Key takeaways
- This is not the doubling-and-halving shortcut that keeps the product constant at every step; here the running product changes, and the discarded remainders are the reason the method works.
- Halve one number repeatedly, discarding remainders, while doubling the other, then add only the doubled values whose halved partner is odd.
- Reading the halving column for oddness writes the first factor in binary, and each row of the doubling column is the second factor times a power of two.
- The algebra is two cases: if n is even, n times m equals half of n times double m, and if n is odd, it equals m plus that same expression applied to n minus one.
- The method needs no multiplication table above two, which is why it survives as a demonstration of binary rather than as a fast route to an answer.
This is not the doubling-and-halving shortcut where 14 × 35 becomes 7 × 70. That one preserves the product at every step and only works when the halving divides exactly. Russian peasant multiplication deliberately breaks the product: it halves with remainders thrown away, which changes the running value, and then repairs the damage by adding back particular rows. The discarded remainders are not a rounding error tolerated by the method — they are the mechanism.
The procedure
Write the two numbers at the head of two columns. Halve the left one repeatedly, discarding any remainder, until it reaches 1. Double the right one the same number of times. Then keep only the rows where the left number is odd, and add the right-hand entries of those rows.
The table below runs the procedure for 37 × 46, halving the left column and doubling the right until the left reaches one, with each row marked according to whether its left-hand value is odd.
| Halve | Double | Keep? |
|---|---|---|
| 37 | 46 | odd — keep |
| 18 | 92 | even |
| 9 | 184 | odd — keep |
| 4 | 368 | even |
| 2 | 736 | even |
| 1 | 1,472 | odd — keep |
Adding the kept rows: 46 + 184 + 1,472 = 1,702.
Check it independently: 37 × 46 = (37 × 40) + (37 × 6) = 1,480 + 222 = 1,702.
A second, smaller case. The same two columns for 23 × 19 produce a different pattern of kept rows, because 23 stays odd for longer on the way down.
| Halve | Double | Keep? |
|---|---|---|
| 23 | 19 | odd — keep |
| 11 | 38 | odd — keep |
| 5 | 76 | odd — keep |
| 2 | 152 | even |
| 1 | 304 | odd — keep |
Total: 19 + 38 + 76 + 304 = 437, which matches 23 × 20 − 23 = 460 − 23 = 437.
Notice what the procedure never asks for. There is no times table beyond doubling, no carrying into a partial product, and no place-value alignment. Everything is halving, doubling and addition.
The reason: the left column is binary
Look at the left column of the 37 × 46 table and mark each row odd or even from the bottom upwards: 1 is odd, 2 is even, 4 is even, 9 is odd, 18 is even, 37 is odd. Write those as 1 and 0 from the bottom up and you get 100101, which is 37 in binary — 32 + 4 + 1.
Now look at what the rows are worth. The top row of the doubling column is 46 × 1, the next is 46 × 2, then 46 × 4, 46 × 8, 46 × 16, 46 × 32. Row k holds 46 × 2ᵏ.
The three rows kept were the ones at 46 × 1, 46 × 4 and 46 × 32 — the same powers of two that add to 37. So the total is 46 × (1 + 4 + 32) = 46 × 37, which is what was asked for.
The same pattern in the second example: 23 is 10111 in binary, or 16 + 4 + 2 + 1, and the kept rows were 19 × 1, 19 × 2, 19 × 4 and 19 × 16. Every binary number is a sum of distinct powers of two, exactly as every decimal number is a sum of multiples of powers of ten, which is the fact NRICH's work on binary numbers builds on. Repeated halving with remainders discarded is one of the standard ways of extracting those powers.
The algebra, in two lines
The method also drops out of a two-case recursion, which is worth seeing because it explains the remainders directly.
If n is even: n × m = (n ÷ 2) × (2m). Nothing is lost, and the row contributes nothing.
If n is odd: n × m = m + ((n − 1) ÷ 2) × (2m). Halving an odd number loses half a unit of n, which is worth one whole m in the product — so that m is banked immediately, which is precisely the rule "keep the row when the left number is odd".
Run it on 23 × 19:
- 23 × 19 = 19 + (11 × 38)
- 11 × 38 = 38 + (5 × 76)
- 5 × 76 = 76 + (2 × 152)
- 2 × 152 = 1 × 304
- 1 × 304 = 304
Collecting the banked terms: 19 + 38 + 76 + 304 = 437.
The recursion and the binary explanation are the same argument in different clothes. The recursion shows why each odd row owes an m; the binary view shows which rows those are.
The older version: doubling in the Egyptian papyri
Multiplication by repeated doubling is recorded in ancient Egyptian mathematical papyri. MacTutor's account at the University of St Andrews sets out the procedure as it appears in the Rhind papyrus: to multiply two numbers the scribe doubles one of them repeatedly, alongside the running powers of two, then selects the doublings whose powers add to the other number and totals them. Their worked case is 41 × 59, where 32 + 8 + 1 accounts for 41 and the corresponding doubled values total 2,419.
MacTutor's comment on that procedure is the one that matters here: the multiplication is achieved with additions only, and it is a very early use of binary arithmetic. The broader overview of Egyptian mathematics makes the reason explicit — the number system in use made general multiplication and division awkward, so methods were devised that needed only addition.
The label "Russian peasant" attached to the halving-and-doubling version is a popular name of much later date, and it is worth treating as a name rather than a claim about where the method was invented. Doubling-based multiplication appears in more than one tradition, and the papyrus evidence long predates the phrase.
When to actually use it
Rarely, for speed. For 37 × 46 the six-row table is slower than splitting into 37 × 40 plus 37 × 6, and much slower than a shortcut if the numbers offer one — near a power of ten, ending in five, or with a shared factor. The methods that genuinely save time are collected in multiplication shortcuts and why they work.
Where it earns its place is narrower and real. It is the clearest demonstration of binary representation available without a computer, it is a working multiplication algorithm for anyone whose times-table recall is unreliable above two, and it is a good exercise in why an algorithm is correct rather than merely effective. The same instinct — asking what a written procedure is actually recording — applies to the grid methods discussed in lattice and line multiplication, which turn out to be place-value bookkeeping in disguise.
One practical caution: the discarded remainders make this method unforgiving of a single misread row. If you mark an odd row as even, the error is a whole power of two times the multiplier, which is large. Check the binary reading against the original number before totalling — in the first example, 32 + 4 + 1 must come back to 37.
Where the app fits
IQ Test & Brain Training Games, our own app at Reign Creative, has no mode that builds this table for you; it is a pencil-and-paper algorithm, and a timed drill is the wrong container for it. What the app does carry is the method in its technique library, written out with its steps, a worked example and its named common mistake, next to the drills that make its two ingredients automatic: Multiplication Rush and Division Dash cover the doubling and halving, and the untimed Accuracy Trainer runs ten questions at a time with an explanation available when one goes wrong. The app itself is described on the IQ Test & Brain Training Games app page.
If you want the method for its own sake, work three or four products on paper, then reconstruct the binary expansion from each table and confirm it matches the original number. That check is the method's real content. For the arithmetic that is actually worth drilling daily, the mental math practice guide puts the methods in order, and the rest of this material sits in the mental math topic hub and under education and brain training articles.
Good to know
Frequently asked questions
How does Russian peasant multiplication work?
Why can you throw away the remainders?
Is this the same as doubling and halving?
What has it got to do with binary?
Sources
- Mathematics in Egyptian Papyri (opens in a new tab)O'Connor, J. J. & Robertson, E. F., MacTutor History of Mathematics, University of St Andrews — accessed
- An overview of Egyptian mathematics (opens in a new tab)O'Connor, J. J. & Robertson, E. F., MacTutor History of Mathematics, University of St Andrews — accessed
- Binary Squares (opens in a new tab)NRICH, Millennium Mathematics Project, University of Cambridge — accessed
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