Binary numbers and their operations play a pivotal role in computer science. They’re foundational to the operation of computers, and many computational problems require an understanding of binary operations. One such challenge is the “Complement of Base 10 Integer” problem on Leetcode. It tests one’s understanding of bitwise operations in the binary realm. This article will take you through the intricacies of this problem, explaining its underlying principles and detailing a solution in Python.

### Table of Contents

- Problem Statement
- Decoding the Problem
- Underlying Principles of Binary Complement
- Strategies to Approach the Problem
- Python Implementation
- Time and Space Complexity Analysis
- Conclusion

### 1. Problem Statement

Every non-negative integer `N`

has a binary representation. For example, `5`

can be represented as “101” in binary and `7`

as “111” in binary. We want to find the complement of the binary representation of `N`

and return its decimal form.

The complement of a binary representation is the number we get when we flip every bit from 0 to 1 and 1 to 0.

**Example:**

```
Input: 5
Output: 2
Explanation: The binary representation of 5 is 101, and its complement is 010 which is 2 in decimal form.
```

### 2. Decoding the Problem

The problem is essentially asking to find the bitwise complement of the number and then convert it back to its decimal (base 10) form. The challenge lies in efficiently flipping the bits and understanding the binary realm.

### 3. Underlying Principles of Binary Complement

- In binary representation, the complement of 0 is 1, and vice versa.
- For any number
`N`

, if we can find a number`mask`

that has all its bits as 1 and is just bigger than or equal to`N`

, then`N XOR mask`

will give us the complement of`N`

. - The reason the XOR operation works here is that
`1 XOR 1 = 0`

and`0 XOR 1 = 1`

.

### 4. Strategies to Approach the Problem

**Find the Mask:**Calculate a number`mask`

that’s just bigger than or equal to`N`

and has all its binary bits as 1.**Use XOR Operation:**Take the bitwise XOR of`N`

and`mask`

to get the complement of`N`

.- The result of the XOR operation is the required complement in decimal form.

### 5. Python Implementation

```
def bitwiseComplement(N):
# If N is 0, its complement is 1
if N == 0:
return 1
# Calculate the number of bits in the binary representation of N
num_bits = len(bin(N)) - 2
# Calculate the mask (a number just bigger than N with all bits as 1)
mask = (1 << num_bits) - 1
# Return the bitwise complement using XOR operation
return N ^ mask
```

### 6. Time and Space Complexity Analysis

**Time Complexity:**- Finding the number of bits in
`N`

will take O(logN) time. - The XOR operation will take O(1) time for computers with a word size of 32 or 64 bits.
- Thus, the overall time complexity is O(logN).

- Finding the number of bits in
**Space Complexity:**- We are using a constant amount of space irrespective of the input size.
- Thus, the space complexity is O(1).

### 7. Conclusion

The “Complement of Base 10 Integer” problem on Leetcode provides a profound understanding of bitwise operations and their utility. The binary complement may seem like a simple bit-flip operation, but the underlying principles and their application have far-reaching implications in computer science, especially in areas like computer networks, data compression, and error detection. This problem serves as a gateway to delve deeper into these areas and provides an excellent hands-on experience for those looking to understand the nuances of binary operations.