3 refers to 3(3^3) = 7,625,597,484,987, not (33)3 = 19,683. The reason for this is because (ab)c can be expressed alternately as ab*c while a(b^c) can't be simplified. The expression of (ab)c is due to the laws of exponents, several laws which you probably learned in algebra, which are:
ab*ac = ab+c
(a*b)c = ac*bc
(ab)c = ab*c
Those laws are important for evaluating expressions with exponents, the highest-level operator that is regularly used in math.
Exponents can be expressed in several ways. In ancient times they were sometimes expressed as ab where b is written in Roman numerals, so for example 3*3*3 would be noted 3III. This then evolved to just ab where b is written in regular numerals. However, exponents when typed are often instead noted a^b with the symbol ^ known as the "caret", and in some programming languages it's denoted a**b since there ^ is used for some logical operators. I'll more often than not use superscripts for exponentiation.
Now that we've familiarized ourselves with the operators (and gotten a sneak peek at what's to come in later pages), let's continue with the English numbers.
Numbers 11 to 999
The rest of the unique names
After the number ten, numbers in English get a little more interesting. The names "one" through "ten" seem to mean nothing in and of themselves. Like the names of one through ten in almost all Indo-European languages (like English, Spanish, Greek, French, Hindi, Russian ... ), they evolved from the original reconstructed Indo-European names for numbers:
1 - oinos
2 - duwo
3 - treyas
4 - kwetores
5 - penkwe
6 - sweks
7 - septm
8 - okto
9 - newn
10 - dekm
Those names are the great-grandparents of the names 1 through 10 in all Indo-European languages, though it's important to note that there's no written record of these names; instead, they were reconstructed by comparing examples of written names in ancient and modern languages. Here are some examples of these names' descendants:
(Note that in the table above, the italic names are transliterated into the Latin alphabet from the language's writing system.)
Of the numbers from 1 to 10, only three of their names have any hint of meaning outside of pure arbitrariness: it is suspected that perhaps the name "five" originally meant "hand" (five fingers on a hand), and maybe the name "ten" originally meant "two hands", and "nine" may be connected to "new". The rest of the numbers have absolutely no clear origin. That isn't too surprising, and it's only a little more arbitrary than the choice of glyphs for these numbers.
The names after ten, however, all have some clear meaning. Although the names "eleven" through "twenty" are all still unique names, most of them have clear derivation. Here's a list of the numbers 11 through 20 with their origin and some facts about them:
Eleven (11) - this number has some interesting properties related to its powers and multiples, and it's the largest number with an "easy" divisibility test in base 10 (see here). Its name is usually believed to mean "one left", meaning that there's one left over from ten. It does not match with the pattern of names "thirteen" to "nineteen".
Twelve (12) - notable for being a highly factorable number for its size (divisible by 2, 3, 4, 6), and a set of twelve is often known as a "dozen", e.g. a package of a dozen eggs. There are many other things that come in twelve: the twelve hours on a clock, the twelve Olympic Greek gods, twelve inches in a foot, etc. Like eleven, "twelve" does not match with the pattern of names "thirteen" to "nineteen". Its name is believed to mean "two left" which makes sense since the name "twelve" somewhat resembles the name "two". It is also a common rule to write numbers "one" through "twelve" as words, and larger numbers as numbers (e.g. 125 not one hundred twenty-five).
Thirteen (13) - this number has a rather inexplicable connotation of bad luck in Western culture, and there are several theories that try to explain why—perhaps it has something to do with having thirteen of something rather than twelve posing bad luck. The name "thirteen" derives clearly from three (thir) plus ten (teen), making it the first number with a very obvious etymology. I don't know why it's "thirteen" and not "threeteen", other than that it may relate to the word "third". Nonetheless this still shows a pattern like with the next few numbers.
Fourteen (14) - there isn't that much very interesting about 14, but it is the number of lines in a sonnet and the number of days in a fortnight. Unlike the name "thirteen" for 13, this number is a better match with the number "four" which it's based on.
Fifteen (15) - the sum of all rows, columns, and diagonals in the 3x3 magic square, and the fifth triangular number. The name of this number comes from "five" plus "teen", but the name is fifteen and not fiveteen because fifteen is easier to pronounce than fiveteen, and thus the name changed to fifteen. This may also have something to do with "fifth" for five.
Sixteen (16) - sixteen is a very special number: as I discussed earlier it's the only integer of the form xy = yx where y and x are distinct integers. It's also known for its association with computing, like all the powers of two are. The name "sixteen" is a perfect match with "six" plus "teen".
Seventeen (17) - one of those prime numbers whose lack of practicality is compensated for by a cult following of sorts that stems from being likely to spring up as a "random" number. This means that many correspondences and facts involving 17 are hard to consider as coincidence, so many people have made webpages or websites devoted to facts about that number. It is both one of my childhood and current favorite numbers, and in fact my number list has a separate page for the number 17. The name is a perfect match with "seven" and "teen".
Eighteen (18) - eighteen may be the very most common "adulthood" age, as it's the age in most countries where you're considered an adult and the voting age in most countries. The name "eighteen" came from "eight" plus "teen", dropping one of the t's from the slightly awkward "eightteen".
Nineteen (19) - 219-1 and 217-1 are both prime numbers (this means that 19 and 17 are Mersenne prime exponents), and 19 and 17 are twin primes (primes that are two numbers apart). 19 and 17 appear to be the largest numbers that are both twin primes and Mersenne prime exponents (though to the best of my knowledge this has not been proven), as the next few of those exponents grow very far apart: they are 31, 61, 89, 107, 127, 521, 607, 1279... The name "nineteen" is a perfect match of "nine" and "teen" and it's the largest number whose name matches that pattern.
Twenty (20) - this number is notable for its usage as a base in several number systems, most famously by the Mayans. This usage is part of the origin of such things as the 2012 apocalypse phenomenon. 20 is also used kind of as a base in other languages; for example, the French word for eighty is "quatre-vingts" (four twenties). The name "twenty" is based on "two" plus a suffix "ty" which multiplies a number by ten. The origin of the suffix "-ty" is not clear, but it may be related to "ten" or "-teen".
After twenty, most of the numbers will use names constructed from other numbers. This kind of thing is present in pretty much every language. This is important because it represents that while we can give a unique name to the first few numbers, after a certain point the names will become tedious to know, and it's both easier and more practical to name numbers from existing numbers.
Building the first block
As I said, after twenty comes names that are constructed from existing names. The first numbers after twenty are twenty-one, twenty-two, twenty-three, twenty-four, twenty-five, twenty-six, twenty-seven, twenty-eight, and twenty-nine. Those names combine the tens place name (twenty) with the ones place name (one, two ... nine).
After 29 comes the name "thirty" for 30, or twenty plus ten. The name "thirty" has a peculiarity that is the exact same as that of "thirteen", which again seems to be tied with the word "third". Of course we can then have thirty-one, thirty-two ... thirty-nine for the next few numbers after 30.
Elementary students will generally see the pattern, and to teach them to count to 100, it's enough after this point to give them these names:
forty = 40 - this name is unusual because it drops the "u" from the name we would expect, "fourty".
fifty = 50 - once again probably made as an easier-to-pronounce version of "fivety", and a number that has gained some cultural significance for being half of 100.
sixty = 60 - another highly factorable number like twelve. It was used as a base by the Babylonians and it's the number of seconds in a minute and minutes in an hour.
seventy = 70 - perfect match with "seven" plus "ty"
eighty = 80 - like "eighteen", this name drops the extra "t" in "eightty"
ninety = 90 - matches with "nine" plus "ty"
With this system, to name any number from 20 to 99, you can just use the tens place's name (thirty, forty ... ninety) plus the ones place's name (one, two ... nine), unless of course the ones place is 0. This is a nice little system that lets you name numbers in quite a versatile range.
But after 99 (ninety-nine), things vary a lot more among the Indo-European languages. The next number after 99 is 100, so we'll be looking at that and how that word came to be.
The name for 100 varies widely across the Indo-European languages. In Romance languages like French or Spanish, the word for 100 resembles the Latin word "centum". In Germanic languages like English, the word resembles "hundred". In Slavic languages like Russian, the word resembles "sto". And in modern Greek, the name is "ekato". What's that all about?
When you Google "define hundred", it gives the definitions of the word "hundred" plus the etymology. This is what it says:
"late Old English, from hund ‘hundred’ (from an Indo-European root shared with Latin centum and Greek hekaton) + a second element meaning ‘number’; of Germanic origin and related to Dutch honderd and German hundert. The noun sense ‘subdivision of a county’ is of uncertain origin: it may originally have been equivalent to a hundred hides of land (see hide3)."
So "hundred" is not as unusual a word as we may think. It too came from Indo-European roots, making it a cousin of the word for 100 in many Indo-European languages; the name came from the root for "100" plus "number". However, there's an odd twist: before the 1800s or so the word "hundred" could refer to 120 (alternately "long hundred") or 100 (alternately "short hundred"). I'm not sure why a hundred would suddenly mean 120 and not 100; my best guess is because 120 is a more highly divisible number than 100 is, which shows that people have always been biased towards highly divisible numbers. Ten as a base is a major exception to this bias, because it's the number of fingers humans have on their hands.
I suspect that the "100 hundred" was used most often to name numbers like "three hundred forty-seven", while the "120 hundred" was used more as just a synonym for 120, in the same way "dozen" is a synonym for twelve. Because of the 100 hundred's usage in naming numbers, 100 eventually dominated 120 for the meaning of "hundred", and so it's now clear how to name numbers after 100. For 101 to 199, you just take "one hundred" plus the appropriate number from 1 to 99, i.e. the number 100 less than the number from 101 to 199 you want to name, so for example 137 is one hundred thirty-seven.
Then, of course, 200 is called "two hundred". It makes sense that the name is two separate words because the name came into use more recently than words like "seventy", and at that time English was certainly far more standardized than it was at the time those other numbers came to be. To name 201 to 299, combine "two hundred" with the appropriate number from 1 to 99, which is the tens and ones place of the number.
Simiarly 300 is called "three hundred" and 301 to 399 are "three hundred" plus the tens and ones place numbers. Then 400 is four hundred, 500 is five hundred, 600 is six hundred, 700 is seven hundred, 800 is eight hundred, 900 is nine hundred. With that, we have the first block of numbers.
To recap the first block, here's a table of all the roots we have so far:
To name any number from 1 to 999, you just use the hundreds root plus the tens root plus the ones root, unless the tens root is 1, in which case you use a name from the last column for both the tens root and ones root. This allows us to name any number from one to nine hundred ninety-nine. But we all know there are numbers beyond that: welcome to the second block.
The Second Block
What's after nine hundred ninety nine? To my knowledge, something like "ten hundred" was almost never used. Instead English made a name for 1000, which is today known as one thousand.
The etymology of the name "thousand" is more obscure than that of smaller numbers. An online etymology dictionary, however, is helpful for finding where such words came from. The dictionary's entry for "thousand" (link) says:
Old English þusend, from Proto-Germanic *thusundi (cognates: Old Frisian thusend, Dutch duizend, Old High German dusunt, German tausend, Old Norse þusund, Gothic þusundi).1
Related to words in Balto-Slavic (Lithuanian tukstantis, Old Church Slavonic tysashta, Polish tysiąc, Russian tysiacha, Czech tisic), and probably ultimately a compound with indefinite meaning "great multitude, several hundred," literally "swollen-hundred," with first element from PIE root *teue- (2) "to swell" (see thigh).
Used to translate Greek khilias, Latin mille, hence the refinement into the precise modern meaning.
1 the unusual letter þ is the former English letter "thorn", which over time instead came to be spelled with "th". It is still used in Icelandic.
This is quite an interesting etymology; it turns out that the word "thousand" originally came from a word referring to any large value. At one point in history, some languages already had their own word for "thousand", the predecessor to English only had hyperbole words (similar to "zillion" today) larger than 100. Therefore, it makes sense to make a word for "ten hundred", and it's interesting how people chose to repurpose a word that didn't refer to a specific number.
The benefit of having a separate word for 1000 is that we can plow through a hefty range of numbers without needing to invent new number names at all. We can just recycle, recycle, recycle, to get some names such as:
one thousand one = 1001
one thousand twenty-three = 1023
one thousand three hundred eleven = 1311
two thousand two = 2002
three thousand = 3000
eleven thousand one hundred = 11,100 (I'm only using commas for convenience here, as at the time numerals didn't use commas yet)
two hundred fifty-two thousand five hundred forty-six = 252,546
Yes, that's right: to name any number of the form a*1000+b where a and b are both positive integers less than 1000, just use the name "herp thousand derp" where "herp" and "derp" are the names of a and b using English numbers. If a is 0 then the name is just "derp" and if b is 0 the name is just "herp thousand".
Building the second block was mercifully simple, with the largest number being nine hundred ninety-nine thousand nine hundred ninety-nine = 999,999. However, there are some quirks worth noting.
First off, notice that 10, 100, and 1000 each get their own name, but 10,000 and 100,000 don't. This is probably because people did just fine combining names of numbers when they're this big. There is some arbitrariness, as many languages like ancient Greek, Chinese, and Hebrew have their own words for 10,000. However, modern Greek uses "deka chiliades" (means ten thousand) for 10,000 due to influence from other languages which don't have a separate word for "ten thousand". But the ancient word "myrio" for 10,000 is still alive in a sense. It echoes in the word "myriad" which has been used for 10,000 (mostly in translations), and in the Greek word for 1,000,000 "ekatommyrio" (literally hundred myriad), upon which the names for larger numbers in Modern Greek are based.
These two blocks of number naming were more than enough for any real use at the time they were made; if you needed 1,000,000 for some purpose you could just say "thousand thousand". However, eventually English yet again borrowed a name for a milestone point in the word of numbers, this time from Italian.
The Third Block and the Word "Million"
An -illion, as the name suggests, is any number whose name ends in -illion, like "billion". The -illions, as we will see, have an unusual history but very much cultural recognition, starting with the peculiar story of the word "million".
By around 1200, English was doing fine with a naming system for numbers up to 999,999. However, some languages went further than that. Italian is the example we're looking at here: it had a peculiar number name called "millione" meaning a thousand thousands. The name combined Italian "mille" meaning a thousand the augmentative suffix -one. -one in Italian can be thought of as equivalent to prefixes like super- or mega-. Therefore, "millione" can be thought of as meaning something like "super-thousand".
English then borrowed the name "millione" from Italian and turned it into "million" around the 1200s; after a few centuries, "million" became a pretty well-known term.
With the word "million" added to the English inventory of number words, a third block was now available to use. There is the first block of the plain old numbers below one thousand, the second block adding the root "thousand", and now a third block adding the root "million". Thus we can name any number from 1,000,000 to 999,999,999 of the form a*106+b*103 + c where a, b, and c are positive integers < 1000 as:
a million b thousand c (if b is zero you drop b and thousand, and if c is zero you drop c)
So for example, let's name 134,217,728:
In 134,217,728, a is 134, b is 217, and c is 728. So the name is:
one hundred thirty-four million two hundred seventeen thousand seven hundred twenty-eight
Also, let's name 2,000,132:
a is 2, b is 0, c is 132
And so the number name is:
two million one hundred thirty-two (since b is 0 we drop b and "thousand")
After 999,999,999 we have 1,000,000,000, which we can call "one thousand million" with the idea of combining names, the same that gives us the name "ten thousand" for 10,000. But wait a minute, isn't 1,000,000,000 a billion? Yes, in modern English that is true, but how the name "billion" came to be is a bit complicated.
The Next -illions: Chuquet and the -illion Scales
The -illions after a million have quite an unusual history, and they didn't start off quite how we know them today. So let's go through those -illions and how they're named.
Chuquet's Latin -illions
Names like the millions were enough to get by in the world for quite some time. But then in the year 1484, something important happened: a French mathematician named Nicolas Chuquet proposed a system to name numbers bigger than a million. Here are the names he proposed:
byllion = 1,000,000,000,000 = 10^12
tryllion = 1,000,000,000,000,000,000 = 10^18
quadrillion = 10^24
quyllion = 10^30
sixyllion = 10^36
septyllion = 10^42
ottyllion = 10^48
nonyllion = 10^54
In addition, he proposed using apostrophes to separate digits in groups of six. For example, for a "quyllion" you would not write 1000000000000000000000000000000 but the easier-to-read 1'000000'000000'000000'000000'000000. The apostrophes were in groups of six to match up with each each name having six more zeros than previous.
What's interesting is that Chuquet was giving names to large numbers for their own sake: this is part of the essence of googology, making large numbers for the sake of it! Better yet, he made somewhat systematic names by taking advantage of the Latin prefixes, which once again is a classic part of googology! His system is most definitely googology because it's a naming system for large numbers with a systematic method. You could say Nicholas Chuquet was a googologist, and he didn't even know it!
Chuquet's names were recognized by some, but didn't catch on just yet. You probably noticed that they neither match the English names for -illions in names nor values. This is where some refined versions of Chuquet's -illions come in.
Peletier's Refined -illions
One person who noticed Chuquet's -illions was another French mathematician from the 1500s named Jacques Pelletier du Mans. He gave an enhanced system for naming large numbers based on Chuquet's, coining these names for numbers:
milliard = 10^9
billion = 10^12
billiard = 10^15
trillion = 10^18
trilliard = 10^21
quadrillion = 10^24
quadrilliard = 10^27
quintillion = 10^30
quintilliard = 10^33
sextillion = 10^36
sextilliard = 10^39
septillion = 10^42
septilliard = 10^45
octillion = 10^48
octilliard = 10^51
nonillion = 10^54
nonilliard = 10^57
In addition, Peletier suggested usage of commas to separate digits in groups of three: for example, 29998559671349 would be written instead as 29,998,559,671,349, which is a lot easier to read. With Peletier's system, that would be about 29.998 billion, although with the modern English names it's about 29.998 trillion. While Chuquet performed googology long before there was a word for it, Petelier can be likened to a googologist who took it in his own hands to improve and refine upon someone else's work.
Peletier's names are very similar to Chuquet's -illions, but they're actually identical to the long scale used in most of Europe. In addition, halfway between each -illion in terms of number of zeros is an -illiard to match the name of a previous -illion. Thus, the millions would be the third block, milliards the fourth, billions fifth, and so on. This certainly matches better with the whole block idea than Chuquet's -illions though.
An Alternate Scale of -illions?
Peletier's illions, although they're identical to the long scale of -illions, are not quite like the -illions as we know them in English. The -illions as they're noted in English came from some French scientists using the Chuquet/Peletier names differently from how they were normally used:
billion = 10^9
trillion = 10^12
quadrillion = 10^15
quintillion = 10^18
sextillion = 10^21
septillion = 10^24
octillion = 10^27
nonillion = 10^30
This system is identical to the short scale used once only in the USA but now in all English-speaking countries and some others. It allows more use for, say, the term "trillion" than the long scale does.
Continuing the -illions
Some people around the 1800s wanted to continue the -illions further from the names that were already coined. In the spirit of Chuquet's names, they continued the series with Latin roots. The names that became recognized in English are as follows (with the short scale values):
decillion = 10^33
undecillion = 10^36
duodecillion = 10^39
tredecillion = 10^42
quattuordecillion = 10^45
quindecillion = 10^48
sexdecillion = 10^51
septendecillion = 10^54
octodecillion = 10^57
novemdecillion = 10^60
vigintillion = 10^63
One off the sequence that surprisigly also got recognized:
centillion = 10^303
And also, there's the googol = 10100 and googolplex = 1010^100, but those are not -illions and weren't named until much later; we'll talk about those a few articles later.
Therefore, when naming numbers strictly using words formally in the English language, we can work with the -illions from million through vigintillion. I like to consider centillion more of a number that would be used for comparison when working with large numbers. But the -illions' usage varies between countries, as we will see.
Short and Long Scales
The short scale of illions is the scale where billion = 10^9, and the long scale is where billion = 10^12. In the rest of this site, we'll use the short scale unless otherwise specified. Some countries use the short scale, and some use the long scale, which can lead to confusion. The short scale is today used in:
Greece (with a variant built upon "ekatommyrio", the Greek word for million)
Some of eastern Europe (including Russia) and central Asia
And the long scale is used in:
Western and central Europe (such as Germany)
Portuguese-speaking countries (except Brazil!)
Some countries use a number system independent of the -illions. For example, many Eastern Asian countries, like China, Japan, and the Koreas, use names for powers of 10,000.
Why is there all this difference? That's actually a good question. The long scale -illions ousted the short scale -illions in most of Europe, but strangely the United States adopted the short scale that French scientists used. Canada is interestingly split on this matter: the English-speaking provinces use the short scale, while Quebec uses the long scale. The United States, from its usage of the short scale, influenced other countries to use the short scale was well, most notably the United Kingdom. The UK once used a variant of the long scale, using names like "thousand million" in place of "milliard". This system was traditionally known as the "British system", along with the American system (the short scale) and the European system (the Peletier long scale). However, America influenced Britain to start using the short scale more and more in place of the long scale. In 1974, the prime minister of Britain officially declared that Britain would use the short scale, and with that the transition completed.
Other countries were, of course, influenced by America's short scale. For instance, although Greek numerals were originally based upon 10,000 like in some Asian countries, Greece adapted their language to use a scale that translates more easily to other European languages, giving these names for powers of 1000:
million = ekatommyrio
billion = disekatommyrio
trillion = trisekatommyrio
quadrillion = tetrakis ekatommyrio
quintillion = pentakis ekatommyrio
sextillion = exakis ekatommyrio
septillion = eptakis ekatommyrio
and so on, continuing with the names of Greek numbers
As I said earlier, this translation of the -illions is the main place where the old Greek name for 10,000 survives; an interesting hybrid between Ancient Greek numerals and the -illions system that most countries use today. It's also of note that Greeks often abbreviate "disekatommyrio" to "dis", "trisekatommyrio" to "tris", "tetrakis ekatommyrio" to "tetrakis", and so on. This is an understandable decision, given that the full Greek names for the -illions are mouthfuls.
Arabic-speaking countries for the most part simply borrowed the names for -illions from English:
million = milion
billion = milyar (an unusual exception obviously influenced by the long scale)
trillion = trilyon
quadrillion = kwadrilyun
So the short vs. long scale is a matter of some confusion around the world, but for our purposes, since this site is in English, the short scale will be used unless otherwise specified, which will be almost never.
Now that we've familiarized ourselves with the history of the -illions, let's get back to the idea of blocks and how numbers are named.
The Fourth Block and Beyond
With the English numbers, the fourth block is the billions, naming numbers like so:
a*109+b*106+c*103+d = a billion b million c thousand d; if any number is 0 you drop the entire number and the power-of-1000 name after it if there is one.
So, for example, 353,000,316,001 would be:
three hundred fifty-three billion three hundred sixteen thousand one (since b is 0, "million", the power-of-1000 name is dropped)
The largest fourth block number is:
nine hundred ninety-nine billion nine hundred ninety-nine million nine hundred ninety-nine thousand nine hundred ninety-nine
which is 999,999,999,999, or 1012-1 for short.
Similarly, the fifth block simply adds the trillions to the scene, so:
a*1012+b*109+c*106+d*103+e = a trillion b billion c million d thousand e; if any number is 0 you drop the number and the power-of-1000 name after it if there is one.
Of course, we can then continue with the sixth block, which is the quadrillions, seventh (quintillions), eighth (sextillions), etc., up to the 22nd block, the vigintillions. In fact, we can generalize this to:
To name a number from 1 to 1066-1, express it as a*1063+b*1060 ... + u*103 + v, where a through v are any positive integer from 0 to 999. Then name it:
a vigintillion b novemdecillion c octodecillion ... t million u thousand v. If any number is 0 you drop the number and the power-of-1000 name after it if there is one.
Therefore, the largest number we can name is 999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999, which is:
Wait, but why do all this? Most people don't use words like "septendecillion", and if they use numbers this big, people almost always use scientific notation to denote numbers, so instead of "seventy-five septendecillion" you'd say 7.5*1055. However, for millions, billions, and trillions, you'd more often use the -illion name when talking about numbers this big. So why bother with those higher -illions?
I don't know, but I can tell you these -illion names are not entirely unused. Millions, billions, and trillions are familiar to most people. The next few are becoming more common in our modern world, although they mostly show up in science. After a trillion, with each step up from the -illion names, the name is less known and people are less likely to know what number you're talking about. Often people will name the lesser-known -illions in terms of more familiar -illions. For example, instead of "septillion" people might say "trillion trillion".
I think the main reason we have all those -illions is just so we have some existence of large numbers within English: words that exist and are quite natural names for anyone who knows how to count in Latin. Also, I think we have them for usage by the enthusiasts of large number—readers and creators of large number material like this site. After all, I like to imagine Chuquet and Peletier took pleasure and pride in coming up with consistent names for numbers larger than a million; they are very early examples of googologists. People have extended those -illions to the point of absurdity, so there are indeed people who care about all those large numbers. We won't go to that just yet, and for now let's review what we've covered.
The English number naming system, as you can see, has many arbitrary choices of usage, and some confusion internationally with large number names. However, they more than suffice for practical use, and most people forget that those quirks exist. You may ask, why did I review the English naming system in detail? I did it for two main reasons: to explore the history of English number names, and as warm-up for later articles where definitions and terminology will likely be new to you. The best way to warm up for such unfamiliar concepts, I believe, is to do something similar with familiar concepts so that the later material is easier to grasp.
Now that you're familiar with the English numbers and their history, next we'll examine their usage in more detail.