Short and Long Scales
For large numbers there are two scales used: the short scale and the long scale.
The naming procedure for large numbers is based on taking the number \(n\) occurring in \(10^{3n+3}\) (short scale) or \(10^{6n}\) (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix –illion. In this way, numbers up to \(10^{3×999+3} = 10^{3000}\) (short scale) or \(10^{6×999} = 10^{5994}\) (long scale) may be named.
The choice of roots and the concatenation procedure is that of the standard dictionary numbers if \(n\) is 20 or smaller. For larger \(n\) (between 21 and 999), prefixes can be constructed based on a system described by John Horton Conway and Richard K. Guy:
| Units | Tens | Hundreds | |
|---|---|---|---|
| 1 | \(\textrm{un}\) | \(^{\textrm{N}} \textrm{ deci}\) | \(^{\textrm{N,X}} \textrm{ centi}\) |
| 2 | \(\textrm{duo}\) | \(^{\textrm{M,S}} \textrm{ viginti}\) | \(^{\textrm{N}} \textrm{ ducenti}\) |
| 3 | \(\textrm{tre }^1\) | \(^{\textrm{N,S}} \textrm{ triginta}\) | \(^{\textrm{N,S}} \textrm{ trecenti}\) |
| 4 | \(\textrm{quattuor}\) | \(^{\textrm{N,S}} \textrm{ quadraginta}\) | \(^{\textrm{N,S}} \textrm{ quadringenti}\) |
| 5 | \(\textrm{quinqua}\) | \(^{\textrm{N,S}} \textrm{ quinquaginta}\) | \(^{\textrm{N,S}} \textrm{ quingenti}\) |
| 6 | \(\textrm{se }^{1,2}\) | \(^{\textrm{N}} \textrm{ sexaginta}\) | \(^{\textrm{N}} \textrm{ sescenti}\) |
| 7 | \(\textrm{septe }^{3,4}\) | \(^{\textrm{N}} \textrm{ septuaginta}\) | \(^{\textrm{N}} \textrm{ septingenti}\) |
| 8 | \(\textrm{octo}\) | \(^{\textrm{M,X}} \textrm{ octoginta}\) | \(^{\textrm{M,X}} \textrm{ octingenti}\) |
| 9 | \(\textrm{nove }^{3,4}\) | \(\textrm{nonaginta}\) | \(\textrm{nongenti}\) |
\(^1\) When preceding a component marked \(^{\textrm{S}}\), “tre” changes to “tres” and “se” changes to “ses”.
\(^2\) When preceding a component marked \(^{\textrm{X}}\), “se” changes to “sex”.
\(^3\) When preceding a component marked \(^{\textrm{M}}\), “septe” and “nove” change to “septem” and “novem”.
\(^4\) When preceding a component marked \(^{\textrm{N}}\), “septe” and “nove” change to “septen” and “noven”.
There appears to be some ambiguity with trecentillion. Is it tre (3) + centi (100) + -illion (103-illion or \(10^{312}\)) or trecenti (300) + -illion (300-illion or \(10^{903}\))?
Trivia
If you alphabetized all the English names of the numbers, the first odd number on the list will be eight billion eighty-five (short scale) or eight billiard eighty-five (long scale).1
Short Scale
| \(\qquad\) Name |
\(\qquad\) Value |
Groups of Three Zeros |
\(\qquad\) Notes |
|---|---|---|---|
| \(\textrm{thousand}\) | \(10^{3}\) | \(1\) | \(\textrm{SI unit: k / kilo-}\) |
| \(\textrm{million}\) | \(10^{6}\) | \(2\) | \(\textrm{SI unit: M / Mega-}\) |
| \(\textrm{billion}\) | \(10^{9}\) | \(3\) | \(\textrm{SI unit: G / Giga-}\) |
| \(\textrm{trillion}\) | \(10^{12}\) | \(4\) | \(\textrm{SI unit: T / Tera-}\) |
| \(\textrm{quadrillion}\) | \(10^{15}\) | \(5\) | \(\textrm{SI unit: P / Peta-}\) |
| \(\textrm{quintillion}\) | \(10^{18}\) | \(6\) | \(\textrm{SI unit: E / Exa-}\) |
| \(\textrm{sextillion}\) | \(10^{21}\) | \(7\) | \(\textrm{SI unit: Z / Zetta-}\) |
| \(\textrm{septillion}\) | \(10^{24}\) | \(8\) | \(\textrm{SI unit: Y / Yotta-}\) |
| \(\textrm{octillion}\) | \(10^{27}\) | \(9\) | |
| \(\textrm{nonillion}\) | \(10^{30}\) | \(10\) | |
| \(\textrm{decillion}\) | \(10^{33}\) | \(11\) | |
| \(\textrm{undecillion}\) | \(10^{36}\) | \(12\) | |
| \(\textrm{duodecillion}\) | \(10^{39}\) | \(13\) | |
| \(\textrm{tredecillion}\) | \(10^{42}\) | \(14\) | |
| \(\textrm{quattuordecillion}\) | \(10^{45}\) | \(15\) | |
| \(\textrm{quindecillion}\) | \(10^{48}\) | \(16\) | |
| \(\textrm{sexdecillion}\) | \(10^{51}\) | \(17\) | |
| \(\textrm{septendecillion}\) | \(10^{54}\) | \(18\) | |
| \(\textrm{octodecillion}\) | \(10^{57}\) | \(19\) | |
| \(\textrm{novemdecillion}\) | \(10^{60}\) | \(20\) | |
| \(\textrm{vigintillion}\) | \(10^{63}\) | \(21\) | |
| \(\textrm{unvigintillion}\) | \(10^{66}\) | \(22\) | |
| \(\textrm{duovigintillion}\) | \(10^{69}\) | \(23\) | |
| \(\textrm{trevigintillion}\) | \(10^{72}\) | \(24\) | |
| \(\textrm{quattuorvigintillion}\) | \(10^{75}\) | \(25\) | |
| \(\textrm{quinvigintillion}\) | \(10^{78}\) | \(26\) | |
| \(\textrm{sexvigintillion}\) | \(10^{81}\) | \(27\) | |
| \(\textrm{septvigintillion}\) | \(10^{84}\) | \(28\) | |
| \(\textrm{octovigintillion}\) | \(10^{87}\) | \(29\) | |
| \(\textrm{novemvigintillion}\) | \(10^{90}\) | \(30\) | |
| \(\textrm{trigintillion}\) | \(10^{93}\) | \(31\) | |
| \(\textrm{untrigintillion}\) | \(10^{96}\) | \(32\) | |
| \(\textrm{duotrigintillion}\) | \(10^{99}\) | \(33\) | |
| \(\textrm{tretrigintillion}\) | \(10^{102}\) | \(34\) | |
| \(\textrm{quattuortrigintillion}\) | \(10^{105}\) | \(35\) | |
| \(\textrm{quintrigintillion}\) | \(10^{108}\) | \(36\) | |
| \(\textrm{sextrigintillion}\) | \(10^{111}\) | \(37\) | |
| \(\textrm{septentrigintillion}\) | \(10^{114}\) | \(38\) | |
| \(\textrm{octotrigintillion}\) | \(10^{117}\) | \(39\) | |
| \(\textrm{novemtrigintillion}\) | \(10^{120}\) | \(40\) | \(39+-illion\) |
| \(\textrm{quinquagintillion}\) | \(10^{153}\) | \(51\) | \(50+-illion\) |
| \(\textrm{novemquinquagintillion}\) | \(10^{180}\) | \(60\) | \(59+-illion\) |
| \(\textrm{centillion}\) | \(10^{303}\) | \(101\) | \(100+-illion\) |
| \(\textrm{novenonagintanongentillion}\) | \(10^{3000}\) | \(1000\) | \(999+-illion\) |
Long Scale
| \(\qquad\) Name |
\(\qquad\) Value |
Groups of Three Zeros |
\(\qquad\) Notes |
|---|---|---|---|
| \(\textrm{thousand}\) | \(10^{3}\) | \(1\) | \(\textrm{SI unit: k / kilo-}\) |
| \(\textrm{million}\) | \(10^{6}\) | \(2\) | \(\textrm{SI unit: M / Mega-}\) |
| \(\textrm{milliard}\) | \(10^{9}\) | \(3\) | \(\textrm{Name unique to long scale.}\) \(\textrm{SI unit: G / Giga-}\) |
| \(\textrm{billion}\) | \(10^{12}\) | \(4\) | \(\textrm{SI unit: T / Tera-}\) |
| \(\textrm{billiard}\) | \(10^{15}\) | \(5\) | \(\textrm{Name unique to long scale.}\) \(\textrm{SI unit: P / Peta-}\) |
| \(\textrm{trillion}\) | \(10^{18}\) | \(6\) | \(\textrm{SI unit: E / Exa-}\) |
| \(\textrm{trilliard}\) | \(10^{21}\) | \(7\) | \(\textrm{Name unique to long scale.}\) \(\textrm{SI unit: Z / Zetta-}\) |
| \(\textrm{quadrillion}\) | \(10^{24}\) | \(8\) | \(\textrm{SI unit: Y / Yotta-}\) |
| \(\textrm{quintillion}\) | \(10^{30}\) | \(10\) | |
| \(\textrm{sextillion}\) | \(10^{36}\) | \(12\) | |
| \(\textrm{septillion}\) | \(10^{42}\) | \(14\) | |
| \(\textrm{octillion}\) | \(10^{48}\) | \(16\) | |
| \(\textrm{nonillion}\) | \(10^{54}\) | \(18\) | |
| \(\textrm{decillion}\) | \(10^{60}\) | \(20\) | |
| \(\textrm{undecillion}\) | \(10^{66}\) | \(22\) | |
| \(\textrm{duodecillion}\) | \(10^{72}\) | \(24\) | |
| \(\textrm{tredecillion}\) | \(10^{78}\) | \(26\) | |
| \(\textrm{quattuordecillion}\) | \(10^{84}\) | \(28\) | |
| \(\textrm{quindecillion}\) | \(10^{90}\) | \(30\) | |
| \(\textrm{sexdecillion}\) | \(10^{96}\) | \(32\) | |
| \(\textrm{septendecillion}\) | \(10^{102}\) | \(34\) | |
| \(\textrm{octodecillion}\) | \(10^{108}\) | \(36\) | |
| \(\textrm{novemdecillion}\) | \(10^{114}\) | \(38\) | |
| \(\textrm{vigintillion}\) | \(10^{120}\) | \(40\) | |
| \(\textrm{unvigintillion}\) | \(10^{126}\) | \(42\) | |
| \(\textrm{duovigintillion}\) | \(10^{132}\) | \(44\) | |
| \(\textrm{trevigintillion}\) | \(10^{138}\) | \(46\) | |
| \(\textrm{quattuorvigintillion}\) | \(10^{144}\) | \(48\) | |
| \(\textrm{quinvigintillion}\) | \(10^{150}\) | \(50\) | |
| \(\textrm{quinvigintilliard}\) | \(10^{153}\) | \(51\) | \(\textrm{Name unique to long scale.}\) |
| \(\textrm{sexvigintillion}\) | \(10^{156}\) | \(52\) | |
| \(\textrm{septvigintillion}\) | \(10^{162}\) | \(54\) | |
| \(\textrm{octovigintillion}\) | \(10^{168}\) | \(56\) | |
| \(\textrm{novemvigintillion}\) | \(10^{174}\) | \(58\) | |
| \(\textrm{trigintillion}\) | \(10^{180}\) | \(60\) | |
| \(\textrm{untrigintillion}\) | \(10^{186}\) | \(62\) | |
| \(\textrm{duotrigintillion}\) | \(10^{192}\) | \(64\) | |
| \(\textrm{tretrigintillion}\) | \(10^{198}\) | \(66\) | |
| \(\textrm{quattuortrigintillion}\) | \(10^{204}\) | \(68\) | |
| \(\textrm{quintrigintillion}\) | \(10^{210}\) | \(70\) | |
| \(\textrm{sextrigintillion}\) | \(10^{216}\) | \(72\) | |
| \(\textrm{septentrigintillion}\) | \(10^{222}\) | \(74\) | |
| \(\textrm{octotrigintillion}\) | \(10^{228}\) | \(76\) | |
| \(\textrm{novemtrigintillion}\) | \(10^{234}\) | \(78\) | \(39+-illion\) |
| \(\textrm{quinquagintillion}\) | \(10^{300}\) | \(100\) | \(50+-illion\) |
| \(\textrm{novemquinquagintillion}\) | \(10^{354}\) | \(118\) | \(59+-illion\) |
| \(\textrm{centillion}\) | \(10^{600}\) | \(200\) | \(100+-illion\) |
| \(\textrm{novenonagintanongentillion}\) | \(10^{5994}\) | \(1998\) | \(999+-illion\) |
Other Named Numbers
| Name | Value | Notes |
|---|---|---|
| \(\textrm{absolute zero °F}\) | \(-459.67\) | |
| \(\textrm{absolute zero °C}\) | \(-273.15\) | |
| \(e\) | \(2.718281828(46)\) | \(\textrm{28 significant digits} \\ \textrm{a.k.a. Euler’s number} \\ \textrm{a.k.a. Napier’s constant}\) |
| \(\textrm{googol}\) | \(10^{100}\) | |
| \(\textrm{googolplex}\) | \(10^{10^{100}}\) | \(\textrm{Also } 10^{\textrm{googol}}\) |
| \(i\) | \(\sqrt{-1}\) | \(\textrm{imaginary number}\) |
| \(\phi, \varphi \textrm{ (phi)}\) | \(1.618033988(75)\) | \(\textrm{golden ratio}\) \({\large {\frac{1 + \sqrt{5}}{2}}}\) |
| \(\pi \textrm{ (pi)}\) | \(3.141592653(59)\) | \(\textrm{28 significant digits}\) |
| \(\textrm{Planck length}\) | \(1.616255(18)×10^{-35} \textrm{ m}\) | \(\ell_\textrm{P}\) |
| \(\textrm{square root of two}\) | \(1.414213562(37)\) | \(\sqrt{2}\) |
| \(\textrm{square root of three}\) | \(1.732050807(57)\) | \(\sqrt{3}\) |
| \(\textrm{square root of five}\) | \(2.236067977(50)\) | \(\sqrt{5}\) |
| \(\textrm{twelfth root of two}\) | \(1.059463094(36)\) | \(\textrm{Proportion between the frequencies of} \\ \textrm{adjacent semitones in the equal} \\ \textrm{temperament scale.}\) \(\sqrt[^{12}]{2}\) |
See Also
There is a guide to pronouncing large numbers (and other numbers) on Clark and Miller’s website: “Numbers in English: The Ultimate Guide“.
References
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Footnotes
- Hat tip to Simon Gamble for the short scale fact. (Source)