Example · thousands scale
ᔭ31↊
qua-ston trey-kan don-dy
Most numeral systems treat negative numbers as an afterthought: a magnitude plus a minus sign. A balanced radix bakes negativity into the digits themselves. Each digit may be negative, zero, or positive—so the sign of a number is distributed across its digits rather than marked separately at the front.
In balanced nonary, every digit lies in the symmetric range −4 … +4, and each position carries a power-of-9 weight. The result is compact, symmetric notation with two written scripts and a disciplined spoken layer for reading numbers aloud.
The nine digits
These are not “digits 0–8.” Each glyph is a signed value centered on zero.
| Value | Name | Nonal | Nonarian |
|---|---|---|---|
| -4 | qua | ᔭ | ջ |
| -3 | ter | Ɛ | ը |
| -2 | dy | ↊ | չ |
| -1 | oon | Ɩ | ի |
| 0 | nil | 0 | օ |
| +1 | ane | 1 | կ |
| +2 | twa | 2 | ե |
| +3 | trey | 3 | մ |
| +4 | fyor | 4 | ձ |
Place value
As in ordinary base 9, the rightmost digit is 9⁰, the next 9¹, then 9², and so on—except each digit may be negative. A string dk…d0 represents Σ di · 9i with each di ∈ {−4,…,+4}.
- 1ᔭ = (+1·9) + (−4·1) = 5
- Ɩ4 = (−1·9) + (+4·1) = −5
- 31↊ = (+3·81) + (+1·9) + (−2·1) = 250 · nonarian մկչ
Spoken layer
Place-value words pair positive- and negative-leaning forms. Within a number, don/den marks the 9¹ place, kan/kun the 9² place, and sten/ston the 9³ (“thousands”) boundary. At comma-scale groupings, mol/mel and bar/bur label whole three-digit groups— analogous to “million” and “billion” in decimal.
| Role | Positive | Negative |
|---|---|---|
| 9¹ place | don | den |
| 9² place | kan | kun |
| 9³ place | sten | ston |
| 9⁶ group | mol | mel |
| 9⁹ group | bar | bur |
Small numbers
0 … +13
- 0 nil · 1 ane · 2 twa · 3 trey · 4 fyor
- 5 don-qua · 6 don-ter · 7 don-dy · 8 don-oon
- 9 don · 10 don-ane · 11 don-twa · 12 don-trey · 13 don-fyor
0 … −13
- 0 nil · −1 oon · −2 dy · −3 ter · −4 qua
- −5 den-fyor · −6 den-trey · −7 den-twa · −8 den-ane
- −9 den · −10 den-oon · −11 den-dy · −12 den-ter · −13 den-qua
Rule A: place words are mandatory wherever they apply—one does not collapse “four-don-three” into “four-three.” Rule B: when a place pairs with ±1, the digit name may be omitted (anedon-twa ≡ don-twa); no other omissions.
Large numbers
Digits group in threes from the right—9³ = 729 plays the role of “thousand.” Group labels (mol/mel, bar/bur) are spoken after the group’s internal reading, with spaces at comma boundaries.
Example · millions scale
4,02ᔭ,31↊
fyor-mol twadon-qua-ston treykan-don-dy