Question

Write a short C++ function, is TwoPower, that takes an int \(i\) and returns true if and only if \(i\) is a power of \(2 .\) Do not use multiplication or division, however.

Short answer

Use bitwise operations: ```cppbool isTwoPower(int i) { return (i > 0) && ((i & (i - 1)) == 0); }```

Step-by-step solution

- Understand the Problem

The task is to write a function in C++ that checks if a given integer is a power of 2 without using multiplication or division.

- Conceptual Approach

A number is a power of 2 if and only if it has exactly one bit set in its binary representation. For instance, 1 (2^0) is 0001, 2 (2^1) is 0010, 4 (2^2) is 0100, and so on. We can use bitwise operations to check this property.

- Use Bitwise AND Operator

If a number is a power of 2, it has only one '1' bit in its binary representation. If we subtract 1 from that number, all the bits after the '1' bit (till the least significant bit) become '1'. For example, 4 in binary is 100, and 3 (4-1) in binary is 011. Therefore, the bitwise AND of these two numbers is zero. Utilizing this property, we can check: if ( i > 0 && (i & (i - 1)) == 0 )

- Write the Function

Translate the conceptual approach into a C++ function. The function will accept an integer and return a boolean indicating if the integer is a power of 2. Here is the code:```cppbool isTwoPower(int i) { return (i > 0) && ((i & (i - 1)) == 0);}```

- Test the Function

You should test the function with various examples to ensure its correctness.For example:```cppint main() { std::cout << isTwoPower(1); // true std::cout << isTwoPower(2); // true std::cout << isTwoPower(3); // false std::cout << isTwoPower(4); // true std::cout << isTwoPower(5); // false return 0;}```

Key concepts

C++ Functions

In C++, a function is a reusable block of code that performs a specific task. Functions help in modularizing code and making it more manageable and readable. Functions are defined using a specific syntax and contain a return type, a function name, a parameter list in parentheses, and a block of code enclosed within curly braces.

Here's the general syntax for a function in C++:
```cpp
returnType functionName(parameter_list) {
// Code to execute
}
```
In the exercise provided, the function `isTwoPower` is designed to take an integer as an input and return a boolean indicating whether the input is a power of two.

Functions provide numerous benefits:

• Code Reusability: Writing a function once allows it to be reused across the program.
  
• Easier Maintenance: Changes to the code can be made once within the function, and those changes automatically take effect wherever the function is called.
  
• Improved Readability: Breaking down complex problems into functions makes code easier to read and understand.
  

Understanding how to write and use functions is fundamental for efficient programming.

Binary Representation

Binary representation is a way of expressing numbers using only two digits: 0 and 1. It is the foundation of computer systems, as computers operate using binary numbers. Every binary number consists of bits (binary digits), and each bit represents a power of 2.
For example:

• The binary number 0001 represents the decimal number 1.
  
• The binary number 0010 represents the decimal number 2.
  
• The binary number 0100 represents the decimal number 4.
  

In the context of checking for powers of two, binary representation is crucial. A number is a power of 2 if exactly one bit in its binary representation is set to 1. This unique property allows us to perform specific bitwise operations to determine if a number is a power of 2.
By subtracting 1 from a power of 2, the resulting binary number has all bits set to 1 up to the position of the original 1 bit, making it easy to use the bitwise AND operation for verification.

Bitwise AND Operator

The Bitwise AND operator is a fundamental bitwise operator in C++ that performs an AND operation on corresponding bits of two integers. It is denoted by the symbol `&`.
When two bits are ANDed:

• 1 & 1 results in 1
  
• 1 & 0 results in 0
  
• 0 & 0 results in 0
  

In the context of this exercise, the Bitwise AND operator is used to determine if a number is a power of 2. By subtracting 1 from a power of 2, the resulting number has all bits set to 1 after the position of the single 1 bit in the original number.
For example, for the number 4 (binary 100):

• 4 - 1 = 3 (binary 011)
  
• 4 & 3 = 100 & 011 = 000
  

If the result is zero, as with `4 & 3`, we can conclude that the number is a power of 2. Otherwise, it is not.
Therefore, the condition `(i & (i - 1)) == 0` effectively checks if the number has exactly one bit set.

Power of Two

A power of two is a number that can be expressed as 2 raised to an integer exponent `n`. In mathematical form: 2^n 2^n 2^n 2^n Thanks,

Examples include:

• 1 (2^0)
  
• 2 (2^1)
  
• 4 (2^2)
  
• 8 (2^3)
  

Powers of two are significant in computer science due to the binary nature of computers. Memory, storage, and processing sizes in computers are often measured in powers of two.
Identifying if a number is a power of two can be very useful. As discussed above, we leverage the unique binary representation of these numbers to make the determination easily and efficiently using bitwise operations.
In the `isTwoPower` function, the expression `(i > 0) && ((i & (i - 1)) == 0)` confirms that the number `i` is greater than zero and has only one bit set, ensuring it is a power of two.