This was actually easier than I thought it would be. Only took me about an hour and a half from start to finish. The input is in Big Endian (I think. Endianness confuses me.) binary on the left-hand side of the image. Output is marked with a piston and torch per bit towards the top of the image. The lone piston and torch are the carry bit for addition on the fourth bit.
The next step for this is probably gonna be changing the input. Instead of binary input, it'll have a single button or switch for numbers zero through seven. Nothing higher than eight because that will produce a number that is larger than four bits.
If this can hold my attention for that long, I may convert this to an eight or sixteen bit calculator.