Number

 The word "number" is a general term which refers to a member of a given (possibly ordered) set. The meaning of "number" is often clear from context (i.e., does it refer to a complex number, integer, real number, etc.?). Wherever possible in this work, the word "number" is used to refer to quantities which are integers, and "constant" is reserved for nonintegral numbers which have a fixed value. Because terms such as real number, Bernoulli number, and irrational number are commonly used to refer to nonintegral quantities, however, it is not possible to be entirely consistent in nomenclature.

To indicate a particular numerical label, the abbreviation "no." is sometimes used (deriving from "numero," the ablative case of the Latin "numerus"), as is the less common "nr." The symbol # (known as the octothorpe) is commonly used to denote "number."

While some authors prefer to include "and" between various parts of a number name, in this work, "and"s are omitted. For example, the number 101 is called "one hundred one" rather than "one hundred and one." According to most definitive sources (Schildberger 2001; The Chicago Manual of Style 2003, p. 381; Mish 2003, p. 852), either is acceptable. However, The Associated Press Stylebook gives implicit examples in which the "and" is omitted (Goldstein 1998, p. 145). So the fact of the matter is that different sources use different conventions, with some sources even being internally inconsistent. For example, Conway and Guy (1996) list the "(and)" as optional on p. 15, while writing out "one hundred and sixty-three" on p. 25.

According to The Chicago Manual of Style (2003, p. 380), in nontechnical written contexts, whole numbers from one to one hundred should always be spelled out, and other whole numbers should be written in terms of numerals. In addition, when a number begins a sentence, it is always spelled out unless it appears awkward, in which case the sentence should be recast. In this work, numbers are sometimes spelled out and sometimes written numerically, depending on which appears clearer.

Note that commas should not be used to separate words that are part of one number (Goldstein 1998, p. 145).

The following table summarizes the English names given to the first few positive numbers (Schildberger 2001, Misch 2003).

	name
1	one
2	two
3	three
4	four
5	five
6	six
7	seven
8	eight
9	nine
10	ten
11	eleven
12	twelve
13	thirteen
14	fourteen
15	fifteen
16	sixteen
17	seventeen
18	eighteen
19	nineteen
20	twenty
21	twenty-one
22	twenty-two
23	twenty-three
24	twenty-four
25	twenty-five
26	twenty-six
27	twenty-seven
28	twenty-eight
29	twenty-nine
30	thirty
31	thirty-one
32	thirty-two
33	thirty-three
34	thirty-four
35	thirty-five
36	thirty-six
37	thirty-seven
38	thirty-eight
39	thirty-nine
40	forty
41	forty-one
42	forty-two
43	forty-three
44	forty-four
45	forty-five
46	forty-six
47	forty-seven
48	forty-eight
49	forty-nine
50	fifty
51	fifty-one
52	fifty-two
53	fifty-three
54	fifty-four
55	fifty-five
56	fifty-six
57	fifty-seven
58	fifty-eight
59	fifty-nine
60	sixty
61	sixty-one
62	sixty-two
63	sixty-three
64	sixty-four
65	sixty-five
66	sixty-six
67	sixty-seven
68	sixty-eight
69	sixty-nine
70	seventy
71	seventy-one
72	seventy-two
73	seventy-three
74	seventy-four
75	seventy-five
76	seventy-six
77	seventy-seven
78	seventy-eight
79	seventy-nine
80	eighty
81	eighty-one
82	eighty-two
83	eighty-three
84	eighty-four
85	eighty-five
86	eighty-six
87	eighty-seven
88	eighty-eight
89	eighty-nine
90	ninety
91	ninety-one
92	ninety-two
93	ninety-three
94	ninety-four
95	ninety-five
96	ninety-six
97	ninety-seven
98	ninety-eight
99	ninety-nine
100	one hundred
101	one hundred one
102	one hundred two
103	one hundred three
104	one hundred four
105	one hundred five
106	one hundred six
107	one hundred seven
108	one hundred eight
109	one hundred nine
110	one hundred ten
111	one hundred eleven
112	one hundred twelve
113	one hundred thirteen
114	one hundred fourteen
115	one hundred fifteen
116	one hundred sixteen
117	one hundred seventeen
118	one hundred eighteen
119	one hundred nineteen
120	one hundred twenty
121	one hundred twenty-one
122	one hundred twenty-two
123	one hundred twenty-three
124	one hundred twenty-four
125	one hundred twenty-five
126	one hundred twenty-six
127	one hundred twenty-seven
128	one hundred twenty-eight
129	one hundred twenty-nine
130	one hundred thirty
131	one hundred thirty-one
132	one hundred thirty-two
133	one hundred thirty-three
134	one hundred thirty-four
135	one hundred thirty-five
136	one hundred thirty-six
137	one hundred thirty-seven
138	one hundred thirty-eight
139	one hundred thirty-nine
140	one hundred forty
141	one hundred forty-one
142	one hundred forty-two
143	one hundred forty-three
144	one hundred forty-four
145	one hundred forty-five
146	one hundred forty-six
147	one hundred forty-seven
148	one hundred forty-eight
149	one hundred forty-nine
150	one hundred fifty

Note that two differing conventions exist for the naming of large numbers (e.g., the American system terms  "a billion," while the British system terms it "a milliard").

American	British	power of 10
million	million
billion	milliard
trillion	billion
quadrillion
quintillion	trillion
sextillion
septillion	quadrillion
octillion
nonillion	quintillion
decillion
undecillion	sexillion
duodecillion
tredecillion	septillion
quattuordecillion
quindecillion	octillion
sexdecillion
septendecillion	nonillion
octodecillion
novemdecillion	decillion
vigintillion
	undecillion
	duodecillion
	tredecillion
	quattuordecillion
	quindecillion
	sexdecillion
	septendecillion
	octodecillion
	novemdecillion
	vigintillion
centillion
	centillion

The numbers of characters (including spaces and dashes) in the English names (written without trailing "and"s) for the numbers 1, 2, 3, ... are 3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 6, 6, 8, 8, 7, 7, 9, 8, 8, 6, 10, 10, ... (OEIS A052360), illustrated above.

The corresponding numbers of letters are 3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 6, 6, 8, 8, 7, 7, 9, 8, ... (OEIS A005589; Eureka 1974), illustrated above.

The corresponding numbers of syllables are 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, ... (OEIS A075774).

Consider the numbers having incrementally largest numbers of letters. This gives the sequence 1, 3, 11, 13, 17, 23, 73, 101, 103, 111, 113, 117, 123, 173, 323, 373, ... (OEIS A052363), which have the corresponding number of digits 3, 5, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 21, ... (OEIS A089585).

REFERENCES:

"Problems Drive." Eureka 37, 8-11 and 33, 1974.

The Chicago Manual of Style, 15th ed. Chicago, IL: University of Chicago Press, 2003.

Barbeau, E. J. Power Play: A Country Walk through the Magical World of Numbers. Providence, RI: Amer. Math. Soc., 1997.

Bogomolny, A. "What is a Number." https://www.cut-the-knot.org/do_you_know/numbers.shtml.

Borwein, J. and Borwein, P. A Dictionary of Real Numbers. London: Chapman & Hall, 1990.

Conway, J. H. On Numbers and Games, 2nd ed. Wellesley, MA: A K Peters, 2000.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996.

Dantzig, T. Number: The Language of Science, 4th rev. ed. New York: Free Press, 1985.

Davis, P. J. The Lore of Large Numbers. New York: Random House, 1961.

De Geest, P. "World!Of Numbers." https://www.worldofnumbers.com/.

Ebbinghaus, H. D.; Hirzebruch, F.; Hermes, H.; Prestel, A; Koecher, M.; Mainzer, M.; and Remmert, R. Numbers. New York: Springer-Verlag, 1990.

Frege, G. Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, 2nd rev. ed. Evanston, IL: Northwestern University Press, 1980.

Goldstein, N. (Ed.). The Associated Press Stylebook and Libel Manual, fully upd. rev. Reading, MA: Addison-Wesley, 1998.

Ifrah, G. From One to Zero: A Universal History of Numbers. New York: Viking, 1987.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

McLeish, J. Number: The History of Numbers and How They Shape Our Lives. New York: Fawcett Columbine, 1992.

Mish, F. C. (Ed.). Merriam-Webster's Collegiate Dictionary, 11th ed. Springfield, MA: Merriam-Webster, 2003.

Munafo, R. "Notable Properties of Specific Numbers." https://home.earthlink.net/~mrob/pub/math/numbers.html.

Phillips, R. Numbers: Facts, Figures & Fiction. Cambridge, England: Cambridge University Press, 1994.

Rosenfelder, M. "Numbers from 1 to 10 in Over 4000 Languages." https://zompist.com/numbers.shtml.

Russell, B. "Definition of Number." Introduction to Mathematical Philosophy. New York: Simon and Schuster, 1971.

Schildberger, G. "English Names for the Numbers from 0 to 1022." Apr. 4, 2001. https://www.research.att.com/~njas/sequences/a000027.txt.

Sloane, N. J. A. Sequences A005589/M2277, A052360, A052363, A075774, and A089585 in "The On-Line Encyclopedia of Integer Sequences."

Smeltzer, D. Man and Number. Buchanan, NY: Emerson Books, 1974.

Weisstein, E. W. "Books about Numbers." https://www.ericweisstein.com/encyclopedias/books/Numbers.html.

Wells, D. W. The Penguin Dictionary of Curious and Interesting Numbers. Harmondsworth, England: Penguin Books, 1986.