Subtractors

Before you read this section, make sure you are comfortable with adders. Subtraction, after all, can be modeled as addition. Observe that, 5 - 2 is equivalent to 5 + (-2).

How Do We Represent Negative Numbers?

We would need an extra bit of information to convey parity. So, 0 for positive and 1 for negative.

Ok great, now what? Say we use three bits of information and one leading bit for parity: 0101 is 5 and 1010 represents -2. Using a regular adder would yield 1111. Hm, not quite right.

Observe that although -2 represented as 1010 does not yield the right result, adding 16 to this representation (so adding 14 instead) does, and this produces an overflow carry we disregard:

                                    0101
                                    1110
                                    ----
                                   10011
        

The two values are synonymous due to modular arithmetic, where we take 2^(number of bits) - (value) to be added instead. So for -2, this could be 14, 30, 46,…, etc., but we choose 14 since we can only represent that given our number of bits.

16 = 2 (mod 14)

Think about modular arithmetic like a clock; -2 can be thought of as -2 o'clock or 10 o'clock. 10 o'clock is more useful for us. This is called two's complement. We can flip all the bits and add 1 (15 - X + 1), which is equivalent to (16 - X) or the modulus. Note that this can be extended to 5 bits (32) and so forth.

So 2 is 0010, but if we do 15 - 2, (or observe that this operation flips all bits), we get 1101. If we add 1, we get 1110, which is what we listed above. Going from a negative digit to a positive digit entails the same process.

In Minecraft, all you would need to do is negate your inputs and add one to an input.