<<–2/”>a href=”https://exam.pscnotes.com/5653-2/”>p>In computer science, representing negative numbers in binary form is crucial for various computational tasks. Two common techniques for this are 1’s complement and 2’s complement representations. Both methods allow binary systems to perform arithmetic operations on signed integers effectively but differ in how they encode negative values. Understanding these techniques is essential for computer scientists and engineers as they form the foundation of numerical computing in digital systems.
| Feature | 1’s Complement Representation | 2’s Complement Representation |
|---|---|---|
| Definition | Inverts all the bits of the binary representation of a number. | Inverts all the bits and adds 1 to the least significant bit. |
| Range of Representation | – (2^(n-1) – 1) to (2^(n-1) – 1) | -2^(n-1) to (2^(n-1) – 1) |
| Zero Representation | Has two representations: +0 (all 0s) and -0 (all 1s). | Has a single representation for zero (all 0s). |
| Positive Numbers | Same as unsigned binary numbers. | Same as unsigned binary numbers. |
| Negative Numbers | Invert all the bits of the positive number. | Invert all the bits of the positive number and add 1. |
| Addition and Subtraction | Requires end-around carry for correction. | Direct binary addition, no correction needed. |
| Sign Bit | Most significant bit indicates the sign (0 for positive, 1 for negative). | Most significant bit indicates the sign (0 for positive, 1 for negative). |
| Overflow Detection | More complex due to dual zero representation. | Simpler, based on sign bit and carry out of the most significant bit. |
| Hardware Complexity | Slightly more complex due to handling end-around carry. | Simpler, as addition and subtraction are straightforward. |
| Usage | Less commonly used in modern systems. | Widely used in most modern digital systems. |
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Q1: What is the main difference between 1’s complement and 2’s complement representation?
A1: The main difference is in how they represent negative numbers. 1’s complement inverts all bits, while 2’s complement inverts all bits and adds 1 to the least significant bit.
Q2: Why is 2’s complement more commonly used in modern systems?
A2: 2’s complement is more commonly used because it simplifies arithmetic operations, has a unique zero representation, and is more efficient for hardware implementation.
Q3: How does 1’s complement handle zero?
A3: 1’s complement has two representations for zero: +0 (all bits are 0) and -0 (all bits are 1), which can cause redundancy and complications in arithmetic operations.
Q4: Can you convert a 2’s complement number to its positive counterpart?
A4: Yes, you can convert a negative 2’s complement number to its positive counterpart by inverting all the bits and adding 1 to the least significant bit.
Q5: How do you detect overflow in 2’s complement arithmetic?
A5: Overflow in 2’s complement arithmetic can be detected by checking the sign bits of the operands and the result. If the sign bit of the result is different from the sign bits of the operands when they are the same, overflow has occurred.
Q6: Is there any advantage of 1’s complement over 2’s complement?
A6: The primary advantage of 1’s complement is its simplicity in manual computation and theoretical analysis. However, it is less efficient for practical digital systems compared to 2’s complement.
Q7: What is the range of numbers that can be represented using 2’s complement?
A7: The range for an n-bit number in 2’s complement is from -2^(n-1) to (2^(n-1) – 1).
Q8: How does end-around carry work in 1’s complement?
A8: In 1’s complement, if an addition operation results in a carry out of the most significant bit, this carry is added back to the least significant bit of the result, known as end-around carry.
Q9: Why does 2’s complement have a wider range for negative numbers?
A9: 2’s complement has a wider range for negative numbers because it includes the most negative number (-2^(n-1)), which does not have a positive counterpart in the representation.
Q10: Are there any specific applications where 1’s complement is preferred over 2’s complement?
A10: 1’s complement is rarely preferred in modern systems due to its inefficiencies. However, it may be used in certain legacy systems or specific theoretical contexts for its simplicity in manual computations.