Main Index Number Theory Arithmetics Multiplication Multiplication tips
 Subject Index
comment on the page

Multiplication by 9, 99, 999, etc.

The basic rule how to multiply a number by 9 is to multiply it by 10 and then subtract the number from the product, that is . We can interpret this rule mathematically in the common decimal representation of numbers as follows: To multiply a one digit number by 9, first subtract 1 from it to get , then subtract from 9 to form . Finally, write the one digit numbers and in this order next to each other:  Really, .

For example, to compute first subtract 1 from 5: . Then subtract the second time and write the product  54.

The above rule has the following graphical representation:

In the middle age when the literacy was not a common property the above rule found the following  “finger” representation:

To multiply by 9  proceed as follows:

1. Spread your two hands out  in front of you (or place them on a desk in front of you).
2. To multiply, say by , fold down the th finger from the left.
3. Read the result: the fingers on the left of the folded down finger gives the number of tens, while the fingers on the right of it give the number units.

The above mathematical proof indicates that the rule remains valid also for factors having more digits. For example, . Open-minded reader immediately sees that this is not the only generalization. More interesting is the following one:

The product   is a special case of the formula  . For practical applications it has sense to suppose that .   

More generally , etc.

Cite this web-page as: