Given a 8-bit processor, calculate 70 - 30 in binary and indicate if there is carry-out and overflow

I did the following exercise but even though I've revised my calculations several times, I don't seem to come up with the right answer:

Given a 8-bit processor and the variables $a=30$ and $b=70$, calculate $b-a$ in 2's complement. Indicate if there is a carry-out and overflow.

a. -40, there's overflow and carry-out

b. 40, there isn't overflow but there is carry-out

c. 40, there isn't overflow or carry-out

d. 296, there is overflow and carry-out

Apparently the right answer is b but I don't arrive to the same answer. This is what I do:

I first convert the previous numbers to the binary system.

$$70_{_{10}}=1000110_{2}$$ $$30_{_{10}}=11110_{2}$$

Since, we're dealing with 2's complements, I need to add a 0 at the beggining of each binary number. There:

$$70_{_{10}}=01000110_{2}$$ $$30_{_{10}}=011110_{2}$$

Since, the exercise wants me to subtract, I need to convert 30 to -30. To do that, I flip all the digits of 30 and add 1 up:

$$30_{10}=011110_{2}\Rightarrow 100001_{2}$$ $$100001_{2} + 1_{2} = 100010_{2}$$ Then, $$-30_{10} = 100010_{2}$$ Therefore, I just need to sum them:

(Since there are more bits in 70 than in -30, I have to add 1s at the beggining of the binary number that represents -30)

Hence, I've found overflow and carry-out and it gives me $100101000_{2}$, which is -216.

Does anyone see the problem?

• Unless 70-30 gives you 40, there is a problem with your calculations. You need to fill out the binary representation of 30 to 8 bits before you take the 2-s complement. – saulspatz Mar 28 '18 at 21:38
• What are the definitions of "overflow" and "carry out" for your processor? These are not standard terms with a fixed mathematical meaning. Moreover, in subtraction, you don't "carry" but rather you "borrow" and you need to say how that is represented in the processor of interest. – Rob Arthan Mar 28 '18 at 21:51
• Bit 8 is the sign bit. Carry-out into bit 9 is discarded. – dan post Mar 28 '18 at 22:09
• @RobArthan As far as I know, when you're summing them, if you sum 1+1, it means that you carry out a 1 and I'd the result has more bits than the processor, overflow occurs – Arnau Mar 28 '18 at 22:42
• "As far as I know" isn't good enough. You need to work from a precise definition. – Rob Arthan Mar 28 '18 at 22:57

$$\underbrace{/\!\!\!1}_{\text{carry out}}\underbrace{0010}_{2}\;\underbrace{1000_2}_{8} = 28_{16} = 40_{10}$$ there is a carry out, but there is no overflow bit.
The overflow checks the most significant bit of the 8 bit result. This is the sign bit. If we add two negative numbers (MSBs$=1$) then the result should be negative (MSB$=1$), whereas if we add two positive numbers (MSBs$=0$) then the result should be positive (MSBs$=0$), so the MSB of the result must be consistent with the MSBs of the summands if the operation was successful, otherwise the overflow bit is set.