French vigesimals
cultural differences in counting

DeChicchis デキキス
1

counting in French

0 zéro30 trente

60 soixante

90 quatre-vingt-dix

1 un

31 trente et un

61 soixante et un

91 quatre-vingt-onze

2 deux32 trente-deux

62 soixante-deux

90 is surprising!

3 trois33 trente-trois

63 soixante-trois

4 quatre

34 trente-quatre

64 soixante-quatre

5 cinq

35 trente-cinq

65 soixante-cinq

6 six

36 trente-six

66 soixante-six

7 sept37 trente-sept

67 soixante-sept8 huit38 trente-huit

68 soixante-huit9 neuf

39 trente-neuf

69 soixante-neuf

10 dix40 quarante

70 soixante-dix

11 onze

41 quarante et un

71 soixante et onze

12 douze

42 quarante-deux

72 soixante-douze

13 treize

43 quarante-trois

73 soixante-treize

14 quatorze

44 quarante-quatre

74 soixante-quatorze

15 quinze

45 quarante-cinq

75 soixante-quinze

16 seize

46 quarante-six

76 soixante-seize

17 dix-sept

47 quarante-sept

77 soixante-dix-sept18 dix-huit

48 quarante-huit

78 soixante-dix-huit

19 dix-neuf

49 quarante-neuf

79 soixante-dix-neuf

20 vingt

50 cinquante

80 quatre-vingts

21 vingt et un

51 cinquante et un

81 quatre-vingt-un

22 vingt-deux

52 cinquante-deux

82 quatre-vingt-deux

23 vingt-trois

53 cinquante-trois

83 quatre-vingt-trois

24 vingt-quatre

55 cinquante-quatre

84 quatre-vingt-quatre

25 vingt-cinq

55 cinquante-cinq

85 quatre-vingt-cinq

26 vingt-six

56 cinquante-six

86 quatre-vingt-six

27 vingt-sept

57 cinquante-sept

87 quatre-vingt-sept28 vingt-huit

58 cinquante-huit

88 quatre-vingt-huit

29 vingt-neuf

59 cinquante-neuf

89 quatre-vingt-neuf

2

counting in French

0 zéro30 trente

60 soixante

90 quatre-vingt-dix

1 un

31 trente et un

61 soixante et un

91 quatre-vingt-onze

2 deux32 trente-deux

62 soixante-deux

90 is surprising!
70 and 80 are too!

3 trois33 trente-trois

63 soixante-trois

4 quatre

34 trente-quatre

64 soixante-quatre

5 cinq

35 trente-cinq

65 soixante-cinq

6 six

36 trente-six

66 soixante-six

7 sept37 trente-sept

67 soixante-sept8 huit38 trente-huit

68 soixante-huit

9 neuf

39 trente-neuf

69 soixante-neuf

10 dix40 quarante

70 soixante-dix

11 onze

41 quarante et un

71 soixante et onze

12 douze

42 quarante-deux

72 soixante-douze

13 treize

43 quarante-trois

73 soixante-treize

14 quatorze

44 quarante-quatre

74 soixante-quatorze

15 quinze

45 quarante-cinq

75 soixante-quinze

16 seize

46 quarante-six

76 soixante-seize

17 dix-sept

47 quarante-sept

77 soixante-dix-sept18 dix-huit

48 quarante-huit

78 soixante-dix-huit

19 dix-neuf

49 quarante-neuf

79 soixante-dix-neuf

20 vingt

50 cinquante

80 quatre-vingts

21 vingt et un

51 cinquante et un

81 quatre-vingt-un

22 vingt-deux

52 cinquante-deux

82 quatre-vingt-deux

23 vingt-trois

53 cinquante-trois

83 quatre-vingt-trois

24 vingt-quatre

55 cinquante-quatre

84 quatre-vingt-quatre

25 vingt-cinq

55 cinquante-cinq

85 quatre-vingt-cinq

26 vingt-six

56 cinquante-six

86 quatre-vingt-six

27 vingt-sept

57 cinquante-sept

87 quatre-vingt-sept28 vingt-huit

58 cinquante-huit

88 quatre-vingt-huit

29 vingt-neuf

59 cinquante-neuf

89 quatre-vingt-neuf

trente 30 = 3×10
quarante 40 = 4×10
cinquante 50 = 5×10
soixante 60 = 6×10
soixante-dix 70 = 6×10 +10
quatre-vingts 80 = 4×20
quatre-vingts-dix 90 = 4×20 +10

3counting in French
different French speakers count differently
1020

30

40

50

60

70

80

90

DIALECT AREA

dix

vingt

trentequarante

cinquantesoixante

soixante-dix

quatre-vingts

quatre-vingt-dix

Paris

dix

vingt

trentequarante

cinquantesoixante

septante

quatre-vingts

nonanteBelgium

dix

vingt

trentequarante

cinquantesoixante

septante

huitante

nonanteSwitzerland

dix

vingt

trentequarante

cinquantesoixante

septante

octantenonanteOccitania

4counting in Frenchsouthern and eastern French speakers count like Italians
1020

3040

5060

708090

DIALECT AREA

dix

vingt

trentequarante

cinquantesoixante

soixante-dix

quatre-vingts

quatre-vingt-dix

Paris

dix

vingt

trentequarante

cinquantesoixante

septante

quatre-vingts

nonanteBelgium

dix

vingt

trentequarante

cinquantesoixante

septante

huitante

nonanteSwitzerland

dix

vingt

trentequarante

cinquantesoixante

septante

octantenonanteOccitania

dieci

venti

trentaquaranta

cinquanta

sessanta

settanta

ottantanovantaMilan

quatre-vingts = 4×20
huitante = 8×10
octante = 8×10
ottanta = 8×10

5quatre-vingts-dix = 4×20 +10
nonante = 8×10
novanta = 8×10

old base-20 pattern
the spread from Normandy• Comparative linguistic analysis suggeststhat a Viking pattern of base-20 number
words was introduced to Normandy, from
where is spread and displaced someRoman base-10 number words.

• This was a re-introduction, because thepre-Roman Gauls also counted in base-20.

• We can still find base-20 numerals inmodern Welsh (ugain "20", deugain "40",pedwar ugain "80") and in modern Danish.

6counting in Danish
50, 70, and 90 show base-20
A decimal system is a counting system based on 10.
A vigesimal system is based on the number 20.
Danish is not as vigesimal as French, but Danish shows signsof a vigesimal system in the higher number words.
Also, the number words for 50, 70, and 90 are anticipatory.
Old Norse and Celtic had many vigesimal features.
Proto-Indo-European counting was decimal, so we wonder:
Why do these Indo-European languages have vigesimal
number words?
Note: a score = 20
(cf. “Four score and seven years ago”)

7

13 tretten
14 fjorten
15 femten
16 seksten
17 sytten
18 atten
19 nitten
20 tyve
21 enogtyve
22 toogtyve
23 treogtyve
24 fireogtyve
25 femogtyve
30 tredive
31 enogtredive
40 fyrre
50 halvtreds
60 tres
70 halvfjerds
80 firs
90 halvfems

3+10
3+10
3+10
3+10
3+10
3+10
3+10
2×10
1+2×102+2×10
3+2×10
4+2×10
5+2×10
3×10
1+3×104 (×10)
3×20
4×20

4 of themhalfway to 3 score
3 scorehalfway to 4 score
4 scorehalfway to 5 score

Decimal, vigesimal, quinary, duodecimal, etc.
Vigesimal counting in Mayan and Ainu
Historically, the counting systems of the world’s languages have been typically based on thenumbers 10 (decimal), 20 (vigesimal), 5 (quinary), and 12 (duodecimal). A base-10 system withsubbases 2 and 5 is called biquinary, and a base-20 system with subbases 4 and 5 is called
quadquinary. To create complex number words from basic number words, languages usemultiplicative, additive, subtractive, and anticipatory strategies.
The Mayan languages have perhaps the most famous vigesimal counting systems, and the written
Maya numerals are quadquinary.
anticipatory vigesimal counting in Qʼeqchiꞌ Mayan
The Ainu language has a vigesimal counting
writtenspoken phrase
meaning of spoken phrase
system, but nowadays the Ainu people speak
kwaqxaqlajuh roxkʼaal 18 on the way to 3×20 (18 of the 3rd 20)
Japanese.
58
Because of influence from English, Spanish,
59
belelajuh roxk’aal
19 on the way to 3×20 (19 of the 3rd 20)
Chinese, and other dominant languages, many60
oxk’aal3×20
languages with original vigesimal counting
systems have become or are becoming decimal. 61
jun xkaak’aal
1 on the way to 4×20 (1 of the 4th 20)
62

kwiib xkaak’aal
8

2 on the way to 4×20 (2 of the 4th 20)

ethnomathematics
studying the relationships between mathematics and culture
Topics studied in ethnomathematics include:
systems of counting
systems of mapping
pastimes and games of skill: board games, string patterns, graphic drawings
symmetry in art, aesthetics of proportion, patterns of weaving, tiling patterns
musical tunings
The mathematics of how different cultures do similar things can be interesting.
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