# Named Numbers

Contents

## 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 $$\textrm{un}$$ $$^{\textrm{N}} \textrm{ deci}$$ $$^{\textrm{N,X}} \textrm{ centi}$$ $$\textrm{duo}$$ $$^{\textrm{M,S}} \textrm{ viginti}$$ $$^{\textrm{N}} \textrm{ ducenti}$$ $$\textrm{tre }^1$$ $$^{\textrm{N,S}} \textrm{ triginta}$$ $$^{\textrm{N,S}} \textrm{ trecenti}$$ $$\textrm{quattuor}$$ $$^{\textrm{N,S}} \textrm{ quadraginta}$$ $$^{\textrm{N,S}} \textrm{ quadringenti}$$ $$\textrm{quinqua}$$ $$^{\textrm{N,S}} \textrm{ quinquaginta}$$ $$^{\textrm{N,S}} \textrm{ quingenti}$$ $$\textrm{se }^{1,2}$$ $$^{\textrm{N}} \textrm{ sexaginta}$$ $$^{\textrm{N}} \textrm{ sescenti}$$ $$\textrm{septe }^{3,4}$$ $$^{\textrm{N}} \textrm{ septuaginta}$$ $$^{\textrm{N}} \textrm{ septingenti}$$ $$\textrm{octo}$$ $$^{\textrm{M,X}} \textrm{ octoginta}$$ $$^{\textrm{M,X}} \textrm{ octingenti}$$ $$\textrm{nove }^{3,4}$$ $$\textrm{nonaginta}$$ $$\textrm{nongenti}$$

$$^1$$ 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
$$\textrm{absolute zero °F}$$ $$-459.67$$
$$\textrm{absolute zero °C}$$ $$-273.15$$
$$e$$ $$2.718281828(46)$$ $$\textrm{28 significant digits} \\ \textrm{a.k.a. Euler’s number} \\ \textrm{a.k.a. Napier’s constant}$$
$$\textrm{googol}$$ $$10^{100}$$
$$\textrm{googolplex}$$ $$10^{10^{100}}$$ $$\textrm{Also } 10^{\textrm{googol}}$$
$$i$$ $$\sqrt{-1}$$ $$\textrm{imaginary number}$$
$$\phi, \varphi \textrm{ (phi)}$$ $$1.618033988(75)$$ $$\textrm{golden ratio}$$
$${\large {\frac{1 + \sqrt{5}}{2}}}$$
$$\pi \textrm{ (pi)}$$ $$3.141592653(59)$$ $$\textrm{28 significant digits}$$
$$\textrm{Planck length}$$ $$1.616255(18)×10^{-35} \textrm{ m}$$ $$\ell_\textrm{P}$$
$$\textrm{square root of two}$$ $$1.414213562(37)$$ $$\sqrt{2}$$
$$\textrm{square root of three}$$ $$1.732050807(57)$$ $$\sqrt{3}$$
$$\textrm{square root of five}$$ $$2.236067977(50)$$ $$\sqrt{5}$$
$$\textrm{twelfth root of two}$$ $$1.059463094(36)$$ $$\textrm{Proportion between the frequencies of} \\ \textrm{adjacent semitones in the equal} \\ \textrm{temperament scale.}$$
$$\sqrt[^{12}]{2}$$