Russian peasant multiplication how does it work

If you are reading this, your browser is not set to run Java applets. The algorithm instructs us to create a column beneath each of the multiplicands. We start by dividing the first number by 2 and dropping the remainder if any and recording the result in the first column.

Then we divide by 2 the recorded number and write the result below. The process of division by two of the successive results continues until 1 is reached. In the second column we write the numbers obtained by successive multiplication by 2 that starts with the second multiplicand. Dividing 85 by 2 and dropping the remainder we get successively: 85, 42, 21, 10, 5, 2, 1. Now, multiply by 2 the second number, 18, as many times: 18, 36, 72, , , , Let's write these in two columns:.

I listed the numbers in two colors for an important reason. We have to look at the first column and mark there the odd numbers. These appear in red. The even numbers are orange - diluted red - to signify their lack of importance for the next step. On the next step, mark the numbers in the second column that are in the same rows with the odd numbers in the first. These are also in red. For the convenience sake, I added a third column where the red numbers from the second one have been separated:.

All that remains at this stage is to add up all the numbers in the third column. As you can see, the result is the same, but is obtained in fewer steps. This is true in general: if tasked with applying the algorithm to finding the product of two numbers a and b, make the smaller number first, the larger one second.

Now for a larger number, you might not know quickly what the right exponent is. Say take , which is a power of 2, but you've forgotten which power. But just write down and write down x. Now divide by 2, getting , and at the same time double x. Keep dividing the left-hand column the one that starts with by 2 and doubling the numbers in the right-hand column which started with x , until you've reduced the left-hand column to 1.

The number at the bottom of the right-hand column is the product you're seeking. It's easy to see why this works, because as you move down both columns simultaneously, the product of the numbers in the two columns is always the same. Looking at the top entries, this product is clearly times x. Looking at the bottom entry gives you the product. But suppose you want to multiply by a number which is not a power of 2.

Say I want to multiply x by Then I go through the same process, except that when you divide the numbers in the left-hand column by 2, just ignore the remainder in case the number is odd. Now to obtain the final answer I need to add x to the number at the bottom of the right-hand column.

In this case, as we move down the two columns, the product at the second step is no longer the same as the product at the first, because 17 became 8, which is not actually half of In general, if you're multiplying x by a and go through this process, every time the number in the left-hand column is odd, the product of the two columns one line down will be smaller than the product before.

To compensate, you need to add in the number which is in the right-hand column at this stage when you get to the end. Create an account. Edit this Article. We use cookies to make wikiHow great. By using our site, you agree to our cookie policy.

Cookie Settings. Learn why people trust wikiHow. Part 1. Think of the numbers you want to multiply. Choose two numbers that you want to multiply, whether to solve a particular problem or just to try out the Russian peasant method of multiplication.

For example, try multiplying x Make two columns. Using a piece of paper and pen, divide the piece of paper into two columns by drawing a line down the middle of the paper. Write one of the numbers you want to multiply at the top the each column. Halve the number in the left column repeatedly. Divide the number at the top of the left column by 2 continually until you get to 1.

Ignore any remainder each time you halve the number. Write each halved number down in the left column, in order. Double the number in the right column until the columns are the same length.

Multiply the number in the second column by 2 until there are the same amount of numbers here as there are in the first column. In this example: Each column should have 8 numbers in it.

This is because it took seven steps of dividing the original number in the right column to reach 1. Cross out rows with an even number in the left column. Using your pen, strike through those horizontal rows which begin with an even number in the left column. In this example: There are 8 rows. You will strike through 5 of them. Strike through the rows beginning with , 36, 18, 4, and 2 in the left column, since these are even numbers. Working left to right, this means striking through the first row , 37 , the third row 36, , the fourth row 18, , the sixth row 4, , and the seventh row 2, Please note that you should strike through even numbers, even if they begin with an odd numeral.

For example, you should strike through the row beginning with since it is an even number, even though begins with an odd numeral, 1. Likewise, you should strike though 36, since it is an even number, even though 36 begins with an odd numeral, 3.

If you prefer, you can just strike through the numbers in the right side that fall into the rows that begin with an even number on the left side as in the picture above. In this example, this means striking through the numbers on the right hand side of the first, third, fourth, sixth, and seventh rows: 37, , , , and Find the sum of the remaining numbers in the right column.

Add the numbers in the right column that you did not strike through. The sum of these numbers is equal to the product you would get from multiplying the original numbers using the standard method. In this example: The remaining numbers in the right column are 74, ,