Range of integer representation

[bob]

You see how integers fit into fixed bits in machines. I show you the limits right away. Unsigned numbers start at zero always. They reach up to two to the power of bits minus one. You grab eight bits and hit two hundred fifty five max. I notice that adding bits stretches the top end fast. Perhaps you wonder why it wraps around on overflow. Now think about what happens when you store bigger values anyway. You force the machine to cut them off oddly. But the range stays strict no matter what. Also maybe larger architectures pick sixty four bits for bigger room. I tell you this choice affects how your programs run on real hardware.
You tackle signed integers next and see the split. Negative values eat into the positive side. Two's complement flips bits and adds one for negatives. I explain the lower bound hits negative two to the power of bits minus one. You reach positive two to the power of bits minus one minus one on the high end. Or notice the asymmetry leaves one extra negative spot. Perhaps that quirk trips you up during calculations sometimes. Now you test with sixteen bits and find negative thirty two thousand seven hundred sixty eight to thirty two thousand seven hundred sixty seven. I watch how this setup lets addition work the same for both signs. You avoid separate circuits that way in the processor. But carry bits still need watching to catch overflows. Also the range grows with more bits yet memory costs rise too.
I recall you ask about thirty two bit systems often. They give unsigned from zero to four billion something. Signed versions go negative two billion to positive two billion roughly. You compare that to sixty four bits and the numbers explode hugely. Perhaps architecture designers pick widths based on speed needs. Now your code runs faster on narrower registers usually. But bigger ranges prevent weird wrap errors in big data tasks. I see how floating points differ but stick to integers here. You experiment with shifts and notice sign bits stay put in arithmetic moves. Or maybe you hit limits in embedded devices with tiny bit counts. The range decides if your counters last long enough before reset. I think hardware manuals list these bounds clearly for coders like you.
You ponder overflow effects in loops that count high. Machines just cycle back without warning often. I catch that in testing when values jump negative suddenly. Perhaps modern chips add checks but older ones skip them. Now you design around the known bounds to stay safe. The topic ties into how compilers pick types for variables too. You match bit widths to expected data sizes always. But picking wrong squeezes your numbers and breaks logic. Also cache lines favor certain widths for quick access. I notice this stuff shapes entire system performance in servers.
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