Say I have a 4 bit ALU, I have a carry flag, overflow flag, and a sign flag(MSB). How would I go about subtracting for example, two signed 8 bit numbers? I take the lower nibble of both numbers and subtract them right, but I don't understand how to know if there needs to be a 5th bit, and carry that over to the LSB of the high nibble of the number, and if so, how to add it considering I am doing this in 2's complement so I already have Carryin being used.. Any help would be appreciated.
This has been asked and answered here many times.
You know the rule about twos complement yes? Invert and add one. Also from grade school
a + b = a + (-b).
We dont have subtract hardware we have add hardware. What you do is a + (-b). Also from grade school we learned about carrying, 9+3 = 2 carry the 1. And from the second column on we have either two or three operands that are added together (a + (-b) + c, c being the carry in). If you think about it we can have a carry in on every column, sometimes it is zero. That is how the hardware works, each column is three in two out, carry in[n], a[n], b[n] the output is result[n] and carry out[n]. and as we know from grade school the carry out of this column is the carry in of the next column. So for a normal add the carry in of the least significant bit is always a zero, but for subtract we want to invert and add one so what we do is invert b and change the carry in of that first bit to a 1 which is the same as
a + (~b) + 1 which equals a + (-b) which equals a - b.
As far as addition and subtract hardware is concerned there is no such thing as signed or unsigned add or subtract. There does exist an unsigned overflow (carry out of the msbit) and a signed overflow (true if carry in and carry out of the msbit are not the same, false if they match).
This works for any number of bits, for example if you have 8 bit hardware but want to do math on 256 bit numbers, just do them 8 bits at a time and apply the carry out to the next 8 bits (add with carry or subtract with borrow instruction). Visualize the single columns one at a time, 4 bits is just four of those columns, 8, 9 bits 37 bits, etc. You can easily take any of those larger numbers draw a vertical line anywhere separating it into two operations all you have to do is what you do for single columns the carry out of the msbit of the thing on the right becomes the carry in of the lsbit of the thing on the left of the dividing line. Apply this to 8 bit math with 4 bit hardware...
So a subtract is an add with a carry in of 1 and the second operand inverted. Now some hardware inverts the carry out (unsigned overflow) on a subtract so that it becomes 1 for borrow and 0 for not borrow (unsigned borrow/overflow). Some dont. So you have to know how this works if you dont have a subtract with borrow instruction. If you have a subtract with borrow it doesnt matter if they invert carry out they will generically invert carry in (on a subtract). If they dont then again they wont on a subtract with borrow. but if you have to use an add with carry to simulate a subtract with borrow you need to possibly not only invert the second operand but invert the carry bit. If you dont have an add with carry then you have to simulate that as well by simply adding 1 or not.