What sir rrr314159 have said, I asked to 8 eletronic teachers about and only one explain to me into that way, other 7 don't answer me.

What he is saying is that an overflow, underflow, carry, ... can happen with only 2 bits, we don't need a byte.

The idea about the leftmost bit being a signal bit (2 complement) stay on limb (limbo) for years, nobody give value to this idea, until one person say: Hey, we can say that the left most bit is a signal, wow, 0 it's positive and 1 it's negative. But you get the point when you think minimalistic way. This way you meet invalid values like -0==+0?

If you deal with bytes, what's the signed number of value 00h?, other one, and about number 80h? I think NaN (not a number) will start making sense. And, this will never happens if we look to the things with unsigned eyes. We lost values like in any other way, but we can expand unsigned numbers ad infinitum, while on signed numbers we are creating a limit, a jail that's the leftmost digit.

I never see on books the easy way to convert betwen decimal to binary, the way I use is: start from number 1, go to left side multiplying to 2.

.. _ _ _ _ _

.. 16 8 4 2 1

So number 10 (now from left to right) in decimal base is 8+2, we just put numbers 1 on that digits, 01010b. We are subtracting 10-8=2, so we continue walking until find another one that is less or equal to remainder. If remainder is 0 we end, if not continue. So to multiply we use shl, to divide we use shr, but now we loose remainder, that's why that and mask.

NaN can happen with interger signed numbers. And here we are talking about how hackers use their buffer overflow exploit. Because we started count by 1 and not by 0. A byte have 256 possibilities, from 0 to 255 (not 256, 0 is a number, should be counted).