Multiply Large Numbers represented as Strings

Given two numbers as strings. The numbers may be very large (may not fit in long long int), the task is to find product of these two numbers.

Examples:

Input : num1 = 4154  
        num2 = 51454
Output : 213739916 

Input :  num1 = 654154154151454545415415454  
         num2 = 63516561563156316545145146514654 
Output : 41549622603955309777243716069997997007620439937711509062916

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

The idea is based on school mathematics.

We start from last digit of second number multiply it with first number. Then we multiply second digit of second number with first number, and so on. We add all these multiplications. While adding, we put i-th multiplication shifted.

The approach used in below solution is to keep only one array for result. We traverse all digits first and second numbers in a loop and add the result at appropriate position.

// C++ program to multiply two numbers represented 
// as strings. 
#include<bits/stdc++.h> 
using namespace std; 
   
// Multiplies str1 and str2, and prints result. 
string multiply(string num1, string num2) 
{ 
    int n1 = num1.size(); 
    int n2 = num2.size(); 
    if (n1 == 0 || n2 == 0) 
    return "0"; 
   
    // will keep the result number in vector 
    // in reverse order 
    vector<int> result(n1 + n2, 0); 
   
    // Below two indexes are used to find positions 
    // in result.  
    int i_n1 = 0;  
    int i_n2 = 0;  
       
    // Go from right to left in num1 
    for (int i=n1-1; i>=0; i--) 
    { 
        int carry = 0; 
        int n1 = num1[i] - '0'; 
   
        // To shift position to left after every 
        // multiplication of a digit in num2 
        i_n2 = 0;  
           
        // Go from right to left in num2              
        for (int j=n2-1; j>=0; j--) 
        { 
            // Take current digit of second number 
            int n2 = num2[j] - '0'; 
   
            // Multiply with current digit of first number 
            // and add result to previously stored result 
            // at current position.  
            int sum = n1*n2 + result[i_n1 + i_n2] + carry; 
   
            // Carry for next iteration 
            carry = sum/10; 
   
            // Store result 
            result[i_n1 + i_n2] = sum % 10; 
   
            i_n2++; 
        } 
   
        // store carry in next cell 
        if (carry > 0) 
            result[i_n1 + i_n2] += carry; 
   
        // To shift position to left after every 
        // multiplication of a digit in num1. 
        i_n1++; 
    } 
   
    // ignore '0's from the right 
    int i = result.size() - 1; 
    while (i>=0 && result[i] == 0) 
    i--; 
   
    // If all were '0's - means either both or 
    // one of num1 or num2 were '0' 
    if (i == -1) 
    return "0"; 
   
    // generate the result string 
    string s = ""; 
       
    while (i >= 0) 
        s += std::to_string(result[i--]); 
   
    return s; 
} 
   
// Driver code 
int main() 
{ 
    string str1 = "1235421415454545454545454544"; 
    string str2 = "1714546546546545454544548544544545"; 
       
    if((str1.at(0) == '-' || str2.at(0) == '-') &&  
        (str1.at(0) != '-' || str2.at(0) != '-' )) 
        cout<<"-"; 
   
   
    if(str1.at(0) == '-' && str2.at(0)!='-') 
        { 
            str1 = str1.substr(1); 
        } 
        else if(str1.at(0) != '-' && str2.at(0) == '-') 
        { 
            str2 = str2.substr(1); 
        } 
        else if(str1.at(0) == '-' && str2.at(0) == '-') 
        { 
            str1 = str1.substr(1); 
            str2 = str2.substr(1); 
        } 
    cout << multiply(str1, str2); 
    return 0; 
} 

Output:

2118187521397235888154583183918321221520083884298838480662480

The above code is adapted from the code provided by Gaurav.

Time Complexity : O(m*n), where m and n are length of two number that need to be multiplied.
Another method:

// Java program to multiply two numbers represented 
// as strings. 
import java.util.Scanner; 
  
public class StringMultiplication 
{ 
    // Driver code 
    public static void main(String[] args)  
    { 
        String num1 = "1235421415454545454545454544"; 
        String tempnum1 = num1; 
        String num2 = "1714546546546545454544548544544545"; 
        String tempnum2 = num2; 
          
        // Check condition if one string is negative 
        if(num1.charAt(0) == '-' && num2.charAt(0)!='-') 
        { 
            num1 = num1.substring(1); 
        } 
        else if(num1.charAt(0) != '-' && num2.charAt(0) == '-') 
        { 
            num2 = num2.substring(1); 
        } 
        else if(num1.charAt(0) == '-' && num2.charAt(0) == '-') 
        { 
            num1 = num1.substring(1); 
            num2 = num2.substring(1); 
        } 
        String s1 = new StringBuffer(num1).reverse().toString(); 
        String s2 = new StringBuffer(num2).reverse().toString(); 
          
        int[] m = new int[s1.length()+s2.length()]; 
                  
                // Go from right to left in num1 
        for (int i = 0; i < s1.length(); i++)  
        { 
            // Go from right to left in num2 
            for (int j = 0; j < s2.length(); j++)  
            { 
                m[i+j] = m[i+j]+(s1.charAt(i)-'0')*(s2.charAt(j)-'0'); 
              
            } 
        } 
          
      
        String product = new String(); 
        // Multiply with current digit of first number 
        // and add result to previously stored product 
        // at current position.  
        for (int i = 0; i < m.length; i++) 
        { 
            int digit = m[i]%10; 
            int carry = m[i]/10; 
            if(i+1<m.length) 
            { 
                m[i+1] = m[i+1] + carry; 
            } 
            product = digit+product; 
              
        } 
          
        // ignore '0's from the right 
        while(product.length()>1 && product.charAt(0) == '0') 
        { 
            product = product.substring(1); 
        } 
          
        // Check condition if one string is negative 
        if(tempnum1.charAt(0) == '-' && tempnum2.charAt(0)!='-') 
        { 
            product = new StringBuffer(product).insert(0,'-').toString(); 
        } 
        else if(tempnum1.charAt(0) != '-' && tempnum2.charAt(0) == '-') 
        { 
            product = new StringBuffer(product).insert(0,'-').toString(); 
        } 
        else if(tempnum1.charAt(0) == '-' && tempnum2.charAt(0) == '-') 
        { 
            product = product; 
        } 
        System.out.println("Product of the two numbers is :"+"\n"+product); 
    } 
} 

Output:

2118187521397235888154583183918321221520083884298838480662480

Related Article :
Karatsuba algorithm for fast multiplication

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