How does a machine with no concept of numbers ADD? It chains full-adders — each takes two bits plus a carry-in, and outputs a sum bit and a carry-out that ripples to the next column, exactly like carrying the one by hand. Stack four and you add 4-bit numbers; stack sixty-four and you have a CPU’s adder. Slide the operands and watch the carry ripple.
A ripple-carry adder chains N full-adders. Each full-adder takes bits aᵢ, bᵢ, and carry-in cᵢ, producing sum sᵢ = aᵢ ⊕ bᵢ ⊕ cᵢ and carry-out cᵢ₊₁ = (aᵢ∧bᵢ) ∨ (cᵢ∧(aᵢ⊕bᵢ)). The carry propagates from the least-significant bit upward — the hardware version of ‘carry the one’. It is built entirely from the AND/OR/XOR gates of [[the-boolean-algebra]]; a final carry-out signals overflow. A fail-loud self-check throws unless the gate logic computes 3+1 = 4 with correct carry propagation. ◆ real digital arithmetic, node-verified.
Ripple-carry is the simplest (and slowest) adder — real CPUs use carry-lookahead to avoid waiting for the ripple; the per-bit full-adder logic and the result are exact. Shown at 4 bits for legibility.