Irrational Numbers?


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

Φ: the Golden Section

Φ: the Golden Section

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?”

Ῥωμανία (Romania).  Copyright (c) 1996-2013 Kelley L. Ross, Ph.D. All Rights Reserved

Ῥωμανία (Romania). Copyright (c) 1996-2013 Kelley L. Ross, Ph.D. All Rights Reserved

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:

Barcelona Cityscape   (c) 2009 Diliff, Wikimedia Commons.

Barcelona Cityscape (c) 2009 Diliff, Wikimedia Commons.

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 (CatalanCatalunya, SpanishCataluña)

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

The Virgin's Mantle--Vierge au manteau, Ville du Puy-en-Velay, 1417, Musée Croatier

The Virgin’s Mantle–Vierge au manteau, Ville du Puy-en-Velay, 1417, Musée Croatier. For background on this Iconography, read this article!

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 6th Century BCE had a flourishing civilization in middle Europe, and by the 5th Century BCE had gone East, West and South, most famously to Iberia, Gaul, Britain, Wales, Scotland and Eire.

Celts in Europe. For Legend, click the image. Atlas of the Celtic World, by John Haywood; London Thames & Hudson Ltd., 2001, pp.30-37 and other sources. QuartierLatin1968,The Ogre,Dbachmann. Wikimedia Commons.

Celts in Europe. For Legend, click the image. Atlas of the Celtic World, by John Haywood; London Thames & Hudson Ltd., 2001, pp.30-37 and other sources. QuartierLatin1968,The Ogre,Dbachmann. Wikimedia Commons.

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!

United Irish Cultural Center, San Francisco.

United Irish Cultural Center, San Francisco.

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 ?

Commentarii de Bello Gallico, an account written by Julius Caesar about his nine years of war in Gaul.

Commentarii de Bello Gallico, an account written by Julius Caesar about his nine years of war in Gaul.

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)
Celtic Nations today. QuartierLatin1968. One note: The Orkneys and Shetlands should not be shaded as part of Scotland - they were formerly Pict and latterly Norse, but never Celtic.

Celtic Nations today. QuartierLatin1968. One note: The Orkneys and Shetlands should not be shaded as part of Scotland – they were formerly Pict and latterly Norse, but never Celtic.

Wikipedia also correctly reports three “Mixed Languages” influenced by Celtic:

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).
Forever 21 Store, San Francisco

Forever 21 Store, San Francisco

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

L'Institut de France, including L'Académie française. Nitot, Wikimedia Commons.

L’Institut de France, including L’Académie française. Nitot, Wikimedia Commons.

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 BelgiumSwitzerland, the Democratic Republic of the CongoRwanda, 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).

The only known photograph of President Lincoln giving his Gettysburg address on November 19, 1863. Library of Congress.

The only known photograph of President Lincoln giving his Gettysburg address on November 19, 1863. Library of Congress.

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.

 Illustration from Sing a Song for Sixpence (1880) by Randolph Caldecott (d. 1886)

Illustration from Sing a Song for Sixpence (1880) by Randolph Caldecott (d. 1886)

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.

Four and twenty blackbirds,

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


neunhundertneunundneunzig  (there is no break in the word, it just won’t fit on the screen!)

Image from

Image from “A Tramp Abroad” (1880) by Mark Twain.

(Kudos to paradoxa4 on the 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?

Googol. Yot, Hidro, Wikimedia Commons.

Googol. Yot, Hidro, Wikimedia Commons.

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 103N+3 name for 103N+3
10 decem 1033 decillion
11 undecim 1036 undecillion
12 duodecim 1039 duodecillion
13 tredecim 1042 tredecillion
14 quattuordecim 1045 quattuordecillion
15 quindecim 1048 quindecillion, quinquadecillion
16 se(x)decim 1051 sexdecillion, sedecillion
17 septemdecim 1054 septendecillion
18 duodeviginti24 1057 octodecillion
19 undeviginti24 1060 novemdecillion, novendecillion
20 viginti 1063 vigintillion
10100 “googol” = ten duotrigintillion
100 centum 10303 centillion
1000 mille 103003 millillion
1000000 decies centena milia 103000003 milli-millillion
1010 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’:”

Schroeder and Lucy discuss googol.

Schroeder and Lucy discuss googol. Charles Schultz.

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 (10100). But the words are fun to play with!

Illustration, anonyme Chronik, „Von der Schöpfung der Welt bis 1384 / From the Creation of the World until 1384“

Illustration, anonyme Chronik, „Von der Schöpfung der Welt bis 1384 / From the Creation of the World until 1384“

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

2 thoughts on “Irrational Numbers?

  1. Well Steven, this is fantastic! I went back to be 12 in Rome and starting high school…Latin was taken very seriously under the dictator Mussolini and I had to memorize the De Bello Gallico in Latin after a full translation…this gives you the climate…beside, my mother was kind of fanatic about languages, so at 10 I was already fluent in written and spoken French with all its funny numbers and German, also with numbers plus guttural accents…When the war started and schools closed because of the bombings, I was so trained about intense study that I continued on my own and in a shot presented the final examinations for High School at 16 and registered in Medicine. Looking back I cant believe it!!!
    Thanks for the great fun in remembering all this!

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