There's not really enough space to make an 8-bit adder (download logisim, it is far more awesome
You can make entire CPUs with it, and then you can make an assembler for your CPU, and then you can run programs on it, and all at a decent speed.)
Making an adder is really not that hard, you don't need to follow any electrical schematics, you just do it the same way you add numbers by hand.
For example, to calculate 0011 + 1010 you sum the bits from least significant to most significant carrying a 1 when necessary.
To calculate a single bit, you have three inputs: A (the bit from the first number), B (the bit from the second number) and C (the carry bit from the previous stage).
From these you need to calculate two outputs: D (the result bit) and E (the carry bit for the next stage).
From these you can write down the truth table:
A B C | D E
===========
0 0 0 | 0 0
0 0 1 | 1 0
0 1 0 | 1 0
0 1 1 | 0 1 <= (B AND C)
1 0 0 | 1 0
1 0 1 | 0 1 \
1 1 0 | 0 1 > A AND (B OR C)
1 1 1 | 1 1 /
From this you can see that D is simply the XOR of the inputs. (ie. if there are an odd number of 1s the result digit will also be a 1)
Finding E is a bit more complicated, but you can see if you look at which rows produce 1 for E that it's 1 when either B AND C are 1, OR when A is 1 AND either B OR C is 1.
So:
D = A xor B xor C
E = (B and C) or (A and (B or C))
You then repeat this set of logic gates for each pair of input bits feeding the carry out from the previous stage into the carry in for the next stage.
For the first stage, the carry in will always be zero, so you can simplify the circuit a bit, and for the last stage there's no need to calculate the carry out unless you want to check for an overflow.
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