# Peasant multiplication

Peasant multiplication is an old algorithm for multiplication. It requires no use of a multiplication table; however, it requires that the user be able to divide by 2. The user must also know how to add.

• Write the two numbers (A and B) you wish to multiply, each at the head of a column.
• Starting with A, divide by 2, discarding any fractions, until there is nothing left to divide. Write the series of results under A.
• Starting with B, keep doubling until you have doubled it as many times as you divided the first number. Write the series of results under B.
• Add up all the numbers in the B-column that are next to an odd number in the A-column. This gives you the result.

Example: 27 times 82

 A-column B-column Add this 27 82 82 13 164 164 6 328 3 656 656 1 1312 1312 Result: 2214

The method works because multiplication is distributive, so:

[itex] \begin{matrix} 82 \times 27 & = & 82 \times (1\times 2^0 + 1\times 2^1 + 0\times 2^2 + 1\times 2^3 + 1\times 2^4)\\

          \ & = & 82 \times (1 + 2 + 8 + 16)\\
\ & = & (82 + 164 + 656 + 1312)\\
\ & = & 2214


\end{matrix} [itex]

This method was known to ancient Egyptians as mediation and duplation, where mediation means halving one number and duplation means doubling the other number. It is still used by peasants in some areas, such as Russia.

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