◌ 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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8. Multiplication
8.1 The core table — the only facts to memorise
Body × body. Ten facts, every product a single glyph:
| × | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
| 4 | 4 | 8 | 12 | 16 |
Everything else in multiplication is built from these ten facts plus two structural moves: ×5 is rotation and ×20 is a place shift.
8.2 ×20 — the place shift
Multiplying by 20 moves every digit one place to the left and writes a 0 in the units. 5F3B × 20 = 5F3B, 0F0B ([8,0] = 160).
8.3 ×5 — rotation
Multiplying a digit by 5 "rotates" it: bodies climb into the flag, and the flag climbs into the next place. Precisely, for a digit with flag level a and body b:
- The body b becomes b flag levels here
- The flag level a becomes a body units one place up, and also leaves a flag levels here (because 5 × 5 = 25 = 20 + 5)
- Total flag levels here: a + b. If this reaches 4 or more, Place Promote (4 levels = 20 = one more unit up)
Examples
- 0F3B × 5: a=0, b=3 → 3 flag levels here → 15F0B = 15 ✓ (the body simply rotated into the flag)
- 5F2B × 5: a=1, b=2 → 1 unit up; 1+2 = 3 levels here → 0F1B, 15F0B = [1,15] = 35 ✓
- 10F4B × 5: a=2, b=4 → 2 units up; 2+4 = 6 levels → promote 4 → one more unit up, 2 levels stay → 0F3B, 10F0B = [3,10] = 70 ✓
If rotation feels unfamiliar, ×5 can always be done as ×4 + ×1 (double, double, add). Rotation is the native shortcut, not a requirement.
8.4 Multiplying by a single digit — the ladder used forwards
Any single-digit multiplier q is a flag plus a body: q = 5a + b. Therefore:
| q × d = a × (5d) + b × d | q × d = a × ( × d) + b × d |
| = (a copies of the rotation of d) + (b copies of d) | a rotated copies + b ordinary copies |
So build the ladder of d — 1d, 2d, 3d, 4d by addition; 5d by rotation; 10d = 5d + 5d; 15d = 10d + 5d — and then q × d is simply one flag rung plus one body rung:
| Multiplier q | Flag rung | + | Body rung |
|---|---|---|---|
| 7 = 5F2B | 5d | + | 2d |
| 13 = 10F3B | 10d | + | 3d |
| 18 = 15F3B | 15d | + | 3d |
Worked: 18 × 14
Ladder of 14: 1d = 10F4B × d = , 2d = 0F1B, 5F3B × d = , 3d = 0F2B, 0F2B × d = , 4d = 0F2B, 15F1B × d = , 5d = 0F3B, 10F0B × d = , 10d = 5F2B, 0F0B × d = , 15d = 10F0B, 10F0B × d =
18 = 15F3B , so 18 × 14 = 15d + 3d:
| 10F0B, 10F0B (15d) | |
| + 0F2B, 0F2B (3d) | + |
| ───────────────── | |
| 10F2B, 10F2B = [12, 12] = 252 ✓ | = |
This is exactly the ladder that division uses in reverse (§9.4). Multiplication picks rungs and adds them; division finds rungs and subtracts them. One tool, two directions.
8.5 Multi-digit multiplication
For a multi-digit multiplier, multiply by each of its digits in turn (§8.4), shift each partial result by that digit's place, and add.
Worked: [1, 8] × [1, 3] (28 × 23 = 644)
By the units digit of the multiplier (3):
| 5F3B × 3: flags 15F, bodies 9B → promote → 5F4B; flags 15F + 5F = 20F → Place Promote | × = + = + + = |
| units = 0F4B, carry 1 up | Units: carry: |
| 0F1B × 3: 3B, + carry 1 = 4B | × + = |
| Partial 1 = 0F4B, 0F4B ([4,4] = 84) | |
By the twenties digit of the multiplier (1):
| [1,8] × 1 = [1,8], shifted one place up | × = → shift one place |
| Partial 2 = 0F1B, 5F3B, 0F0B ([1,8,0] = 560) | |
Add the partials:
| 0F0B, 0F4B, 0F4B | |
| + 0F1B, 5F3B, 0F0B | + |
| ────────────────── | |
| 0F1B, 10F2B, 0F4B (twenties: 4B + 5F3B = 5F7B → promote → 10F2B) | = Twenties: + = [ + ] → |
| = [1, 12, 4] = 644 ✓ | |