3-bit binary adder: three full adders, live

A 3-bit binary adder adds two 3-bit numbers A2 A1 A0 and B2 B1 B0 plus a carry-in, giving a 3-bit sum S2 S1 S0 and a carry-out. It is three full adders in a row, each passing its carry to the next: a ripple-carry adder.

Binary addition works column by column, exactly like decimal addition: each column adds its two bits and the carry from the column to its right, writes the low bit of the result and carries the high bit left. A full adder does one column, so three of them chained add three bits. For example 5 + 6: 101 + 110 gives S = 011 with Cout = 1, which is 1011 = 11. The largest sum, 7 + 7 + 1 = 15, needs all four output bits. The carry has to ripple through every stage before the top sum bit is final, which is why wide adders use faster schemes such as the carry look-ahead adder. Each FA block here is the verified full adder, saved as a part; open the circuit in the lab to look inside. Smaller and larger versions: the 2-bit adder, the 4-bit binary adder and the 8-bit adder.

Live circuit: click the inputs to toggle them and watch the outputs update. It runs the real simulator, right here on the page.

Open this circuit in the lab →

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Free to embed on a course page, blog or worksheet: paste this code. The circuit stays live, and the small digiwleea badge links back here.

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To cite it: digiwleea, “3-bit binary adder”, https://digiwleea.wleeaf.dev/tools/build/adder-3bit/

What you need

3× full adder: one per bit: Si = Ai XOR Bi XOR Ci, carry to the next stage

Step by step

Place inputs A2 to A0, B2 to B0 and Cin, three full adders and probes S2 to S0 and Cout.
Bit 0: A0, B0 and Cin into the first full adder; its sum is S0.
Bits 1 and 2: Ai, Bi and the carry-out of the stage below into the next full adder; its sum is Si.
The last carry-out is Cout, the fourth bit of the result.
Try 101 + 110: S shows 011 and Cout is 1, so the result is 1011 = 11.

3-bit adder: sample rows

A	B	Cin	Cout S2 S1 S0	Decimal
011	001	0	0100	3 + 1 = 4
101	110	0	1011	5 + 6 = 11
111	001	0	1000	7 + 1 = 8
010	010	1	0101	2 + 2 + 1 = 5
111	111	1	1111	7 + 7 + 1 = 15

The full truth table has 128 rows (six data bits and a carry-in); all of them are checked on the live circuit against real addition before the page is built.

You can also read the full theory in the lesson on the full adder. Prefer to build it yourself from scratch? Open the digiwleea lab and place the gates on the canvas, or start from the logic gate simulator.

More builds: Arithmetic

2-bit adder 4-bit incrementer Full subtractor using a decoder 4-bit decrementer Full adder using a decoder 4-bit ripple carry adder full adder using two half adders 4-bit carry lookahead adder full adder 4-bit carry lookahead generator half adder 4-bit carry-select adder

All 246 build guides, by topic

Each of these is one step from a transistor to a working CPU. Build every gate from CMOS transistors and work all the way up in the free digiwleea lab, with the free theory reference alongside.

Open the 3-bit binary adder in the lab →

Frequently asked

How do you design a 3-bit binary adder?

Chain three full adders. The first adds A0, B0 and the carry-in; each next one adds its bits and the carry from the stage below. The sums are S0 to S2, and the last carry is the carry-out.

How many full adders are needed for a 3-bit adder?

Three, one per bit. If there is no carry-in, the lowest stage can be a half adder instead.

What is the largest sum of a 3-bit adder?

7 + 7 = 14, or 15 with a carry-in of 1, which is 1111: three sum bits plus the carry-out.

Why is it called a ripple-carry adder?

Because the carry ripples from the lowest stage to the highest, one full adder at a time, so the top sum bit is only correct after the carry has passed through every stage.