Multiplication
Part of Mathematics
Efficient methods for multiplying numbers by hand, including ancient algorithms that work without memorized tables.
Why This Matters
Multiplication is the workhorse of practical arithmetic. Every trade calculation, every material estimate, every recipe scaling depends on it. How many bricks do you need for a wall 12 meters long, 2 meters high, with bricks 25 cm long and 10 cm tall? That requires two multiplications. How much grain do you need to feed 47 people for 90 days at 600 grams per day per person? That requires three. A community that must rely on tedious repeated addition for these problems will make costly errors and waste time.
In a post-collapse world, you cannot assume everyone has memorized multiplication tables up to 12×12. The Russian peasant method (also called Ethiopian multiplication or binary multiplication) requires knowing only how to double and halve numbers — skills almost anyone can acquire quickly. Combined with basic table memorization for the most common small numbers, these methods cover every practical calculation you will face.
Understanding multiplication also unlocks area calculation, unit conversion, and eventually algebra — all of which are needed for construction, agriculture planning, and resource management. This article covers both the method of memorization (for small numbers) and the algorithmic approaches (for any numbers), so that every community member can be functional regardless of prior education.
The Multiplication Table: Priority Memorization
Full memorization of 1–12 times tables is the goal, but if time is limited, prioritize this order:
Tier 1 (memorize first): 2×, 5×, 10× — these follow simple patterns Tier 2: 3×, 4×, 9× — teachable with tricks Tier 3: 6×, 7×, 8× — require rote work but most common in practice Tier 4: 11×, 12× — useful for construction (inches per foot, dozens)
Patterns That Reduce Memorization Load
Twos: Add the number to itself. 7×2 = 7+7 = 14.
Fives: Count by fives (5, 10, 15…) or use the clock. Every minute mark on a clock face is a multiple of 5. 7×5 = 35 (between the 6 and 7 on a clock face, at the 35-minute position).
Tens: Append a zero. 7×10 = 70.
Nines trick: The digits of any multiple of 9 (up to 9×10) sum to 9. For 9×N where N is 1–10: the tens digit is (N−1), the units digit is (9−(N−1)). Example: 9×7 — tens digit = 6, units digit = 3, answer = 63. Check: 6+3=9. ✓
Elevens: For 11×N where N is 1–9, write the digit twice. 11×7 = 77. For 11×11 = 121, 11×12 = 132 — these are worth memorizing directly.
Squares: 5×5=25, 6×6=36, 7×7=49, 8×8=64, 9×9=81. Memorizing the squares reduces the unique facts needed to learn all other products.
The Russian Peasant Method
This algorithm requires no memorized tables beyond knowing how to double (add a number to itself) and halve (divide by 2, discarding any remainder). It works for any two numbers.
The method:
- Write the two numbers side by side
- Halve the left number (discard any remainder), double the right number
- Repeat until the left number reaches 1
- Cross out any row where the left number is even
- Sum all remaining right-side numbers
Example: 23 × 17
| Left (halve) | Right (double) | Keep? |
|---|---|---|
| 23 | 17 | Yes (odd) |
| 11 | 34 | Yes (odd) |
| 5 | 68 | Yes (odd) |
| 2 | 136 | No (even) |
| 1 | 272 | Yes (odd) |
Sum of kept right values: 17 + 34 + 68 + 272 = 391
Check: 23 × 17 = 391. ✓
Why it works: This is essentially binary multiplication. The left column halving decomposes the left number into powers of 2. Odd numbers contribute their binary digit; even numbers do not. The right column doubling scales the right number accordingly. The sum reconstitutes the product.
Practice example: 45 × 13
| Left | Right | Keep? |
|---|---|---|
| 45 | 13 | Yes |
| 22 | 26 | No |
| 11 | 52 | Yes |
| 5 | 104 | Yes |
| 2 | 208 | No |
| 1 | 416 | Yes |
Sum: 13 + 52 + 104 + 416 = 585. Check: 45 × 13 = 585. ✓
Long Multiplication (The Standard Algorithm)
For those comfortable with small multiplication facts, the standard long multiplication algorithm is faster for large numbers.
Example: 348 × 27
3 4 8
× 2 7
-------
2 4 3 6 (348 × 7)
+ 6 9 6 0 (348 × 20, shift one place left)
-------
9 3 9 6
Step by step:
- Multiply 348 by the units digit (7): 8×7=56 (write 6, carry 5); 4×7=28+5=33 (write 3, carry 3); 3×7=21+3=24. First partial product: 2436.
- Multiply 348 by the tens digit (2): same process, but write the result shifted one place to the left (or append a zero). 8×2=16 (write 6, carry 1); 4×2=8+1=9; 3×2=6. Second partial product: 6960.
- Add partial products: 2436 + 6960 = 9396.
Key habits:
- Write partial products neatly, aligned by column
- Track carries carefully — this is where most errors happen
- After finishing, estimate with rounding to check: 350 × 27 ≈ 9,450. Answer 9,396 is plausible.
Estimation Before Calculation
Before any multiplication, estimate the answer. This catches gross errors and confirms you have not made a factor-of-ten mistake.
Round to nearest convenient number:
- 47 × 93 ≈ 50 × 90 = 4,500 (actual: 4,371)
- 128 × 15 ≈ 130 × 15 = 1,950 (actual: 1,920)
If your calculated answer is far outside the estimated range, recalculate. A good estimator is often more valuable than a slow exact calculator.
Multiplying by Special Numbers
By 25: Multiply by 100, then divide by 4. 36 × 25 = 3600 ÷ 4 = 900.
By 50: Multiply by 100, then divide by 2. 44 × 50 = 4400 ÷ 2 = 2,200.
By 9: Multiply by 10, then subtract the original. 47 × 9 = 470 − 47 = 423.
By 11: For two-digit numbers N×11: write the first digit, add the two digits of N together for the middle, write the last digit. If the middle sum exceeds 9, carry. Example: 54×11 — first digit 5, middle = 5+4=9, last digit 4 — answer: 594. Example: 67×11 — first 6, middle = 6+7=13 (write 3, carry 1), last 7+1=8 — answer: 737.
Practical Worked Examples
Building materials: A wall requires 18 courses of brick, each course using 24 bricks. Total bricks = 18 × 24.
- Estimate: 20 × 24 = 480
- Russian peasant: 18 is even — but let’s do 24 × 18
- 24 → 48 → 96 → 192 → 384 → 432
- Actually: 18 × 24 = (18 × 20) + (18 × 4) = 360 + 72 = 432
- Answer: 432 bricks
Food calculation: 34 people need 650 grams of grain per day for 60 days.
- Step 1: 34 × 650 = 34 × 600 + 34 × 50 = 20,400 + 1,700 = 22,100 grams per day
- Step 2: 22,100 × 60 = 22,100 × 6 × 10 = 132,600 × 10 = 1,326,000 grams = 1,326 kg
Area of a field: 47 meters × 31 meters
- Russian peasant:
- 47 | 31 (odd) ✓
- 23 | 62 (odd) ✓
- 11 | 124 (odd) ✓
- 5 | 248 (odd) ✓
- 2 | 496 (even) ✗
- 1 | 992 (odd) ✓
- Sum: 31 + 62 + 124 + 248 + 992 = 1,457 square meters
Teaching Multiplication in a Community
When teaching children or adults who have never memorized tables:
- Start with the Russian peasant method — it works immediately and builds confidence
- Add table facts gradually — begin with 2s, 5s, and 10s
- Use physical objects — arrange 4 rows of 6 stones; count total; write 4×6=24
- Practice estimation daily — “roughly how many?” should become instinctive
- Post a multiplication chart — visible reference reduces errors and aids memorization through exposure
A community where most adults can multiply two-digit numbers reliably and a few specialists can handle three-digit multiplication covers virtually all practical needs for construction, agriculture, and trade.