Here is a circuit for a 4-bit adder that uses only 2 Memory Banks . The picture shows the input and output for the circuit as well. The I/O takes up much more space than the circuit itself – the block on the right is the actual circuit. That’s all it takes!
The design is extremely simple. The two inputs go to the two banks in parallel. One bank is programmed for the 4-bit output sum. The other bank calculates the carry output bit. For a simple 4-bit core, that’s all you need. For a larger word size, the higher order modules are more complex to implement due to the carry, but they work on the same principle. Also, your numerical representation can affect the circuit. This circuit assumes unsigned/positive integers for simplicity.
You program the primary memory bank with:
Compute the carry with:
That circuit works well enough if that's all you need. Usually an addition requires more than 4 bits, however. That means we need to have another, more complex circuit for higher 'nibbles' of data. (A nibble is a 4-bit chunk of data, since a byte is 8-bits. LOL but seriously!) This circuit needs to have a carry IN, as well as a carry OUT.
The diagram below is for a full 8-bit addition module. At the top of the picture is the basic adder, with its carry out. This works for the lowest nibble of data since it needs no carry IN. The portion in the dotted box is the FULL ADDER which includes a carry IN and the carry OUT. You can extend this design for arbitrarily large data widths by duplicating this portion and chaining the carry OUT from the module for the lower data to the carry In of the next higher module.
There are no connections between crossed lines in this schematic, only at the 'T's.
Each square box represents a Memory Bank. The ones with a "+" in it perform the primary addition and are programmed as listed above. The ones with a "C" in them perform the carry addition and are programmed with the proper data shown above.
Adding the carry IN more than doubles the complexity of the circuit since it now performs two additions. The two operands are added and the carry is added to that result. (When building it, the carry could be added to one of the operands first and there would be no difference.) When either of these additions produce a carry out, it must be propagated to the next circuit, or to the output. Therefore the two carries must be OR'd together.
The full adder can be found in my Computer Circuits world here
Other (temp holding)
To implement a logic compare output bits: 0 is =, 1 is <, 2 is >. Bit 3 could be any, such as 'complement'. program with: 1444444444444444214444444444444422144444444444442221444444444444222214444444444422222144444444442222221444444444222222214444444422222222144444442222222221444444222222222214444422222222222144442222222222221444222222222222214422222222222222142222222222222221
A subtraction circuit depends more on your numerical representation scheme. I.e., whether you use signed/unsigned/1’s complement/2’s complement integers. All those options make it tough to present in a simple message reply. However, you can usually subtract by complementing the ‘subtrahend’ and adding them. What ‘complement’ means depends on the representation. I’ll cover subtraction in more detail when I get to it.