how to convert two's complement to decimal

Charlotte

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02-02-2025

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Two's complement is a method used to represent signed integers (positive and negative numbers) in binary format. Understanding how to convert a two's complement binary number to its decimal equivalent is crucial in computer science and digital electronics. This comprehensive guide will walk you through the process step-by-step.

Understanding Two's Complement

Before diving into the conversion process, let's briefly review the concept of two's complement. In this system, the most significant bit (MSB) represents the sign: 0 for positive and 1 for negative. Positive numbers are represented the same as in unsigned binary. Negative numbers, however, are represented differently. To find the two's complement representation of a negative number:

1. Find the binary representation of the positive magnitude: Ignore the minus sign and convert the number to binary as usual.

2. Invert the bits: Change all 0s to 1s and all 1s to 0s.

3. Add 1: Add 1 to the result of step 2.

Step-by-Step Conversion: Two's Complement to Decimal

Let's illustrate the conversion process with an example. Suppose we have the 8-bit two's complement number 10001101. Here's how to convert it to decimal:

1. Identify the Sign Bit: The MSB is 1, indicating a negative number.

2. Find the Magnitude:

• Invert the bits: 10001101 becomes 01110010.
• Add 1: 01110010 + 1 = 01110011

3. Convert the Magnitude to Decimal: Now, treat 01110011 as an unsigned binary number and convert it to decimal:

(0 * 2⁷) + (1 * 2⁶) + (1 * 2⁵) + (1 * 2⁴) + (0 * 2³) + (0 * 2²) + (1 * 2¹) + (1 * 2⁰) = 64 + 32 + 16 + 2 + 1 = 115

4. Apply the Sign: Since the original number was negative, the final decimal equivalent is -115.

Another Example: Positive Two's Complement Number

Let's consider a positive number, say 00110110.

1. Identify the Sign Bit: The MSB is 0, indicating a positive number.

2. Convert to Decimal Directly: We can directly convert this to decimal as it's already in standard binary representation.

(0 * 2⁷) + (0 * 2⁶) + (1 * 2⁵) + (1 * 2⁴) + (0 * 2³) + (1 * 2²) + (1 * 2¹) + (0 * 2⁰) = 32 + 16 + 4 + 2 = 54

Therefore, the decimal equivalent of 00110110 is 54.

Handling Different Bit Sizes

The process remains the same regardless of the number of bits (8-bit, 16-bit, 32-bit, etc.). The only difference is the range of numbers that can be represented. For example, an 8-bit two's complement number can represent values from -128 to 127.

Common Mistakes to Avoid

• Forgetting to add 1 after inverting the bits: This is a crucial step for accurately representing negative numbers.
• Incorrectly identifying the sign bit: Always remember that the MSB determines the sign.
• Treating negative numbers directly as unsigned binary: This will lead to incorrect results.

Conclusion

Converting two's complement to decimal might seem complex initially, but with practice, it becomes straightforward. By carefully following the steps outlined above, you can confidently convert any two's complement binary number to its decimal equivalent. Remember to always check your work and account for the sign bit. This understanding is fundamental to working with binary data in various computing applications.