➕ Half and Full Adders

Written and maintained by the PhDino author · Last reviewed 21 September 2026 · Checked against 3 independent reference calculations · how PhDino checks its numbers

The basic building blocks that add binary digits together, and how they chain into multi-bit addition.

A half adder takes two single bits and adds them, producing a sum bit and a carry-out bit — it's called "half" because it has no way to accept a carry-in from a previous, less-significant bit position, which is exactly what's needed to add numbers wider than a single bit. A full adder adds that missing third input (carry-in), making it possible to chain full adders together — each one's carry-out feeding the next one's carry-in — to add binary numbers of any width, one bit position at a time.

This chaining is literally how a basic ripple-carry adder circuit works inside real arithmetic logic units: n single-bit full adders wired in a chain compute the sum of two n-bit binary numbers, with the carry propagating ("rippling") from the least significant bit up to the most significant.

Key formula

Sum = (A + B + Carry-in) mod 2
Carry-out = 1 if (A + B + Carry-in) ≥ 2, else 0

Notes & limitations

• Setting carry-in to 0 makes this behave exactly like a half adder, since a half adder is just the carry-in-always-zero special case of a full adder — this calculator handles both with the same single formula.
• A ripple-carry adder built this way is simple but not the fastest possible adder circuit — the carry has to physically propagate through every bit position in sequence, which is why faster (but more complex) designs like carry-lookahead adders exist for applications where addition speed matters more than gate count.