Track first digit during a long multiplication in c - c

Question

I have an array of integer element range up to 10^5, and I have to find the first element after the total multiplication.
Example:
Array : 2,4,6,7
multiplication result: 336 and the first element is 3.
Obviously I cannot multiply the elements with the range up to 10^5.
How can I track only the first digit during multiplication?

Answer 1

We can also find the first digit with another method.
Suppose p be the final value after multiplying all the elements.
So, we have to find
P = a[0]*a[1]*a[2]*a[3]*.......*a[n-1]
for n sized array then we can take log with base 10 on both the side after that our expression changes to
log(p) = log(a[i])+log(a[1])+log(a[2])+.....+log(a[n-1])
Now, to find the first digit we have to get the fractional part of this variable sum which can be done in this way
frac = sum - (integer)sum
and at the last step calculate the 10^frac and convert it to the integer value which is our required first digit.
This algorithm is better in comparison to time complexity.
int getFirstDigit(long a[], long n) {
double p;
for(int i=0;i<n;i++) {
p = p+log10(a[i]);
}
double frac = p - (long)p;
int firdig = (int)pow(10,frac);
return firdig;
}

Answer 2

In c or c++ make integer data type as long double such that first digit of number is before decimal point and rest are after decimal point.
Above can be done as follows:-
long double GetFraction(int number){
int length = (int) log(number) + 1; // this will give number of digits in given number. And log is log base 10.
long double fraction = (long double) number / (10^(length - 1);
return fraction;
}
Example :-
Let number = 12345
length = log(12345) + 1 = 5;
fraction = (long double) 12345 / (10^4) = 1.2345
Now for all integers in array find fraction as mention above and multiply them as follow:-
int GetFirstDigit(int arr[] , int size){
if(size == 0)
return 0;
long double firstDigit = 1.0;
for(int i = 0 ; i < size ; i++){
firstDigit = firstDigit*GetFraction(arr[i]);
if(firstDigit >= 10.00) // You have to shorten your number otherwise it will same as large multiplication and will overflow.
firstDigit/=10;
}
return (int) firstDigit;
}
Disclaimer:- This is my approach and I don't have any formal proof about accuracy of result. But I have verified result for integer up to 10^9 and array size up to 10^5

Answer 3

Please donot forget to note that this is just an attempt to make you understand the logic and that you need to make changes in the code as per your requirement. I strongly suggest you make this a subroutine in your program and parse the arguments to it from the main thread in your program.
#include <stdio.h>
void main()
{
int num1, num2;
printf("Enter ur lovely number:\n");
scanf("%d",&num1);
num2=num1;
while(num2)
{
num2=num2/10;
if(num2!=0)
num1=num2;
}
printf("The first digit of the lovely number is %d !! :P\n ",num1);
}

Answer 4

Try this approach,
Take integer as input let us say int x1, now copy this in a double let us say double x2, and suppose you have previous product as double y, initially y = 1 . now use this loop,
while(x1!<10){
x1 = x1/10;
x2 = x2/10; //this will make double in standard form x*10^y without 10^y part
}
ex x1 = 52, then x2 will be converted to 5.2.
Now let us assume y = 3 and x is 5.2.
then product now is 15.6, again reduce this to 1.56 and repeat the process. in the end you will have the only digit before the decimal as the first digit of the product of all the numbers.