Math Help - About digits in a number

1. About digits in a number

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.

2. 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 $10^{m-1}$ and $10^m$. If y has n digits then y must lie between $10^{n-1}$ and $10^n$. Then the product xy must lie between $10^{m+n-2}$ and $10^{m+n}$, and therefore has either m+n–1 or m+n digits.

3. 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.