3) Let R1 and R2 be two 4-bit registers that store numbers in 2's complement form. For the operation R1 + R2, which one of the following values of R1 and R2 gives an arithmetic overflow?
Question
- Let R1 and R2 be two 4-bit registers that store numbers in 2's complement form. For the operation R1 + R2, which one of the following values of R1 and R2 gives an arithmetic overflow?
Solution
An arithmetic overflow occurs when the result of an operation is larger than the maximum value that can be stored in the registers. In 2's complement form, a 4-bit register can store values from -8 to 7.
Let's consider the possible values of R1 and R2:
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R1 = 7 (0111 in binary), R2 = 1 (0001 in binary). The sum is 8, which cannot be represented in a 4-bit register with 2's complement. So, this pair causes an arithmetic overflow.
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R1 = -8 (1000 in binary), R2 = -1 (1111 in binary). The sum is -9, which also cannot be represented in a 4-bit register with 2's complement. So, this pair also causes an arithmetic overflow.
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Any other pair of values will result in a sum that can be represented in a 4-bit register with 2's complement, so they will not cause an arithmetic overflow.
So, the pairs of values that cause an arithmetic overflow are R1 = 7, R2 = 1 and R1 = -8, R2 = -1.
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