Question

Left-shifting an unsigned integer by 1 bit is equivalent to multiplying the value by 2. Write function power 2 that takes two integer arguments, number and pow, and calculates number \(* 2\) pow Use a shift operator to calculate the result. The program should print the values as integers and as bits.

Short answer

Shift left by `pow` positions to multiply by \(2^{\text{pow}}\).

Step-by-step solution

Understanding the Problem

We are asked to create a function that computes \( \text{number} \times 2^{\text{pow}} \) using bitwise shift operators. We'll print the result in both integer and binary formats.

Using Bitwise Shift Operator

The left shift operator (<<) shifts the bits of the operand to the left. Left-shifting a number by \( n \) positions is equivalent to multiplying it by \( 2^n \). Thus, \( \text{number} << \text{pow} \) computes \( \text{number} \times 2^{\text{pow}} \).

Implementing the Function

Write a function `power2(number, pow)` where `result = number << pow`. This line of code effectively multiplies `number` by \( 2^{\text{pow}} \).

Printing the Results

Once calculated, print the integer result with `print(result)`. Then, convert the result to binary using Python's built-in `bin()` function, `print(bin(result))`, to show the binary representation.

Complete Function Example

Here is a complete function: ```python def power2(number, pow): result = number << pow print(f"Integer result: {result}") print(f"Binary result: {bin(result)}") ```

Key concepts

Shift Operators

Shift operators are fundamental in programming, especially in low-level programming languages like C++. They manipulate bits in data types like integers.

There are two primary shift operators: the left shift (<<) and the right shift (>>). The left shift operator moves the bits of a number to the left, effectively doubling the number with each single shift.

For example, when you apply a left shift on the integer 3 ( 3 in binary is 00000011 d) by 1 position (3 << 1), it becomes 6 ( 6 in binary is 00000110 ).

In the programming task, we specifically used this operator to multiply a given number by powers of two by simply shifting the number left by the desired power of two.

Binary Representation

Understanding binary representation is key to grasping bitwise operations. In binary, data is represented using only two digits: 0 and 1. Each binary digit (or bit) has a place value or weight, much like our decimal system.

For example, the binary number 1010 translates to the decimal number 10.

In this binary number: - The rightmost '0' has a place value of 2⁰ (1 in decimal). - The second '1' has a place value of 2¹ (2 in decimal). - The third '0' has a place value of 2² (4 in decimal). - The leftmost '1' has a place value of 2³ (8 in decimal). > Therefore, the full calculation is 1×8 + 0×4 + 1×2 + 0×1 = 10 in decimal.

Working with binary is efficient for computers and vital for optimizing performance when executing bitwise operations like the shift. This ability to visualize data in binary form allows programmers to perform operations like shifts easily and predictably.

Function Implementation

Function implementation involves writing code that performs a specific task. In this case, the task is computing a number multiplied by a power of two using shift operators.

The function here is simple: it accepts two arguments. - The first argument is `number`, representing the integer you wish to manipulate. - The second, `pow`, is the exponent in the power of two.

The core operation within the function is `result = number << pow`. This shift operation moves the number's binary representation to the left by `pow` bits.
This means the number is effectively multiplied by 2 raised to the power of `pow`. After performing the operation: - The result is printed in integer form. - It is also printed in binary format using Python’s `bin()` function, illustrating how the number looks at the bit level. This implementation effectively demonstrates how easily computations reliant on powers of two can be optimized through bitwise operations.

Power of Two Calculations

Calculating powers of two efficiently is essential for various computing tasks, especially those dealing with large data sets or requiring fast computation.

Mathematically, a power of two can be represented as 2 raised to some integer exponent, say n, or as 2ⁿ.

In programming, however, bitwise shift operators offer a fast and efficient means to achieve this. When we left shift a number by `n` positions, we multiply it by 2 raised to the `n`th power.

For instance, calculating 3 multiplied by 2 raised to the power of 4 (or 3 × 2⁴), using bitwise operations, simply becomes `3 << 4`. - The operation effectively configures the bits, equating to a decimal calculation of 3 × 16 = 48.

In practical applications, this method uses minimal computational resources, making it ideal for performance-centric programming environments. By leveraging shifts for powers of two, programmers achieve faster computations and less processor load compared to traditional multiplication.