Consider Z = X - Y, where X, Y and Z are all in sign-magnitude form. X and Y are each represented in

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GATE-CS is a competitive examination conducted in India. This question is from the 2019 GATE-CS Digital Logic paper.

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Question

Consider Z = X - Y, where X, Y and Z are all in sign-magnitude form. X and Y are each represented in n bits. To avoid overflow, the representation of Z would require a minimum of:

Accepted Answer

Correct Answer: (C) n+1 bitsConcept: Sign-Magnitude Arithmetic and OverflowIn sign-magnitude representation with n bits: 1 bit is the sign bit and n-1 bits represent the magnitude. The maximum positive magnitude representable is 2^(n-1) - 1.When computing Z = X - Y in sign-magnitude, subtraction can be equivalent to addition of magnitudes (when signs differ). The worst case occurs when X is the largest positive number and Y is the most negative number (or vice versa): |Z| can be as large as (2^(n-1) - 1) + (2^(n-1) - 1) = 2^n - 2. This value requires n bits to represent the magnitude (since 2^n - 2 needs n bits: log₂(2^n-2) ≈ n). Including the sign bit, Z requires n+1 bits total. Therefore, n+1 bits are needed to guarantee no overflow.Definition: Overflow occurs when the result of an arithmetic operation requires more bits than available to represent it correctly. In sign-magnitude, adding two numbers of opposite signs can produce a result whose magnitude is the sum of the magnitudes.

Consider Z = X - Y, where X, Y and Z are all in sign-magnitude form. X and Y are each represented in n bits. To avoid overflow, the representation of Z would require a minimum of:

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Consider Z = X - Y, where X, Y and Z are all in sign-magnitude form. X and Y are each represented in n bits. To avoid overflow, the representation of Z would require a minimum of:

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