Two-bit binary adder, built from NAND gates

2026-08-09

Two bits plus two bits, with a carry in and a carry out. Flip the switches on the left rail and the answer appears on the right.

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Underneath there are exactly two parts: an ADDER for the low bit and another for the high one, with the low adder’s carry-out wired into the high adder’s carry-in. That single wire is the whole idea of ripple-carry arithmetic, and it is also its weakness — the high bit cannot settle until the low bit has, so a wide adder built this way gets slower in proportion to its width.

What one bit of addition costs

Click either ADDER and it opens as a workspace of its own. Inside are two XORs, two ANDs and an OR; open the XOR and you find four NANDs. Follow it all the way down and the whole board is 30 NAND gates — about 120 transistors in CMOS to add two two-bit numbers.

The sum bit is A XOR B XOR CIN. The carry-out is high when at least two of the three inputs are high, which the adder computes as (A AND B) OR ((A XOR B) AND CIN).

Things worth trying

• Set both inputs to 11 with the carry in high: the result is 111, three bits from two two-bit numbers, which is why COUT exists at all.
• Chain further: the ADD-4 module in the library is the same trick with four stages, and ADDSUB-4 adds a SUB line that flips every B bit and rides into the carry-in — turning A + B into A - B with no extra adder.

Built out of NAND gates and nothing else. The sandbox, the JSON module format and the rest of the library are described in Digital logic — from NAND to a computer.

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