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4. The result of n-bit 2's complement addition is always correct (regardless of carry out) as long as the result falls into the range of n-bit 2's complement. a) What is the range of 3-bit 2's complement? b) What is the range of n-bit 2's complement? c) Perform (10+9) in 5-bit 2's complement, is there overflow?

4. The result of n-bit 2's complement addition is always correct (regardless of carry out) as long as the result falls into the range of n-bit 2's complement. a) What is the range of 3-bit 2's complement? b) What is the range of n-bit 2's complement? c) Perform (10+9) in 5-bit 2's complement, is there overflow?

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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### Understanding n-bit 2's Complement Arithmetic The concept of 2's complement is crucial in digital systems for representing signed numbers and performing arithmetic operations. Below are some key concepts and practical problems related to n-bit 2's complement numbers. #### Question Analysis: **4. The result of n-bit 2's complement addition is always correct (regardless of carry out) as long as the result falls into the range of n-bit 2's complement.** This statement emphasizes the importance of understanding the range and behavior of 2’s complement numbers during arithmetic operations. **a) What is the range of 3-bit 2's complement?** In an n-bit 2's complement system, the range is from \(-2^{(n-1)}\) to \(2^{(n-1)} - 1\). - For 3 bits, the range is from \(-2^{(3-1)} = -4\) to \(2^{(3-1)} - 1 = 3\). **b) What is the range of n-bit 2's complement?** In general, for an n-bit system, the range is: \[ -2^{(n-1)} \text{ to } 2^{(n-1)} - 1 \] This provides a way to determine the limits of representable values in any n-bit 2's complement system. **c) Perform (10+9) in 5-bit 2's complement, is there overflow?** 1. **Convert to 5-bit 2's Complement:** - 10 in 5 bits: 01010 - 9 in 5 bits: 01001 2. **Addition:** - 01010 (10) - +01001 (9) - ------ 10011 (Addition result) 3. **Check for Overflow:** - In 5-bit 2's complement, the range is \(-16\) to \(15\). - The result `10011` is -13, which is within the range. - Therefore, there is no overflow, but the arithmetic result should be carefully interpreted to confirm whether it's as expected. Understanding the addition and associated rules of overflow detection ensures reliability in computations using 2's complement representations.
Transcribed Image Text:### Understanding n-bit 2's Complement Arithmetic The concept of 2's complement is crucial in digital systems for representing signed numbers and performing arithmetic operations. Below are some key concepts and practical problems related to n-bit 2's complement numbers. #### Question Analysis: **4. The result of n-bit 2's complement addition is always correct (regardless of carry out) as long as the result falls into the range of n-bit 2's complement.** This statement emphasizes the importance of understanding the range and behavior of 2’s complement numbers during arithmetic operations. **a) What is the range of 3-bit 2's complement?** In an n-bit 2's complement system, the range is from \(-2^{(n-1)}\) to \(2^{(n-1)} - 1\). - For 3 bits, the range is from \(-2^{(3-1)} = -4\) to \(2^{(3-1)} - 1 = 3\). **b) What is the range of n-bit 2's complement?** In general, for an n-bit system, the range is: \[ -2^{(n-1)} \text{ to } 2^{(n-1)} - 1 \] This provides a way to determine the limits of representable values in any n-bit 2's complement system. **c) Perform (10+9) in 5-bit 2's complement, is there overflow?** 1. **Convert to 5-bit 2's Complement:** - 10 in 5 bits: 01010 - 9 in 5 bits: 01001 2. **Addition:** - 01010 (10) - +01001 (9) - ------ 10011 (Addition result) 3. **Check for Overflow:** - In 5-bit 2's complement, the range is \(-16\) to \(15\). - The result `10011` is -13, which is within the range. - Therefore, there is no overflow, but the arithmetic result should be carefully interpreted to confirm whether it's as expected. Understanding the addition and associated rules of overflow detection ensures reliability in computations using 2's complement representations.
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