◌ 0B: no body contribution. The dotted rectangle marks a zero body contribution; the dotted square marks a zero flag contribution. Square brackets group the parts of one numeral. These placeholders are teaching aids, not the numeral zero or characters to insert into the final answer. Bare flags show flag contributions; complete numbers keep their normal bodies.
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10. Vigesimal Fractions
10.1 Place value continues below the units
There is nothing new to learn. The places do not stop at the units — they continue to the right, each worth one twentieth of the place before it:
| Place | Worth |
|---|---|
| Twenties | 20 |
| Units | 1 |
| Twentieths | ⁄ 1/20 |
| Four-hundredths | ⁄ 1/400 |
| Eight-thousandths | ⁄ 1/8000 |
A vigesimal point (, two low stacked dots) separates the units from the twentieths, exactly as the twenties are separated from the units by position. A digit in the twentieths place can be any Ńdẹ́bẹ́ numeral 0–19, just like any other place.
Commentary for base-10 readers: this is the same idea as a decimal point, with twentieths where you are used to tenths. 0.1 in vigesimal means 1/20 (= 0.05 in base 10); 0.(10) in vigesimal means 10/20 = one half.
10.2 Division simply keeps going
When a division leaves a remainder and the dividend's digits are exhausted, append a 0 digit in the next place down and keep forming windows. This is Place Demotion: the remainder moves down one place, gaining twenty-fold room, and the ladder continues to apply. Every digit produced after the vigesimal point is a fractional digit.
Worked entirely in Ńdẹ́bẹ́: 17 ÷ 3
Ladder of 3: 1d = 0F3B × d = · 2d = 5F1B × d = · 3d = 5F4B × d = · 4d = 10F2B × d = · 5d = 15F0B × d = · 10d = 0F1B, 10F0B × d = · 15d = 0F2B, 5F0B × d =
Units window: 15F2B
- Flag stage: 5d = 15F0B → flags tie, body 2B > 0B → fits → 5F. Remove → 0F2B .
- Body stage: 1d = 0F3B > 0F2B → no fit → 0B ◌.
- Digit [ + ◌] → (5), remainder 0F2B . Digits exhausted → place the vigesimal point and continue.
Twentieths window: append 0 → 0F2B, 0F0B
- Flag stage: 15d = 0F2B, 5F0B → leftmost tie; units 0F < 5F → too big. 10d = 0F1B, 10F0B → leftmost 2B > 1B → fits → 10F.
- Remove: 0F2B, 0F0B − 0F1B, 10F0B − → units need Place Demotion (worked in §7) → remainder 10F0B .
- Body stage: 4d = 10F2B → flags tie, 0B < 2B → too big. 3d = 5F4B → 10F > 5F → fits → 3B. Remove: 10F0B − 5F4B − → Flag Demotion → 5F5B − 5F4B = 0F1B ( + ) − = .
- Digit 10F3B [ + ] → = 13, remainder 0F1B .
Four-hundredths window: append 0 → 0F1B, 0F0B
- Flag stage: 15d, 10d too big (leftmost 1B < 2B; then units 0F < 10F). 5d = 15F0B (one digit) → the window has two digits → fits → 5F.
- Remove: 0F1B, 0F0B − 15F0B − → Place Demotion → units 15F + 5F − 15F = 5F → remainder 5F0B .
- Body stage: 2d = 5F1B → flags tie, 0B < 1B → too big. 1d = 0F3B → 5F > 0F → fits → 1B. Remove: 5F0B − 0F3B − → Flag Demotion → 0F5B − 0F3B = 0F2B ( + ) − = .
- Digit 5F1B [ + ] → = 6, remainder 0F2B .
The remainder 0F2B has appeared before (after the units window) — the sequence now repeats.
Answer: 5 . (13) (6) (13) (6) … — the block (13)(6) recurs forever. (Base-10 check: 17 ÷ 3 = 5.666…, and 13/20 + 6/400 + 13/8000 + … = 0.666… ✓)
10.3 Rounding
For the nonnegative values here, use round-half-up. Add one to the last retained digit when rounding up; if that digit is 19, write 0 and carry to the left, repeating through any further 19s.
Round up if the next vigesimal digit is 10 or above; round down if it is 9 or below. Ten is half the base, so this is the exact midpoint — and since 10 has its own distinct flag, the threshold is visible in the glyph: a next digit carrying 10F or 15F rounds up; one carrying 0F or 5F rounds down.
- 17 ÷ 3 to one vigesimal place: next digit is 6 (flag 5F) → round down → 5.(13)
- To two places: next digit is 13 (flag 10F) → round up → 5.(13)(7)
10.4 Fractions a base-20 society finds clean
A fraction terminates in vigesimal exactly when its denominator, after reducing the fraction to lowest terms, divides a power of 20 — that is, when it is built only from 2s and 5s. The same fractions terminate in base 10, but several need fewer fractional digits in base 20:
| Fraction | Vigesimal | Note |
|---|---|---|
| ⁄ 1/2 | 0.(10) | one digit |
| ⁄ 1/4 | 0.5 | one digit — 1/4 = 0.25 in base 10 needs two |
| ⁄ 1/5 | 0.4 | one digit |
| ⁄ 1/8 | 0.(2)(10) | two digits — needs three in base 10 |
| ⁄ 1/10 | 0.2 | one digit |
| ⁄ 1/16 | 0.(1)(5) | two digits — needs four in base 10 |
| ⁄ 1/20 | 0.1 | one digit |
| ⁄ 3/4 | 0.(15) | one digit |
| ⁄ 1/3 | 0.(6)(13)(6)(13)…… | recurring, as in every base without a factor of 3 |