Math Night Module: Methods of Multiplication

Vertically and Crosswise

URDHVATIRYAGBHYAM
This sutra, or verse, is the short and simple algorithm for multiplying two numbers, as written in the Vedas. The Vedas are ancient Hindu texts passed down at first in an oral tradition, composed betweeen the 16th centuiry BC and the 5th century BC. They contained knowledge in many different areas, including mathematics, written in verse. We owe the numeral system we use to this tradition-- the numerals we use and the concept of zero originated in India The Vedas contain 16 sutras, or verses, which are algorithms written in a line or two of Sanskrit verse.
Suppose you want to multiply 7x78.

Vertically: 8x7=56
Crosswise: 70x7=490
Add the 9 in the tens place to the 5 in the tens place to get 546.

Suppose you want to multiply 34x67.

First vertically: 3x6=18 (actually 30x60=1800)
Then crosswise: 6x4+3x7=24 + 21 = 45 (actually 60x4+30x7=450)
Finally vertically: 4x7=28.

The answer is 2,278. With practice you can do this in your head, even if you are working with more than 2 digits.

MULTIPLE METHODS OF MULTIPLICATION
In the US we are taught one standard algorithm for multiplication. It looks like this:

There are actually many more ways to multiply. Here are a few from different times in history and different parts of the world: ancient India, ancient Egypt, and renaissance Scotland. Try using these methods on a couple of mutiplication problems--you may find a method you like better than our usual one!
Multiply like an Egyptian
This is 53x72, ancient Egypt-style. Here's the same problem in our usual notation:

Why does this work?
You can write 53 = 1 + 4 + 16 + 32.

Now multiply:
53x72 = (1 + 4 + 16 + 32)x72

This is the same as
1x72 + 4x72 + 16x72 + 32x72
How to multiply like an Egyptian
1. Make two columns of numbers. In the left hand column, start with 1 and double it over and over again. In the right hand column, write the larger of the two numbers you are multiplying next to the 1 in the left hand column. Double it over and over again.
2. Circle the numbers in the left-hand column which add up to the smaller of the numbers you're mulitplying.
3. Add up the numbers in the right hand column which are next to the numbers you have circled. This sum is the answer to your multiplication problem.

John Napier was a Scottish baron, inventor, and mathematician who lived during the 16th century (1550-1617). Logarithms ("for the more easie working of questions in arithmetike and geometrie") are his most famous contribution to math. He also invented a calculating device known as "Napier's bones", based on the Gelosia method of 12th century India.

Each "bone" is a list of the first nine multiples of a number between 1 and 9. There is an index "bone" for reference.

To multiply 4x759 using the bones, line up the 7, 5, and 9 bones. Put the index bone next to them. Look at the 4 row only.

Add the numbers in the diamonds. You get 2 in the 1000's place, 8 + 2 = 10 in the 100's place, and 6 in the 1's place.

Answer: 2000 + 1000 + 30 + 6 = 3036.

What if you want to multiply a number by a two digit number, such as 64x759?

Multiply 4x759 as shown above.

Multiply 6x759. Use the same bones but look at the 6 row.

Multiplying 64x759 is the same as multiplying 60x759 + 4x759. Multiply 4554 by 10 and add it to 4x759=3036.

64x759=3036 + 45540 = 48576.