Add two numbers without using + or - operator

Sum of Two Integers without Using + or - operators

Blind 75. Binary

Given two integers a and b, return the sum of the two integers without using the operators + and -.

Input: a = 1, b = 2

Output: 3

How to add two numbers without using + operators?

Before we solve the problem.

let's first understand what are some real world application of solving this problem?

what can you do with the technique learnt here?

Note: Bitwise operation we will discuss later while solving this problem.

1.  Indexing, compression, and query optimisation in database engines
2. Graphics processing:
3. Pixel manipulation and color blending in image processing.
4. Embedded systems and microcontrollers:
5. Where hardware resources are limited, bitwise operations can be more efficient.
6. Database Indexing:
7. Where: In companies like Amazon for their product database or Google for their search index.
8. Quickly finding and sorting through millions of items.
9. Compression in Streaming Services:
10. Where: Netflix or YouTube for video streaming.
11. Application: Reducing video file sizes without losing much quality.
12. Bitwise: Allows for efficient manipulation of video data at the bit level.

Bitwise operations are typically faster than arithmetic operations.

It's faster than traditional comparison methods for large datasets. They're often executed in a single CPU cycle.

From looking at the question:

  1. Without using + operators how to add numbers?

  2. Why is - even mentioned?

A instinct we can have, is how does computer does the sum operation in it's core?

when we use +, only we understand it,

What is the logic that computer computes on?

• Also we know that in the computer language there is 0 and 1 which is binary.

So let's think about this, thus the concept of bitwise operator, in other words binary calculation works?

let's see how numbers add up in binary

Decimal  Binary  

 1     1  

 2     10

 3     11

 4     100

 5     101

 6     110

• decimal (2 + 3), let's see how these works in binary

1.  2 is 10, 3 is 11 from the above table of numbers

a. 

  1 0

  1 1

----------

   1   

b. 

 1  

  1 0

  1 1

----------

  1 0 1   

  here, we have carry over

  101 is 5.

• 3 is 11 in binary, let's add 3 + 3

a.  

  1

  1 1

  1 1

----------

   0 

here we have cary over, we need to learn how to handle cary over

also when there is a cary over there is introduction 0 and 1 shift to the left

b. 

  1

  1 1

  1 1

----------

   0 

  here, 1 + 1 = 10, and we have 1 (carry over)

  notice here we calculated, without the carry over then 

  calculated number + carryover

     10 + 1 = 11

 1 1

  1 1

  1 1

----------

  1 0

again, we have 1 cary over going to the left

 1 1

  1 1

  1 1

----------

 1 1 0  => 6 in decimal

1.  so the conclusion is how do we handle cary over?
2.  add the number in place without carry over.
3.  and how do we make a shift to the number?

 in bitwise operation, what are similar operators that can gear us up to handle these steps?

 peseudo code:

 function add(a, b) {

  caluclation loop {

    // identify carry

    // Add without carry

    // Shift carry

  }

    return total

}

now let's look at our bitwise operator pool:

1. AND (&): identifyies where carry over happend for example 1 & 1 = 10. In binary addition, a carry occurs when both bits are 1.

2. OR (|): Combines bits, resulting in 1 if either bit is 1.

3. XOR (^) (Exclusive OR) adds bits without handling carry. It returns 1 if bits are different, 0 if they're the same.

4. NOT (~). Inverts all bits. Can be used in subtraction algorithms.

5. Right Shift (>>),Can be used for division by powers of 2. Example: 1000 >> 1 = 0100

6. Left Shift(<<), We use this to move the carry to the next significant bit position.

form this pool, we match AND, XOR and Left Shift

function add(a,b){

  while(b !==0){

    // Identify carry

    let carry = a & b

    // Xor, add numbers without cary over

    a = a ^ b

    // left shift

    b = carry << 1

  }

  return a

}

in short: Algorithm steps:

• Identify carry (AND)
• Add without carry (XOR)
• Shift carry (Left Shift)
• Repeat until no carry

this is how this problem can be solved.

Why it's important?

in short:

• Efficiency in large-scale systems
• Optimization for speed and memory
• Low-level control in hardware
• Solving complex computational problems