Math Problem Statement
What is the 8-bit two's complement for the following integer? −53
Solution
To find the 8-bit two's complement for , follow these steps:
Step 1: Find the binary representation of (positive integer)
First, convert into binary:
26 \div 2 = 13 \, \text{R}0 \\ 13 \div 2 = 6 \, \text{R}1 \\ 6 \div 2 = 3 \, \text{R}0 \\ 3 \div 2 = 1 \, \text{R}1 \\ 1 \div 2 = 0 \, \text{R}1$$ So, $$53$$ in binary is $$110101$$. To represent it in 8 bits, pad with zeros: $$53 = 00110101_2$$ ### Step 2: Find the one's complement To find the one's complement, invert all the bits (change $$0$$s to $$1$$s and $$1$$s to $$0$$s): $$\text{One's complement of } 00110101_2 = 11001010_2$$ ### Step 3: Add 1 to get the two's complement Now, add $$1$$ to the one's complement: $$11001010_2 + 1_2 = 11001011_2$$ ### Conclusion: The 8-bit two's complement representation of $$-53$$ is: $$11001011_2$$ Would you like more details or have any questions? Here are 5 related questions to explore: 1. How do you convert a positive decimal to binary? 2. What is the process for calculating one's complement? 3. How do you check if an 8-bit binary number is negative in two's complement? 4. What is the two's complement representation for positive numbers? 5. How does two's complement handle overflow in 8-bit arithmetic? **Tip:** Always ensure your final two's complement result is in the correct bit length (e.g., 8-bit, 16-bit) by padding with leading zeros if necessary!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Binary Numbers
Two's Complement
Negative Number Representation
Formulas
Binary Conversion
One's Complement
Two's Complement Calculation
Theorems
Two's Complement Theorem
Suitable Grade Level
Grades 9-12