The Binary Adder

Write out the only four cases and read the answer off.

1. 0+0, 0+1, 1+0, 1+1 in binary.
2. Note that the last one is 10 in binary — two digits.
3. Tabulate the RIGHT digit (sum) and the LEFT digit (carry) separately.
4. Compare each column with the truth tables from the relay blueprint.

The sum column is exactly XOR and the carry column is exactly AND. Nothing has been invented — arithmetic has been recognised as logic you already have.

That is the entire conceptual step, and it is worth not rushing past. Addition looked like a different kind of operation from AND and OR, and it is not. Once numbers are written in binary, arithmetic IS Boolean algebra, which is why a machine built from switches can do sums.

Wire four full adders in a row and catch the wrong answer on its way to being right.

1. Build a 4-bit ripple-carry adder from logic ICs, with LEDs on every sum output and every carry.
2. Set the inputs to 0111 + 0001 — a sum that forces a carry through every stage.
3. Trigger the oscilloscope on the input change and watch the most significant sum bit.
4. Measure the time from input change to the final output settling.

For a brief moment the output shows a wrong answer, then corrects, possibly several times, before settling. Those intermediate values are real voltages on real wires — the machine genuinely holds wrong results while the carry travels.

Measure the delay per stage and multiply: a 32-bit ripple adder is 32 gate delays deep, and the clock cannot tick faster than that. The adder sets the processor's speed limit.

Combine the two halves of this batch: an adder that computes and flip-flops that remember.

1. Connect the sum outputs of your 4-bit adder to a register of four D-type flip-flops.
2. Feed the register outputs back to one input of the adder.
3. Put the number to be added on the other input, and pulse the clock.
4. Pulse it repeatedly with 1 on the input and watch the register count.

The register now holds a running total, and each clock adds the input to it. That is an accumulator — the central component of every early computer, and the reason so many machines had exactly one place where arithmetic happened.

Notice what became possible the moment memory joined arithmetic: results can be fed back into the next operation, so a sequence of steps can build on each other. Without the register the adder answers one question; with it, the machine can work.