Arithmetic Shifts and Integer Range Errors

A logical right shift fills new high positions with zeroes. An arithmetic right shift of a two’s-complement value repeats its sign bit. For eight-bit −6, pattern 11111010, shifting two positions logically gives 00111110=62, while arithmetic shifting gives 11111110=−2. These are different results from the same source bits.

Under this two’s-complement arithmetic-right-shift model, the signed result follows division by 2² with rounding down: −6/4=−1.5 becomes −2. It is not truncation towards zero. Read the question’s convention rather than importing an unrelated language’s shift rule. A left shift inserts low zeroes in both the usual logical and arithmetic pattern operation, but can lose high bits.

For eight-bit unsigned storage the range is 0 through 255. The true sum 255+1=256 exceeds the maximum; retaining only eight bits gives zero but loses the exact result. A result below the permitted lower bound is also a range failure: 0−1 cannot be represented as a non-negative unsigned value, even though a wrapped machine pattern may be 11111111.

Classroom discussions may call falling below a lower bound underflow. In machine arithmetic, signed range failures can also be described as overflow conditions. In the later IEEE floating-point topic, underflow has a different emphasis: tiny nonzero magnitudes approach the format’s subnormal/zero limit. State the representation and intended meaning instead of treating every negative value or every discarded fractional bit as underflow.

Eight-bit signed +64 is 01000000. A one-place left shift retains 10000000, interpreted as −128. The mathematical doubled result +128 lies above the signed maximum +127, so the pattern is not an exact signed multiplication result. The same retained pattern is unsigned 128 if the task uses that different interpretation.

A safe shift argument names the original value, width, signedness, fill rule, discarded bits and decoded result. Sign extension while widening a negative value is different from shifting it within the same width. Use examples at zero, the signed endpoints and just beyond each range to check whether a claimed conversion or arithmetic operation meets its contract.

Attempt the task before opening its worked response. Record each change of state, explain the rule used, and check the result against the original requirements. Code examples labelled Python 3 use that language’s conventions; pseudocode states its own assumptions.