Add two numbers without using arithmetic operators

We want to add 2 numbers without using arithmetic operator ( + , - , / , * ) . The trick here is to use properties of XOR. 

So, let's try to understand how it works.

We'll use Half adder logic here. Points to note down are:

• Sum of two bits can be obtained by performing XOR (^) of the two bits.
• Carry bit can be obtained by performing AND (&) of two bits.

The above logic can be used to add 2 single bits. Now, we are going to extend this logic for integers. Let's say we have to 2 integers, X and Y. In order to get sum of X and Y we'll first convert both of them into their binary repersentation.

Here there can be 2 cases:

• X and Y don't have any common position in which bits of both are set.
• Or they might have positions in which bits are set in both.

For the first case, we can easily get sum of X and Y by performing XOR. But for the second case, we need to use bitwise AND( & ). Bitwise AND of X and Y will give all carry bits. Bitwise AND of X and Y gives all carry bits. We will calculate (X & Y) << 1 and add it to X ^ Y to get the required result.

Here is a function of implementation of this apporach.

int Add(int x, int y)  
{  
    // Loop until there is no carry left 
    while (y != 0)  
    {  
        // carry now contains common set bits of x and y  
        
        int carry = x & y;  
  
        // Sum of bits of x and y where at least one of the bits is not set  
        
        x = x ^ y;  
  
        // Carry is shifted by one so that adding  
        // it to x gives the required sum  
        y = carry << 1;  
    }
  
    return x;  
}

Here is Recursive version of above function.

int Add(int x, int y) 
{ 
	if (y == 0) 
		return x; 
	else
		return Add( x ^ y, (x & y) << 1); 
} 

Here is a complete example with Main function.

// C++ Program to add two numbers
// without using arithmetic operator
 
#include <bits/stdc++.h> 
using namespace std; 

int Add(int x, int y) 
{ 
	if (y == 0) 
		return x; 
	else
		return Add( x ^ y, (x & y) << 1); 
} 


int main() 
{ 
	cout << Add(11, 23); 
	return 0; 
}

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