Egyptian Multiplication

Select the two numbers to multiply:

n

37

m

21

mode

Egyptian

Russian peasant

explain

n: 37	m: 21
1	21
2	42
4	84
8	168
16	336
32	672
37	777

First column:	1	+	4	+	32	=	37	✓
Final answer:	21	+	84	+	672	=	777

Egyptian multiplication: Under column headings

{n,m}

, put

{1,m}

as the first row of the table, then double each row to get the next row, continuing down as long as the numbers in the first column are less than or equal to

n

. Now strike out (here shown in gray) enough of the rows so that the remaining first-column numbers add up to

n

. (There is only one way to do this, as you can easily see: adding the left column from the bottom to the top, but throwing away any entries that would make the sum too big.) Now add the numbers remaining in the second column. The answer found will be

n×m

.

Russian peasant multiplication: Start with

{n,m}

and generate each row from the previous one by halving the first number (discarding any remainder) while doubling the second, until the first-column entry becomes 1. Then strike out any rows with even numbers in the first column, and add the remaining second-column numbers.

The two algorithms are closely related, and enabling the "explain" checkbox will show expansions of the second-column entries which may aid in understanding why the algorithms work. Hint: try writing

n

in binary!