# About digits in a number

• Mar 12th 2008, 01:34 AM
san4unall
Hi,

Could anyone please tell me if there is any formula or method to find the number of digits in the result of multiplication of 2 numbers?
For e.g. if a 12 digit number is multiplied by a 16 digit no. then how many digits will be there in the result?

Thanks.
• Mar 12th 2008, 09:14 AM
Opalg
Quote:

Originally Posted by san4unall
Could anyone please tell me if there is any formula or method to find the number of digits in the result of multiplication of 2 numbers?
For e.g. if a 12 digit number is multiplied by a 16 digit no. then how many digits will be there in the result?

If a number x has m digits then x must lie between \$\displaystyle 10^{m-1}\$ and \$\displaystyle 10^m\$. If y has n digits then y must lie between \$\displaystyle 10^{n-1}\$ and \$\displaystyle 10^n\$. Then the product xy must lie between \$\displaystyle 10^{m+n-2}\$ and \$\displaystyle 10^{m+n}\$, and therefore has either m+n–1 or m+n digits.
• Mar 13th 2008, 07:12 AM
Henderson
The line between m+n and m+n-1 digits is clear in most cases, depending on the first digits.

If the first digits are:
1 and anything 1-4
2 and 2
Then there will be m+n-1 digits.

If the first digits are:
1 and anything 5-9
2 and 3
2 and 4
3 and 3
Then it could be either m+n or m+n-1 digits, depending on later digits.

If the first digits are anything else,
Then there are m+n digits.