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Questions and Answers
What does the leftmost bit represent in sign/magnitude notation?
What does the leftmost bit represent in sign/magnitude notation?
How many unique patterns can be represented with m binary digits?
How many unique patterns can be represented with m binary digits?
What is a potential complication of using sign/magnitude representation?
What is a potential complication of using sign/magnitude representation?
What technique divides the available patterns into positive and negative portions?
What technique divides the available patterns into positive and negative portions?
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What might be required to compare two sign/magnitude numbers for equality?
What might be required to compare two sign/magnitude numbers for equality?
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In the context of radix complementation, what point does an integer A being positive imply about its sign bit?
In the context of radix complementation, what point does an integer A being positive imply about its sign bit?
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Which representation allows for two distinct values for zero?
Which representation allows for two distinct values for zero?
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What is true about the number of patterns used in signed representations as compared to unsigned representations?
What is true about the number of patterns used in signed representations as compared to unsigned representations?
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What is the largest positive number that can be represented using four bits in sign magnitude representation?
What is the largest positive number that can be represented using four bits in sign magnitude representation?
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Using four bits, which two's complement representation corresponds to the decimal value -1?
Using four bits, which two's complement representation corresponds to the decimal value -1?
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What pattern emerges when all four bits in the two's complement representation are set to 1?
What pattern emerges when all four bits in the two's complement representation are set to 1?
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When representing negative integers using two's complement, why can one additional negative integer be represented compared to positive integers?
When representing negative integers using two's complement, why can one additional negative integer be represented compared to positive integers?
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What is the formula used to derive the value of a negative integer in two's complement?
What is the formula used to derive the value of a negative integer in two's complement?
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If a positive number is represented in n-bit two's complement, what can be concluded about its sign bit?
If a positive number is represented in n-bit two's complement, what can be concluded about its sign bit?
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What is the effect of performing arithmetic operations that cross the endpoints of the number circle in two's complement?
What is the effect of performing arithmetic operations that cross the endpoints of the number circle in two's complement?
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When adding a positive integer and a negative integer in two's complement, which outcome is guaranteed?
When adding a positive integer and a negative integer in two's complement, which outcome is guaranteed?
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What is the two’s complement representation of -2 in a 4-bit binary system?
What is the two’s complement representation of -2 in a 4-bit binary system?
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How do you determine the decimal value of a binary number represented in two’s complement?
How do you determine the decimal value of a binary number represented in two’s complement?
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What is the two’s complement binary representation of +5 in 5 bits?
What is the two’s complement binary representation of +5 in 5 bits?
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What is the result of adding 1 to the flipped bits of 00111 when finding -7 in two's complement?
What is the result of adding 1 to the flipped bits of 00111 when finding -7 in two's complement?
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If the two’s complement binary value is 1010, what is its decimal value?
If the two’s complement binary value is 1010, what is its decimal value?
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What is the total number of distinct values represented in a signed n-bit two's complement system?
What is the total number of distinct values represented in a signed n-bit two's complement system?
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What must be done to represent a negative decimal value 'A' in two’s complement?
What must be done to represent a negative decimal value 'A' in two’s complement?
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Given the binary value 11100 in two's complement, what is its decimal equivalent?
Given the binary value 11100 in two's complement, what is its decimal equivalent?
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What is the decimal value of the octal number 27?
What is the decimal value of the octal number 27?
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What is the result of adding the binary numbers 00101 and 00110?
What is the result of adding the binary numbers 00101 and 00110?
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In converting the hexadecimal number 1D to decimal, what bit representation is used?
In converting the hexadecimal number 1D to decimal, what bit representation is used?
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When adding two’s complement values, what happens to the carry out of the most significant bit?
When adding two’s complement values, what happens to the carry out of the most significant bit?
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If applying two's complement to the binary number 00110, what is the flipped version?
If applying two's complement to the binary number 00110, what is the flipped version?
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What is the decimal equivalent of the unsigned binary number 011101?
What is the decimal equivalent of the unsigned binary number 011101?
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How is the value 2^m related to the operation of adding two binary numbers?
How is the value 2^m related to the operation of adding two binary numbers?
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In Two's complement addition, what is the result of adding +7 and -2?
In Two's complement addition, what is the result of adding +7 and -2?
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What is the two's complement representation of the number 0?
What is the two's complement representation of the number 0?
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What happens when you add two positive numbers and the result is negative in two's complement?
What happens when you add two positive numbers and the result is negative in two's complement?
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What is the result of adding 5 and 14 in two's complement with 5 bits?
What is the result of adding 5 and 14 in two's complement with 5 bits?
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To compute D = Y - X using two's complement, what is the first step?
To compute D = Y - X using two's complement, what is the first step?
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What is indicated when an overflow register is activated during addition?
What is indicated when an overflow register is activated during addition?
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Which situation will NOT generate an overflow when performing addition in two's complement?
Which situation will NOT generate an overflow when performing addition in two's complement?
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What is the final step when using two's complement to find the negative of a binary number?
What is the final step when using two's complement to find the negative of a binary number?
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What representation is used to indicate both 0 and –0 in two's complement?
What representation is used to indicate both 0 and –0 in two's complement?
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Study Notes
Signed Number Representations
- Unsigned number representations offer 2^m unique patterns for m binary digits, while signed representations divide these into positive and negative patterns.
- Common techniques for signed number representation include sign/magnitude, two’s complement, and binary coded decimal.
Sign/Magnitude Notation
- Utilizes the leftmost (most significant) bit as the sign bit: 0 for positive and 1 for negative.
- Remaining m-1 bits represent the magnitude in unsigned binary.
- Results in two representations for zero: positive zero (0000) and negative zero (1000), complicating equality checks in computing.
Two’s Complement Method
- Based on modular arithmetic; allows representation of signed quantities using n bits.
- For positive integers, the sign bit is 0 and the remaining bits represent the magnitude.
- Negative integers utilize 1 as the sign bit and adapt the remaining bits to allow for straightforward arithmetic operations.
- Formula for value calculation:
- Val = -2^(n−1) * An−1 + Σ(2^i * Ai) for i = 0 to n-2.
- Two's complement ensures a single representation for zero and simplifies arithmetic operations.
Shortcut for Two’s Complement Conversion
- For positive A:
- Use sign-magnitude representation with the leftmost bit as 0.
- For negative A:
- Calculate binary for +A.
- Flip bits and add 1 to derive two's complement.
Checking Decimal Value from Two’s Complement
- If the leftmost bit (sign bit) is 0, interpret as positive; compute as unsigned binary.
- If the sign bit is 1, flip bits, add 1, and treat as negative.
Arithmetic Operations
- Rules for binary addition simplified to a 2x2 table.
- Hardware adder processes two binary numbers using bitwise addition and handles carries effectively.
- Example of adding +7 (00111) and -2 (11110 after conversion) results in valid arithmetic without requiring special sign considerations.
Handling Overflow
- An overflow condition occurs when adding two large numbers exceeds the maximum representable value.
- Check for overflow by assessing two positive numbers resulting in a negative value or two negative numbers resulting in a positive value.
Performing Subtraction
- Subtraction is executed by converting to addition of the negative counterpart.
- To find D = Y - X, compute D = -X + Y by flipping bits of X and adding 1.
Summary of Arithmetic Results
- Addition results should discard overflow bits, tracking carries with a carry register while monitoring for overflow conditions with an overflow register.
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Description
This quiz focuses on signed number representations in computer science, building upon the concepts of unsigned number systems. You will explore the limitations of representing signed quantities and how they relate to binary digits. Test your understanding of these fundamental concepts in the context of digital systems.
