First digit of the product of two very large numbers [closed]

问题

There are 2 large integers, x and y such that z=x*y overflows. I would like to compute the first digit of x. Directly doing so isn't possible as the result overflows.

I was wondering if there was a certain technique for this.

Example: 10000000000000000000000000000000000 * 20579725928294522859735727575, here the first digit is 2 as I can see it directly, however, besides this special case, is there a way?

 #include <iostream>
using namespace std;

int main() {
long long int x,y,z;
cin>>x>>y;
z=x*y;
while(z>9)
{
    z/=10;
}
cout<<z;
    return 0;
 }

Above is a code that fails to work. For x=10000000000000000000000000000000000 y=20579725928294522859735727575

It gives output 5 as z overflows, and thus z=5240626126797720724 which is clearly wrong. I am looking to find a way out of this.

回答1:

You can do the multiplication with strings, like I learned in school many years ago

123456789987654321 * 998877665544332211

                    123456789987654321
                  * 998877665544332211
--------------------------------------
                    123456789987654321
                   123456789987654321
                  246913579975308642
                 246913579975308642
                370370369962962963
               370370369962962963
              493827159950617284
             493827159950617284
            617283949938271605
           617283949938271605
          740740739925925926
         740740739925925926
        864197529913580247
       864197529913580247
      987654319901234568
     987654319901234568
   1111111109888888889
+ 1111111109888888889
--------------------------------------
  123318230178465034444583004253633731

回答2:

In general; the most significant digit of the result depends on all the other digits, so you can't determine the most significant digit without determining all the other digits too.

For example, consider 3333334 * 3, where the carry would ripple all the way from the least significant digit all the way to the most significant digit.

Note that this is really:

     4 * 3 * 1
   + 3 * 3 * 10
   + 3 * 3 * 100
   + 3 * 3 * 1000
   + 3 * 3 * 10000
   + 3 * 3 * 100000
   + 3 * 3 * 1000000

..where the problem is that each addition can cause carries.

For multiple digits it gets worse. For example, 3333334 * 31 becomes:

   + 4 * 1 * 1
   + 3 * 1 * 10
   + 3 * 1 * 100
   + 3 * 1 * 1000
   + 3 * 1 * 10000
   + 3 * 1 * 100000
   + 3 * 1 * 1000000
   + 4 * 3 * 10
   + 3 * 3 * 100
   + 3 * 3 * 1000
   + 3 * 3 * 10000
   + 3 * 3 * 100000
   + 3 * 3 * 1000000
   + 3 * 3 * 10000000

回答3:

you can use Boost's Multiprecision Library to create big enough integers that would not overflow. But as to figuring out what the first digit is, you can turn your integer into string and then turn the first character of the string into integer. Having said that I fail to see why you would need such thing. If you are working with such big numbers that computation on them causes overflow, you should use something other than integer (Like BCD or boost) instead of regular int.

Sam

来源：https://stackoverflow.com/questions/37682728/first-digit-of-the-product-of-two-very-large-numbers

标签

c++

math

integer-overflow