Some Thoughts About Numeric Reform

Note: This page is part of a short series. Please read the page on Phonetic Reform first for background on these reforms.

English number names are both long and inconsistent compared to number names in many other languages, and therefore ripe for reform. Studies suggest that these failings lead to decreased mathematical performance. Chinese speakers can process nine numbers every two seconds due to the shorter names, where English speakers can only process seven. There is also experimental evidence that consistent agglutination systems for higher numbers (e.g. “ten-one” instead of “eleven”) make it easier for children to learn to count.

In my own personal work as a statistician, and before reading the evidence on international comparisons, I had already years ago independently adopted a primitive version of these reforms to speed up my counting of items, so that I could work faster. (I used [won] [tʊʊ] [tii] [fʊə] [fai] [si] [se] [nei] [nai] [ten].) Even this small change noticeably sped up my counting, even as a competent mathematician, so I can easily believe that young children gain even more benefit from such a difference.

Though these thoughts were the initial motivation behind the reforms, they were not the only factors involved in their design. Here are some important values in a counting system that I had to take into account:

  • distinctiveness: the names should be phonetically distinct from each other, to aid communication;
  • patterns: the names should follow patterns, to aid identification of groups of names;
  • continuity: the names should remain close to the original English names, to aid transition from the old system;
  • consistency: the grammar of agglutinated number names should follow a consistent schema, to aid learning.
  • brevity: the names should be short, to aid speed of calculation;

My counting number names are: [zoʊ] (zero), [um] (one), [tʊʊ] (two), [tii] (three), [fʊ] (four), [fi] (five), [so] (six), [se] (seven), [nuə] (eight), [na] (nine). Then the set numbers, which are used conjunctively without “and”: [dein] (ten), [keən] (hundred), [θaʊn] (thousand), [mail] (million), [#ail] (*illion). “Eleven” is [dein um], “seventy seven” is [se dein se], “nine thousand eight hundred and seventy six” is [na θaʊn nuə keən se dein so], and so on.

I have never (even as a young child) liked the existing system of billions and trillions and so on, because they add in new words too frequently to be efficient (the American system is even worse than the British system for this). I think that they would be better if they followed a binary pattern: “million” is six 0s, “billion” is twelve 0s, “trillion”, etc. Since I can make my own system function however I like, this is therefore how they work in my system.

The names follow the counting numbers followed by [ail]: 1,000,000 is [mail], as above, then 1,000,000,000,000 is [tʊʊwail], 1,000,000,000,000,000,000,000,000 is [tiiyail], 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 is [fʊhail], and so on. By the time we reach [deinail] there are over three thousand zeroes, at which point everyone has started using scientific notation long before anyway, and this is vastly larger than the highest standard continuous set number in English, a vigintillion with a mere sixty-three zeroes. 2,000,000,000,000 is [tʊʊ tʊʊwail], distinguished from 2,000,000 which is [tʊʊ mail].

Distinctiveness

The initial distinction is that counting numbers ([um] to [na]) have two phonemes, and the main set numbers ([dein], [keən], etc) have three phonemes.

The initial consonant of every counting number name comes as part of a distinct pair, and the initial consonant of every set number name is unique. The terminal consonant of the lower set names ([dein], [keən], [θaʊn]) is distinct from the terminal consonant of the higher set names ([mail], etc).

Each name also has a unique vowel phoneme, making it very hard to confuse them for each other. Not only is each vowel phoneme unique, but where the consonants are paired, the vowels are from separated positions of the vowel ring, to further enhance their distinctiveness. Compare [ʊʊ] and [ii], [ʊ] and [i], [o] and [e], [uə] and [a].

“One” [um] is made further distinct by reversing the normal consonant-vowel pattern to be vowel-consonant.

Patterns

The two/three phoneme distinction between counting numbers and set numbers is supported by the medium matching the message: the smaller type of number is shorter; the larger type of number is longer. The pattern reinforces the distinction.

Adjacent counting numbers have the same initial consonant. The lower set numbers have the same terminal consonant, and likewise for the higher set numbers.

The higher set numbers follow a clear pattern based on the ordinal binary. One set of six zeroes is (abbreviated from) [umail]; two sets of six zeroes are [tʊʊwail]; four are [tiiyail]; and so on.

Continuity

Not that I ever expect anything I do to have any practical effect on the world, but it is an important principle that changes should be made in such a way as to smooth the hypothetical transition to the new system. After all, as with the phonetic reform, my aim is to reform English, not to create a wholly new language. I therefore aimed to keep the names reminiscent of their existing values as far as possible. This decision itself supported several other aims.

One is the pattern of paired initial consonants and the unique and contrasting vowel phonemes for the counting numbers: these had developed organically in English and were able to be retained. I erred on the side of achieving greater continuity for the lower, more common numbers.

The names of the set numbers were likewise influenced by their names in English and related European languages. “Ten” could not begin with ‘t’ to ensure it stayed distinctive from [tʊʊ] and [tii], but ‘d’ is a common substitute in many European languages (dix, diez, dieci, etc), and the vowel and ending mirror German zehn. For “hundred” I wanted to avoid the easily dropped ‘h’ to maintain distinctiveness with [um], and could not use ‘s’ because of [so] and [se], so reverted to Latin ‘k’, choosing [keən] for continuity with Latin centum while maintaining distinctiveness from [se] and also from other common English words such as “cane” [kein], “cone” [coʊn], etc. [θaʊn] is a clear analogue of “thousand”. For “million” I kept the ‘m’ and ‘l’ with an ‘i’ included, but needed a phoneme that had not been used elsewhere, so [mil]/[fi], [miil]/[tii], and [meil]/[dein] were out, and [moil] sounded inelegant.

I retained base 10. I thought about changing it, but so long as most humans have ten fingers, it makes a very practical base for our species to use. Again, the aim is to improve English, not create something original for its own sake. 10 is also a useful order of magnitude: much lower, and you need too many set numbers; much higher, and you need too many counting numbers. Ten is in the Goldilocks zone, and it matches our fingers, so it’s good enough.

Consistency

The grammar of English numbers is inconsistent. We use set numbers ordinarily for milions, thousands, and hundreds, but for tens we use [-tii], warp the associated counting number (e.g. “thirty” instead of “three ten”), and insist on a prefatory “and”. This is especially bad as the tens are the set numbers we most often have cause to use, and even when we use higher set numbers we often need the tens too.

The tens have therefore been reformed to, in effect, match the hundreds and thousands. They end with [dein]; they use the normal counting number; and they do not need an “and”.

The grammar of the higher set numbers has also been made more consistent. It is all very well that bi- is Latin for “twice”, and so on, but that makes the counting names less descriptive, and therefore less comprehensible to a learner. So they have been changed in the way described above to ensure that they have a consistent and comprehensible grammar.

Brevity

It is clear that the new names are much more concise than their natural language equivalents. There is further benefit gained by removing the superfluous conjunctions that follow most set numbers. We can quantify the gains of this brevity, which vary.

[tʊʊ] is exactly the same and [eit]/[nuə] is essentially the same length; [ten] has as many phonemes as [dein] but is a short rather than long vowel, so could be considered a loss, while [fʊə]/[fʊ] is the reverse. Now the gains: [won]/[um] and [θrii]/[tii] both gain us a phoneme; [faiv]/[fi] and [nain]/[na] both gain us a phoneme and shorten the vowel. [siks]/[so] gains us two phonemes; [sevən]/[se] gains us three phonemes, including an extra syllable. [hundrəd]/[keən] gains us four phonemes and a syllable; [θaʊzənd]/[θaʊn] gains us three phonemes and a syllable; [milyən]/[mail] gains us three phonemes and a syllable (or four phonemes and two syllables if we use [miliiyən]); all of these also benefit from losing the conjunction.

So, in an extreme case, 7777 is [sevən θaʊzənd sevən hundrəd and sevəntii sevən] with 38 phonemes and 14 syllables, or [se θaʊn se keən se dein se] with 17 phonemes and 7 syllables, a 55% reduction in phoneme count. The benefit of speed ought to be immediately apparent to all.

Characters

Finally, while I was at it, I also reformed the numeric characters. Each number from 1 to 5 has a number of marks to indicate its number. Then 6 to 9 are 1 to 4 with a bar across the top. Zero is noted with a moderate circle, not the full oval of 0. Here are the numbers 1-10.

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This page is part of a short series. You may also be interested in my thoughts about Phonetic Reform and Grammatical Reform.

sum θoətiz abaʊt nyʊʊmerik rifʊəm

noət: ðis peij bii paət ov a ʃʊət siəriiz. kom pliiz riid ðə peij on fənetik rifʊəm umθ foə bakgraʊnd on ðiiz rifʊəmiz.

ingliʃ numbə neimiz bii boʊθ loŋg and inkənsistənt kəmpeərəð tə numbə neimiz in meniiy uðə langwijiz, and ðeəfoə raip foə rifʊəm. studiiyiz səjest ðat ðeez feiliŋgiz liid tə dikriisəð maθmatikəl pəfʊəməns. cainiiz spiikəriz kan proʊses na numbəriz evrii tʊʊ sekəndiz dyʊʊ tə ðə ʃʊətə neimiz, weər ingliʃ spiikəriz kan oʊnlii proʊses se. ðeə bii oəlsoʊw eksperimentəl evidəns ðat kənsistənt aglʊʊtineiʃən sistəmiz foə haiyə numbəriz (e.g. “ten-one” insted ov “eleven”) meik hiiy iiziiyə foə caildiz tə luən tə caʊnt.

in miis oʊn puəsənəl wuək az a statistiʃən, and bifʊə riidiŋ ðiiy evidəns on intənaʃənəl kəmparisəniz, mii oəlredii yiəriz agoʊw indipendəntliiy adoptivid a primitiv vuəʒən ov ðiiz rifʊəmiz tə spiid up miis kaʊntiŋ ov aitəmiz, soʊ ðat mii kʊd wuək fastə. (mii yʊʊzid [won] [tʊʊ] [tii] [fʊə] [fai] [si] [se] [nei] [nai] [ten].) iivən ðis smoəl ceinj noʊtisəblii spiidid up miis kaʊntiŋg, iivən az a kompətənt maθmətiʃən, soʊ mii kan iizilii biliiv ðat yuŋ caildiz gein iivən mʊə benifit from suc a difrəns.

ðoʊ ðiiz θoətiz biiyid ðiiy iniʃəl moʊtiveiʃən bihaind ðə rifʊəmiz, dii biiyid not ðiiy oʊnlii faktəriz involvəð in diis dizain. hiə bii sum impʊətənt valyʊʊwiz in a kaʊntiŋ sistəm ðat mii hafid tə teik intʊʊw akaʊnt:

  • distinktivnəs: ðə neimiz ʃʊd bii fəneticlii distinkt from iic uðə, tʊʊw eid kəmyʊʊnikeiʃən;
  • patəniz: ðə neimiz ʃʊd foloʊ patəniz, tʊʊw eid aidentifikeiʃən ov grʊʊpiz ov neimiz;
  • kontinyʊʊwitii: ðə neimiz ʃʊd rimein kloʊs tə ðiiy ərijinəl ingliʃ neimiz, tʊʊw eid tranziʃən from thiiy oʊld sistəm;
  • kənsistənsii: ðə gramər ov aglʊʊtineitəð numbə neimiz ʃʊd foloʊw a kənsistənt skiimə, tʊʊw eid luəniŋ.
  • brevitii: ðə neimiz ʃʊd bii ʃʊət, tʊʊw eid spiid ov kalkyəleiʃən;

miis kaʊntiŋ numbə neimiz bii: [zoʊ] (zero), [um] (one), [tʊʊ] (two), [tii] (three), [fʊ] (four), [fi] (five), [so] (six), [se] (seven), [nuə] (eight), [na] (nine). ðen ðə set numbəriz, wic bii yʊʊzəð kənjunktivlii wið-aʊt “and”: [dein] (ten), [keən] (hundred), [θaʊn] (thousand), [mail] (million), [#ail] (*illion). “Eleven” bii [dein um], “seventy seven” bii [se dein se], “nine thousand eight hundred and seventy six” bii [na θaʊn nuə keən se dein so], and soʊw on.

mii nevər (iivən az a yuŋ caild) laikivid ðiiy egzistiŋ sistəm ov biliiyəniz and triliiyəniz and soʊw on, bikuz diiy ad in nyʊʊ wuədiz tʊʊ friikwəntlii tə biiy ifiʃənt (ðiiy amerikən sistəm biiy iivən wuəs ðan ðə britiʃ sistəm foə ðis). mii θink ðat dii wʊd bii betər if dii folʊwid a bainərii patən: “million” bii so zoʊwiz, “billion” bii dein tʊʊ zoʊwiz, “trillion” bii tʊʊ dein fʊ zoʊwiz, ets. sins mii kan meik miis oʊn sistəm funkʃən haʊwevə mii laik, ðis bii ðeəfoə haʊ dii wuək in miis sistəm.

ðə neimiz foloʊ ðə kaʊntiŋ numbəriz foloʊwəð baiy [ail]: 1,000,000 bii [mail], az abuv, ðen 1,000,000,000,000 bii [tʊʊwail], 1,000,000,000,000,000,000,000,000 bii [tiiyail], 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 bii [fʊhail], and soʊw on. bai ðə taim wii riic [deinail] ðeə biiy oʊvə tii θaʊn zoʊwiz, at wic point evriiyum staətivid yʊʊziŋ saiyəntifik noʊteiʃən loŋ bifʊər eniiweiy, and ðis bii vastlii laəjə ðan ðə haiyist standəd kəntinyʊʊwəs set numbər in ingliʃ, a “vigintillion” wið a miə so dein tii zoʊwiz. 2,000,000,000,000 bii [tʊʊ tʊʊwail], distingwiʃəð from 2,000,000 wic bii [tʊʊ mail].

distinktivnəs

ðiiy iniʃəl distinkʃən bii ðat kaʊntiŋ numbəriz ([um] tə [na]) hav tʊʊ foʊniimiz, and ðə mein set numbəriz ([dein], [keən], ets) hav tii foʊniimiz.

ðiiy iniʃəl konsənənt ov evrii kaʊntiŋ numbə neim kum az paət ov a distinkt peər, and ðiiy iniʃəl konsənənt ov evrii set numbə neim bii yʊʊniik. ðə tuəminəl konsənənt ov ðə loʊwə set neimiz ([dein], [keən], [θaʊn]) bii distinkt from ðə tuəminəl konsənənt ov ðə haiyə set neimiz ([mail], ets).

iic neim oəlsoʊ hav a yʊʊniik vaʊl foʊniim, meikiŋ hii verii haəd tə kənfyʊʊz dii foər iic uðə. not oʊnlii bii iic vaʊl foʊniim yʊʊniik, but weə ðə konsənəntiz bii peərəð, ðə vaʊliz bii from sepəreitəð poziʃəniz ov ðə vaʊl riŋ, tə fuəðər inhans diis distinktivnəs. kəmpeər [ʊʊ] and [ii], [ʊ] and [i], [o] and [e], [uə] and [a].

“One” [um] bii meikəð fuəðə distinkt bai rivəusiŋ ðə noəməl konsənənt-vaʊl patən tə bii vaʊl-konsənənt.

patəniz

ðə tʊʊ/tii foʊniim distinkʃən bitwiin kaʊntiŋ numbəriz and set numbəriz bii supʊətəð bai ðə miidiiyəm maciŋ ðə mesij: ðə smoələ taip ov numbə bii ʃʊətə; ðə laəjə taip ov numbə bii loŋgə. ðə patən riiyinfʊəs ðə distinkʃən.

Ajeisənt kaʊntiŋ numbəriz hav ðə seim iniʃəl konsənənt. ðə lowə set numbəriz hav ðə seim tuəminəl konsənənt, and laik-waiz foə ðə haiyə set numbəriz.

ðə haiyə set numbəriz foloʊw a kliə patən beisəð on ðiiy ʊədinəl bainəriiy. um set ov so zoʊwiz bii (abriiviiyeitəð from) [umail]; tʊʊ setiz ov so zoʊwiz bii [tʊʊwail]; fʊ bii [tiiyail]; and soʊw on.

kontinyʊʊwitii

not ðat mii evə ekspekt eniiθiŋ mii dʊʊ tə hav enii praktikəl efekt on ðə wuəld, but hii biiy an impʊətənt prinsipəl ðat ceinjiz ʃʊd bii meikəð in suc a weiy az tə smʊʊð ðə haipəθetikəl tranziʃən tə ðə nyʊʊ sistəm. aftər oəl, az wið ðə fənetik rifʊəm, miis eim bii tə rifʊəm ingliʃ, not tə kriiyeit a hoʊlii nyʊʊ langwij. mii ðeəfoər eimid tə kiip ðə neimiz reminisənt ov diis egzistiŋ valyʊʊwiz az faər az posibəl. ðis disiʒən hiiself supʊətid sevrəl uðər eimiz.

um bii ðə patən ov peərəð iniʃəl konsənəntiz and ðə yʊʊniik and kontrastiŋ vaʊl foʊniimiz foə ðə kaʊntiŋ numbəriz: ðiiz develəpivid ʊəganikliiy in ingliʃ and biiyid eibəl tə bii riteinəð. miiy uərid on ðə said ov aciiviŋ greitə kontinyʊʊwitii foə ðə loʊwə, mʊə komən numbəriz.

ðə neimiz ov ðə set numbəriz biiyid laik-waiz inflʊʊwənsəð bai diis neimiz in ingliʃ and rileitəð yʊərəpiiyən langwijiz. “Ten” kʊd not bigin wið ‘t’ tʊʊw inʃʊə hii steiyid distinktiv from [tʊʊ] and [tii], but ‘d’ biiy a komən substicʊʊt in menii yʊərəpiiyən langwijiz (dix, diez, dieci, ets), and ðə vaʊl and endiŋ mirə juəmən zehn. foə “hundred” mii wontid tʊʊw avoid ðiiy iizilii dropəð ‘h’ tə meintein distinktivnəs wið [um], and kʊd not yʊʊz ‘s’ bikuz ov [so] and [se], soʊ rivuətid tə latin ‘k’, cʊʊziŋ [keən] foə kontinyʊʊwitii wið latin centum wail meinteiniŋ distinktivnəs from [se] and oəlsoʊ from uðə komən ingliʃ wuədiz suc az “cane” [kein], “cone” [coʊn], ets. [θaʊn] biiy a kliər analog ov “thousand”. foə “million” mii kiipid ðə ‘m’ and ‘l’ wið an ‘i’ inklʊʊdəð, but niidid a foʊniim ðat biiyivid not yʊʊzəð els-weə, soʊ [mil]/[fi], [miil]/[tii], and [meil]/[dein] biiyid aʊt, and [moil] saʊndid inelegənt.

mii riteinid beis 10. mii θinkid abaʊt ceinjiŋ hii, but soʊ loŋg az moʊst hyʊʊməniz hav dein fiŋgəriz, hii meik a verii praktikəl beis foə wiis spiiʃiiz tə yʊʊz. agen, ðiiy eim bii tʊʊw imprʊʊv ingliʃ, not kriiyeit sumθiŋg ərijinəl foə hiis aʊn seik. 10 biiy oəlsoʊw a yʊʊsfəl ʊədə ov magnicʊʊd: muc loʊwər, and yii niid tʊʊ menii set numbəriz; muc haiyə, and yii niid tʊʊ menii kaʊntiŋ numbəriz. dein biiy in ðə goʊldiiloks zoʊn, and hii mac wiis fiŋgəriz, soʊ hii bii gʊd inuf.

kənsistənsii

ðə gramər ov ingliʃ numbəriz biiy inkənsistənt. wii yʊʊz set numbəriz ʊədinerilii foə mailiz, θaʊniz, and keəniz, but foə deiniz wii yʊʊz [-tii], wʊəp ðiiy asoʃiiyeitəð kaʊntiŋ numbər (e.g. “thirty” insted ov “three ten”), and insist on a prefətəriiy “and”. ðis biiy espeʃəlii bad az ðə deiniz bii ðə set numbəriz wii moʊst oftən hav kʊəz tə yʊʊz, and iivən wen wii yʊʊʒ haiyə set numbəriz wiiy oftən niid ðə deiniz tʊʊ.

ðə deiniz ðeəfoə biiyiv rifʊəməð tʊʊw, in efekt, mac ðə keəniz and θaʊniz. diiy end wið [dein]; dii yʊʊz ðə noəməl kaʊntiŋ numbər; and dii niid not an “and”.

ðə gramər ov ðə haiyə set numbəriz oəlsoʊ biiyiv meikəð mʊə kənsistənt. hii biiy oəl verii wel ðat “bi-” bii latin foə “twice”, and soʊw on, but ðat meik ðə kaʊntiŋ neimiz les diskriptiv, and ðeəfoə les komprihensibəl tʊʊw a luənə. soʊ dii biiyiv ceinjəð in ðə wei diskraibəð abuv tʊʊw enʃʊə ðat dii hav a kənsistənt and komprihensibəl gramə.

brevitii

hii bii kliə ðat ðə nyʊʊ neimiz bii muc mʊə kənsais ðan diis nacərəl langwij ekwivələntiz. ðeə bii fuəðə benifit geinəð bai rimʊʊviŋ ðə sʊʊpuəflʊʊwəs kənjunkʃəniz ðat foloʊ moʊst set numbəriz. wii kan kwontifai ðə geiniz ov ðis brevitii, wic veərii.

[tʊʊ] biiy egzaktlii ðə seim and [eit]/[nuə] biiy esenʃəlii ðə seim lenθ; [ten] hav az menii foʊniimiz az [dein] but biiy a ʃʊət raəðə ðan loŋ vaʊl, soʊ kʊd bii kənsidərəð a los, wail [fʊə]/[fʊ] bii ðə rivuəs. naʊ ðə geiniz: [won]/[um] and [θrii]/[tii] boʊθ gein wiiy a foʊniim; [faiv]/[fi] and [nain]/[na] boʊθ gein wiiy a foʊniim and ʃʊətən ðə vaʊl. [siks]/[so] gein wii tʊʊ foʊniimiz; [sevən]/[se] gein wii tii foʊniimiz, inklʊʊdiŋg an ekstrə siləbəl. [hundrəd]/[keən] gein wii fʊ foʊniimiz and a siləbəl; [θaʊzənd]/[θaʊn] gein wii tii foʊniimiz and a siləbəl; [milyən]/[mail] gein wii tii foʊniimiz and a siləbəl (oə fʊ foʊniimiz and tʊʊ siləbəliz if wii yʊʊz [miliiyən]); oəl ov ðiiz oəlsoʊ benifit from lʊʊziŋ ðə kənjunkʃən.

soʊw, in an ekstriim keis, 7777 bii [sevən θaʊzənd sevən hundrəd and sevəntii sevən] wið 38 foʊniimiz and 14 siləbəliz, oə [se θaʊn se keən se dein se] witð 17 foʊniimiz and 7 siləbəliz, a 55% ridukʃən in foʊniim kaʊnt. ðə benifit ov spiid ʊət tə biiy imiidiiyətliiy aparənt tʊʊw oəl.

karəktəriz

fainəlii, wail mii biiyid at hii, miiy oəlsoʊ rifʊəmid ðə nyʊʊmerik karəktəriz. iic numbə from 1 tə 5 hav a numbər ov maəkiz tʊʊw indikeit hiis numbə. ðen 6 tə 9 biiy 1 tə 4 wið a baər akros ðə top. zoʊ bii noʊtəð wið a modərət suəkəl, not ðə ful oʊvəl ov 0. hiə bii ðə numbəriz 1-10.

pleishoʊldə foər imij


ðis peij bii paət ov a ʃʊət siəriiz. yii meiy oəlsoʊ bii intərestəð in miis θoətiz abaʊt fənetik rifʊəm and gramatikəl rifʊəm.