Summing Large Numbers

Question

I have being doing some problems on the Project Euler website and have come across a problem. The Question asks,"Work out the first ten digits of the sum of the following one-hundred 50-digit numbers." I am guessing there is some mathematical way to solve this but I was just wondering how numbers this big are summed? I store the number as a string and convert each digit to a long but the number is so large that the sum does not work.

Is there a way to hold very large numbers as a variable (that is not a string)? I do not want the code to the problem as I want to solve that for myself.

Solution 1

I was just wondering how numbers this big are summed?

You can use an array:

long LargeNumber[5] = { < first_10_digits>, < next_10_digits>....< last_10_digits> };

Now you can calculate the sum of 2 large numbers:

  long tempSum = 0;
  int carry = 0;
  long sum[5] = {0,0,0,0,0};

  for(int i = 4; i >= 0; i--)
  {
    tempSum = largeNum1[i] + largeNum2[i] + carry; //sum of 10 digits

    if( i == 0)
      sum[i] = tempSum; //No carry in case of most significant digit
    else
      sum[i] = tempSum % 1000000000; //Extra digits to be 'carried over'

    carry = tempSum/1000000000;
  }

  for( int i = 0; i < 5; i++ )
    cout<<setw(10)<<setfill('0')<<sum[i]<<"\n"; //Pad with '0' on the left if needed

Is there a way to hold very large numbers as a variable (that is not a string)?

There's no primitive for this, you can use any data structure (arrays/queues/linkedlist) and handle it suitably

I am guessing there is some mathematical way to solve this

Of course! But,

I do not want the code to the problem as I want to solve that for myself.

Solution 2

You may store the digits in an array. To save space and increase performance in the operations, store the digits of the number in base 10^9. so a number 182983198432847829347802092190 will be represented as the following in the array

arr[0]=2092190 arr[1]=78293478 arr[2]=19843284 arr[3]=182983

just for the sake of clarity, the number is represented as summation of arr[i]*(10^9i) now start with i=0 and start adding the numbers the way you learnt as a kid.

Solution 3

I have done in java, Here I am taking to numbers N1 and N2, And I have create an array of size 1000. Lets take an example How to solve this, N1=12, N2=1234. For N1=12, temp=N1%10=2, Now add this digit with digit N2 from right to Left and store the result into array starting from i=0, similarly for rest digit of N1. The array will store the result but in reverse order. Have a looking on this link. Please check this link http://ideone.com/V5knEd

import java.util.*;
import java.lang.*;
import java.io.*;

/* Name of the class has to be "Main" only if the class is public. */
class Ideone
{
    public static void main (String[] args) throws java.lang.Exception  {
        Scanner scan=new Scanner(System.in);
        int A=scan.nextInt();
        int B=scan.nextInt();
        int [] array=new int[1000];
        Arrays.fill(array,0);
        int size=add(A,B,array);
        for(int i=size-1;i>=0;i--){
            System.out.print(array[i]);
        }
    }
    public static int add(int A, int B, int [] array){
        int carry=0;
        int i=0;
        while(A>0 || B>0){
            int sum=A%10+B%10+carry+array[i];
            array[i]=sum%10;
            carry=sum/10;
            A=A/10;
            B=B/10;
            i++;
        }
        while(carry>0){
            array[i]=array[i]+carry%10;
            carry=carry/10;
            i++;
        }
        return i;
    }
}