A question from a High School student whom I am tutoring in French, during his tutoring session Saturday morning, inspired today’s Posting: thank you!
First, let me apologize to the mathematicians out there. This won’t be about your irrational numbers.
Rational and Irrational Numbers in Math
The math whizzes in the audience would tell us, as does Kenneth Rosen in his 2007 Discrete Mathematics and its Applications, that “[i]n mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.” The set of Rational Numbers is designated by the symbol ℚ or Q. All integers are rational numbers. The decimal expression of a rational number always either ends after a finite number of digits or begins a regular series of repetition. (Thank you, Wikipedia!)
An Irrational Number, on the other hand, “is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.” The most famous irrational is π (3.1416……forever, non-repeating), the golden ratio φ (see the Fibonacci Sequence) and the square root of two √2. The first proof of their existence is attributed to a Pythagorean (possibly Hippasus of Metapontum) Ἵππασος, Híppasos; 5th century BCE.
Real Numbers, by the way, are any number that is a quantity along a continuous line. They include Rational and Irrational Numbers, and Transcendent Numbers (π is sometimes called one of these). Surreal Numbers include the Reals, as wells as Infinite and infinitesimal Numbers.
I know very little about higher math, but the terminology and semiotics of math do fascinate me.
But we’re not talking about that today.
“Irrational Numbers” in French!
My jumping off point for another meaning for “irrational numbers” was my student’s question: “Why are the French numbers from 70-99 so strange?”
For non-Francophones, we can quickly summarize. French number names are generally pretty regular Romance Language words, evolved from Latin. We usually think of the Romance Languages as Spanish, French, Italian and Portuguese. Some people know the fun cocktail party fact that Romanian is a Romance Language (notice the name?). But the current and usual count is 23:
- Portuguese
- Spanish
- Asturian-Leonese
- Aragonese
- Catalan
- Gascon
- Provençal
- Gallo-Wallon
- French, both Languedoc and Langues d’oïl
- Franco-Provençal (Arpitan)
- Romansh
- Ladin
- Friulian
- Venetian
- Lombard
- Corsican
- Italian
- Napolitan–Sicilian
- Sardinian
- Dalmatian
- Istro-Romanian
- Aromanian
- Daco-Romanian
Whether or not you realize it, you heard quite a bit of Catalan spoken in the summer of 1992, as it was the first language for all announcements at the Barcelona Olympics. Barcelona is the capital of Catalonia (Catalan: Catalunya, Spanish: Cataluña)
The evolution of Latin to French numbers went fairly smoothly:
Number | Latin numerals | Latin Name | French Name |
1 | I | ūnus | un |
2 | II | duo | deux |
3 | III | trēs | trois |
4 | IV | quattuor | quatre |
5 | V | quīnque | cinq |
6 | VI | sex | six |
7 | VII | septem | sept |
8 | VIII | octō | huit |
9 | IX | novem | neuf |
10 | X | decem | dix |
11 | XI | ūndecim | onze |
12 | XII | duodēcim | douze |
13 | XIII | trēdecim | treize |
14 | XIV | quattuordecim | quatorze |
15 | XV | quīndecim | quinze |
16 | XVI | sēdecim | seize |
17 | XVII | septendecim | dix-sept |
18* | XVIII | duodēvīgintī | dix-huit |
19* | XIX | ūndēvīgintī | dix-neuf |
20 | XX | vīgintī | vingt |
21 | XXI | vīgintī ūnus | vingt-et-un |
30 | XXX | trīgintā | trente |
40 | XL | quadrāgintā | quarante |
50 | L | quīnquāgintā | cinquante |
60 | LX | sexāgintā | soixante |
70* | LXX | septuāgintā | soixante-dix |
80* | LXXX | octōgintā | quatre-vingts |
90* | XC | nonāgintā | quatre-vingt-dix |
100 | C | centum | cent |
Notice the number names *starred. In the first case, for 18 and 19, French carried on using the 10+ pattern. Latin, on the other hand, had earlier switched to its subtractive principle that we have spoken about in the past, as evidenced in Roman numerals (IV = 4, that is, 5-1, IX = 9 i.e. 10-1) and their calendar. The Latin 18 and 19 literally mean “two from twenty,” and “one from twenty.”
In the second case, for 70, 80, 90, Latin carries on logically, while French suddenly changes to an additive principle:
70 soixante-dix
71 soixante et onze
72 soixante-douze
73 soixante-treize
74 soixante-quatorze
75 soixante-quinze
76 soixante-seize
77 soixante-dix-sept
78 soixante-dix-huit
79 soixante-dix-neuf
And then from a Base-10 (decimal) system to a Base-20 (vigesimal) system:
80 quatre-vingts
81 quatre-vingt-un
82 quatre-vingt-deux
83 quatre-vingt-trois
84 quatre-vingt-quatre
85 quatre-vingt-cinq
86 quatre-vingt-six
87 quatre-vingt-sept
88 quatre-vingt-huit
89 quatre-vingt-neuf
90 quatre-vingt-dix
91 quatre-vingt-onze
92 quatre-vingt-douze
93 quatre-vingt-treize
94 quatre-vingt-quatorze
95 quatre-vingt-quinze
96 quatre-vingt-seize
97 quatre-vingt-dix-sept
98 quatre-vingt-dix-huit
99 quatre-vingt-dix-neuf
Oh là là là là là là là!! Tiens! Qu’est-ce que se passe? Oh my, oh my! Look! What’s happening?
Decimal and Vigesimal Systems
In the Middle Ages, evolving French had two numbering systems. The decimal system (by 10s) was inherited from Latin, as we have seen above:
Latin: vīgintī (20); trīgintā (30); quadrāgintā (40); quīnquāgintā (50); sexāgintā (60); septuāgintā (70); octōgintā (80); nōnāgintā (90)
French: vingt (20); trente (30); quarante (40); cinquant (50); soixante (60)
The base-20 system (vigesimal) was inherited from the Celtic languages (for example Breton, still spoken in Northwestern France). If we wrote them in modern spelling, they would be:
- vingt (20)
- vingt-dix (30)
- deux-vingts (40)
- deux-vingt-dix (50)
- trois-vingts (60)
- trois-vingt-dix (70)
- quatre-vingts (80)
- quatre-vingt-dix (90).
Celts? In France?
Yep! Actually the Celts have been just about everywhere in central to western Europe. Possibly originating in the Caucasus or Carpathian Mountains, by the 6^{th} Century BCE had a flourishing civilization in middle Europe, and by the 5^{th} Century BCE had gone East, West and South, most famously to Iberia, Gaul, Britain, Wales, Scotland and Eire.
That is, they journeyed as far west as they could. St. Brendan may have made it to the New World (several scholars believe his “Voyage” has a real voyage behind its colorful descriptions), but the rest of the Celts had to wait until the re-discovery of the New World by Columbus (on this very weekend, 521 years ago!!) to immigrate to Boston, New York, Chicago, Toronto, San Francisco and other places and begin running them!
The Celts were a force to be reckoned with in Gaul. Just remember that the Romans called them “Gauls,” in France and that they gave Julius Caesar a lot of trouble. Can we say, Vercingetorix ?
Today, there are six surviving Celtic languages. “Speakers” indicates Native Speakers + those with some skills in the language.
- Galige (Irish) – ca. 1,888,000 speakers – Eire, UK, US
- Gàidhlig (Scotts) – ca. 92,000 speakers – Scotland & Cape Breton Island (NS, Canada)
- Cymraeg (Welsh) – ca. 722,000 speakers – Wales, England, Chubut Province Argentina, US, Canada
- Brezhoneg (Breton) – ca. 356,000 speakers – Brittany, France
- Kernewek (Cornish) – ca. 3,000 – Cornwall, England
- Gaelg (Manx) – ca. 1,823 speakers – The Isle of Man (British Crown Dependency)
Wikipedia also correctly reports three “Mixed Languages” influenced by Celtic:
- Shelta, Based largely on Irish with influence from an undocumented source (some 86,000 speakers in 2009).
- Some forms of Welsh-Romani or Kååle also combined Romany itself with Welsh language and English language forms (extinct).^{[35]}
- Beurla-reagaird, Highland travellers language
Proto-Celtic is the common ancestor, and comes is part of the Indo-European family of languages. Many other Celtic languages are now extinct, including those spoken in Spain and Portugal.
Vigesimal Celtic Numbering
As it turns out, some of the Celtic Languages use a vigesimal system, base-20, including today, Welsh, Irish and Scots:
- Welsh: Deugain = 2 times 20 i.e. 40, trigain = 3 times 20 i.e. 60.
- Irish: 30 = fiche a deich (formerly fiche agus deich), i.e. twenty and ten. 40 = dhá fhichead, i.e. two twenties (still in the decimal system as daichead), trí fichid = 60 (three twenties), ceithre fhichid = 80 (four twenties).
- Scots: deich ar fhichead = 30 (ten over twenty), dà fhichead = 40 (two twenties), dà fhichead ‘s a deich = 50 (two twenty and ten), trì fichead = 60 (three twenties), up to naoidh fichead = 180 (nine twenties).
Decimal number names are now being taught in the schools in these Countries, but the old names also persist.
Vigesimal systems are fairly wide-spread in human cultures. One of the best known (thanks to the 2012 rumors) is Mayan. In many languages, the name for 20 is “special,” not related to the normal naming convention. Even in Latin, viginti is different. Logically, one would have expected duginti. Greek also has a “special” 20, and we can also see some parallels and variations in the various stages of Greek for all the numbers.
Proto-Indo-European 20 has been variously reconstructed as *wīḱm̥t-; originally perhaps *widḱomt- or *duidḱmti. The first two show where the Latin viginti comes from. The English twenty is cognate with the German zwanzig.
The Celtic 20s in French Numbers
In the 17th century, when the French Academy began to try to standardize the language, they chose the most common usage of
the best authors of the day for these numbers, which turned out to be decimal up to 60, and then vigesimal from 70-99.
In Belgium, Switzerland, the Democratic Republic of the Congo, Rwanda, the Aosta Valley and the Channel Islands, the decimal system is used in French for 70-99, and so you get archaisms (old usages) such as septante (70); octante (80), or huitante (80); nonante (90) or neuvante (90).
Believe it or not, we have a similar usage in English. Remember how Lincoln’s Gettysburg address begins “Four score and seven years ago our fathers brought forth on this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all men are created equal…”? That means 4×20 + 7 = 87 years ago. “Score” was our word for multiples of Twenty. Two Score = 40. Two Score and Ten = 50. This is now archaic in English, but we still see it in historical documents.
Four and Twenty Blackbirds
While we’re on the subject of language about numbers, how about the old Nursery Rhyme:
Sing a song of sixpence,
A pocket full of rye.
Baked in a pie.
Even though we normally say Twenty-Four, we can understand what “Four and Twenty” means. It is an echo of our language’s Germanic roots. As speakers and students of modern German know, counting in German is like English, until you get to 21:
eins | 1 |
zwei | 2 |
drei | 3 |
vier | 4 |
fünf | 5 |
sechs | 6 |
sieben | 7 |
acht | 8 |
neun | 9 |
zehn | 10 |
elf | 11 |
zwölf | 12 |
dreizehn | 13 |
vierzehn | 14 |
fünfzehn | 15 |
sechzehn | 16 |
siebzehn | 17 |
achtzehn | 18 |
neunzehn | 19 |
zwanzig | 20 |
Notice now that we have the “Four and Twenty Blackbirds” approach. The ones number is said before the tens number, linked by “und” (and).
21 to 100 | |
einundzwanzig | 21 |
zweiundzwanzig | 22 |
dreiundzwanzig | 23 |
vierundzwanzig | 24 |
fünfundzwanzig | 25 |
sechsundzwanzig | 26 |
siebenundzwanzig | 27 |
achtundzwanzig | 28 |
neunundzwanzig | 29 |
dreißig | 30 |
einunddreißig | 31 |
vierzig | 40 |
einundvierzig | 41 |
fünfzig | 50 |
sechzig | 60 |
siebzig | 70 |
achtzig | 80 |
neunzig | 90 |
hundert | 100 |
German does get a little carried away with its word building procedures. For example, the number 999,999,999 is
Neunhundertneunundneunzigmillonenneunhundertneunundneunzigtausend
neunhundertneunundneunzig (there is no break in the word, it just won’t fit on the screen!)
(Kudos to paradoxa4 on the wordreference.com forum for correctly parsing this!) Mark Twain parodied German’s idiosyncracies in his essay “The Awful German Language” in 1880’s A Tramp Abroad.
I have to say that elf (11) and zwölf (12) are my two favorite German numbers, they just sound so cool!
In Austria and Bavaria, zwei becomes zwo, and zwanzig becomes zwozig. Other German speakers will sometimes use these variants to distinguish zwei from drei. In the southern German variant, ja (ya – yes) becomes jo (yo – yes). So Rocky is saying, “Jo, Adrian,” “Yes, Adrian.” (!)
Our Cardinal Numbers
And how about our names for cardinal numbers? Coming from Old English, our number names are from the Germanic family, from one to 999,999 (Nine hundred ninety-nine thousand, nine hundred (and) ninety-nine). These were so commonly used that we kept them, even after the Norman Conquest. The Latin 1,000,000 is attested at decies centena milia, but how many times did the ancient world have to count that high?
In today’s world, however, we need much larger numbers. The “Short Scale” is used exclusively in the U.S., and increasingly in British English:
- million
- billion
- trillion
- quadrillion
- quintillion
- sextillion
- septillion
- octillion
- nonillion
What do you notice? Through Old French, from Italian, we have adopted the Latin names, augmenting them with the suffix –on(e).
Higher still, the most common counting scheme is given by Robert Munafo (in some very interesting pages! ) as
N | N in Latin ^{3},^{18} | 10^{3N+3} | name for 10^{3N+3} |
10 | decem | 10^{33} | decillion |
11 | undecim | 10^{36} | undecillion |
12 | duodecim | 10^{39} | duodecillion |
13 | tredecim | 10^{42} | tredecillion |
14 | quattuordecim | 10^{45} | quattuordecillion |
15 | quindecim | 10^{48} | quindecillion, quinquadecillion |
16 | se(x)decim | 10^{51} | sexdecillion, sedecillion |
17 | septemdecim | 10^{54} | septendecillion |
18 | duodeviginti^{24} | 10^{57} | octodecillion |
19 | undeviginti^{24} | 10^{60} | novemdecillion, novendecillion |
20 | viginti | 10^{63} | vigintillion |
10^{100} | “googol” = ten duotrigintillion | ||
100 | centum | 10^{303} | centillion |
1000 | mille | 10^{3003} | millillion |
1000000 | decies centena milia | 10^{3000003} | milli-millillion |
10^{10 to the 100} | “googolplex” (i.e. 10 to a googol. |
As Munafo explains, Googol and Googolplex were invented just for the expression of these huge numbers:
“In 1938, mathematician Edward Kasner asked his nephew Milton Sirotta (who was 9 years old at the time) to invent a name for the number 1 with a hundred zeros written after it, and the nephew chose the name googol. As Kasner’s telling of the story goes, ‘He was very certain that this number was not infinite, and therefore equally certain that it had to have a name.’
“The name googolplex was invented at the same time, and also by Milton Sirotta if we trust the grammatical implication of the wording in [his account]. An article by Carl Bialik indicates that relatives and ‘family archives’ point to the invention of these words being in the 1920’s, and gives more details.
“Regarding its definition, Kasner wrote: ‘It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. […] It would never do to have [contemporary boxing champion Primo] Carnera [be] a better mathematician than Dr. Einstein, simply because he had more endurance. [and so it was decided that googolplex should be] a specific finite number, with so many zeros after the 1 that the number of zeros is a googol.’
“A generation later, googol had been popularized enough to be able to reach audiences of the widely-distributed Peanuts comic strip in which Schroeder estimates Lucy’s odds of eventually marrying him as ‘Oh, I’d say about ‘googol’ to one’:”
If you love language about numbers, please treat yourself to Robert Munafo’s pages (© 1996-2013 Robert P. Munafo) !
Scientists and Mathematicians usually just use the exponent notation system (10^{100}). But the words are fun to play with!
I should note that this post comes on the day of perhaps the most infamous number in the western world, 13! It was not a happy day in 1307: On Friday, 13 October, King Philip IV (the Fair) of France rounded up the Knights Templar in his realm, and arrested them on trumped-up charges of Heresy. They were subsequently tortured into “confessing,” or died. Pope Clement V later disbanded the Order. Philip’s motives were primarily economic, as he wanted their “treasure” which vanished with their persecution.
Today, with the recent discovery and publication of the Chinon Parchment by Barbara Frale, which details Clement V’s exoneration of the Templar leadership and attempts to thwart Philip’s villainy, the Roman Church admits that the arrest, murder and disbanding of the Templars was a mistake.
We usually pat ourselves on the back and think how lucky we are not to live in those benighted times. A quick look around the world, however, and even in our own country, will show that people are still persecuted for their differences in religious belief, and religion is still used as a mask for economic and political oppression. O Tempora! O Mores!
We’ve taken quite a journey around the world of numbers and the words that represent them. I hope it has been fun! See you next time!
Steven A. Armstrong
Tutor, Editor, Consultant