Each of the numbers from 1 to 20 has a simple word that represents its basic identity. These words are simple consonant-vowel syllables; these should be generally accessible to speakers of numerous widespread languages, and contrast in place and manner of articulation as much as possible.
A table of the numbers and their names is below. The numbers are named as follows: "Ye" is 1; "bi" is 2; "sa" is 3; "te" is 4; "fu" is 5; "go" is 6; "mi" is 7; "pa" is 8; "ze" is 9; "du" is 10; "vo" is 11; "ki" is 12; "hu" is 13; "be" is 14; "su" is 15; "to" is 16; "fi" is 17; "ga" is 18; "me" is 19; and "pu" is 20.
The consonants are pronounced roughly as in English, and the vowels are pronounced as in Spanish or Japanese. A different orthography might be appropriate for English speakers: the word for 2 rhymes with "bumblebee", and the words for both 14 and 19 rhyme with "meh", but the words for 2 and 14 have different vowels. It'd probably be almost impossible to get English speakers to pronounce everything properly, though ;)
Each of these numbers can be used as the basis for a number system. To indicate that a number is the base, append "-n" to that number's name. For example, the word "du" represents the idea of 10. Thus, "dun" represents 10 within the context of the decimal (base-10) system. Effectively, appending "-n" to "du" means "10 raised to the 1st power".
This may not seem terribly useful. But the system can regularly extrapolate a word for any position in the number system as an exponent of the base. The decimal word for 100, for example, means literally "10 raised to the 2nd power". To do this, append "-l" to the word for the base, and follow it with the word for 2 ("bi") appended by "-n". So the word for a hundred is "dulbin". The word for 1,000 is "dulsan".
Overall, the system is strict place-value notation, similar to East Asian numbers. Numbers larger than the base are said as they are written in positional notation, multiplying and adding as necessary. Within the context of decimal math, "dun ye" (literally, "ten and one") is 11. "Bi dulbin dun bi" (literally, "two hundreds, ten, and two") is 212. "Sa dulsan ye dulbin bi dun sa" (literally, "three thousands, one hundred, two tens, and three") is 3,123. Since "dun" is the word for 10 in a base-10 context, the decimal system is internally known as the "dunal system".
This system's main feature is that it can predictably extend any base up to 20. The octal, or base-8, system, is the "panal system", since "pan" is the word for 8 in a base-8 context. "Te pan fu" (literally, "four 8s and five") is decimal 37. "Mi palbin fu pan sa" (literally, "seven 64s, five 8s, and three") is decimal 491. In hexadecimal, "te tolbin" (literally, "four 256s") is decimal 1,024.
This is more limited than scientific notation, of course. But without resorting to scientific notation, in vigesimal, the system can count to more than decimal 2 octillion. In decimal, the highest it can count is one short of one sextillion.
For convenience and clarity, binary can be handled in a somewhat different manner, the short binal system. In short binal, 1 is "ye", 2 is "bin", and 4 is "tel". Subsequent numbers are created by appending "-b" to the name of the power of 2. For example, "sab" is 8 (rather than "bilsan"), and "teb" is 16 (rather than "bilten"). Thus, "teb sab tel ye" is decimal 29.
One clear drawback is the high level of rhyming. These words are more similar than the same words in English; the example of decimal 212 above adequately demonstrates how repeating the same sounds could be confusing. Of course, explicitly repeating the base might make it easier to break the number up. And the difference in scale between "dulsan", "dulgon", and "dulzen" might be more readily apparent than between "thousand", "million", and "billion", given that people often fail to easily conceptualize the difference in degree. Certainly, this is a limited system deficient for the purposes that John Nystrom envisioned (replacing the number system of a natural language); but it may be useful within the scope of its intent.
|7||Mi||mi||mi||mi||tel bin ye|
|10||Du||ye dun||du||du||sab bin|
|11||Vo||ye dun ye||vo||vo||sab bin ye|
|12||Ki||ye dun bi||ki||ki||sab tel|
|13||Hu||ye dun sa||hu||hu||sab tel ye|
|14||Be||ye dun te||be||be||sab tel bin|
|15||Su||ye dun fu||su||su||sab tel bin ye|
|16||To||ye dun go||ye ton||to||teb|
|17||Fi||ye dun mi||ye ton ye||fi||teb ye|
|18||Ga||ye dun pa||ye ton bi||ga||teb bin|
|19||Me||ye dun ze||ye ton sa||me||teb bin ye|
|20||Pu||bi dun||ye ton te||ye pun||teb tel|
|21||||bi dun ye||ye ton fu||ye pun ye||teb tel ye|
|22||||bi dun bi||ye ton go||ye pun bi||teb tel bin|
|23||||bi dun sa||ye ton mi||ye pun sa||teb tel bin ye|
|24||||bi dun te||ye ton pa||ye pun te||teb sab|
|25||||bi dun fu||ye ton ze||ye pun fu||teb sab ye|
|100||||ye dulbin||go ton te||fu pun||gob fub tel|
|256||||bi dulbin fu dun go||ye tolbin||ki pun to||pab|
|400||||te dulbin||ye tolbin ze ton||ye pulbin||pab mib teb|