An adder is a digital circuit that performs addition of numbers. In many computers and other kinds of processors adders are used in the arithmetic logic units or ALU. They are also utilized in other parts of the processor, where they are used to calculate addresses, table indices, increment and decrement operators, and similar operations.

Although adders can be constructed for many number representations, such as binary-coded decimal or excess-3, the most common adders operate on binary numbers. In cases where two's complement or ones' complement is being used to represent negative numbers, it is trivial to modify an adder into an adder–subtractor. Other signed number representations require more logic around the basic adder.

Inputs Outputs
A B C S
0 0 0 0
1 0 0 1
0 1 0 1
1 1 1 0
Half adder using NAND gates only.

Logic diagram for a full adder.
Full adder in action. A full adder gives the number of 1s in the input in binary representation.
Schematic symbol for a 1-bit full adder with Cin and Cout drawn on sides of block to emphasize their use in a multi-bit adder

A full adder adds binary numbers and accounts for values carried in as well as out. A one-bit full-adder adds three one-bit numbers, often written as A, B, and Cin; A and B are the operands, and Cin is a bit carried in from the previous less-significant stage.[2] The full adder is usually a component in a cascade of adders, which add 8, 16, 32, etc. bit binary numbers. The circuit produces a two-bit output. Output carry and sum typically represented by the signals Cout and S, where ${\displaystyle {\text{sum}}=2\times C_{\text{out}}+S}$ in decimal system.

A full adder can be implemented in many different ways such as with a custom transistor-level circuit or composed of other gates. One example implementation is with ${\displaystyle S=A\oplus B\oplus C_{\text{in}}}$ and ${\displaystyle C_{\text{out}}=(A\cdot B)+(C_{\text{in}}\cdot (A\oplus B))}$.

In this implementation, the final OR gate before the carry-out output may be replaced by an XOR gate without altering the resulting logic. Using only two types of gates is convenient if the circuit is being implemented using simple IC chips which contain only one gate type per chip.

A full adder can also be constructed from two half adders by connecting ${\displaystyle A}$ and ${\displaystyle B}$ to the input of one half adder, then taking its sum-output(S) as one of the inputs to the second half-adder and ${\displaystyle C}$${\displaystyle in}$ as its other input, and finally the carry-outputs from the two half-adders are connected to an OR gate. The sum-output from the second half-adder is the final Sum-output (${\displaystyle S}$) of the Full-Adder and the output from the OR gate is the final Carry-output (${\displaystyle C}$${\displaystyle out}$). The critical path of a full adder runs through both XOR-gates and ends at the sum bit ${\displaystyle s}$. Assumed that an XOR-gate takes 1 delays to complete, the delay imposed by the critical path of a full adder is equal to

${\displaystyle T_{\text{FA}}=2\cdot T_{\text{XOR}}=2D.}$

The critical path of a carry runs through 1 XOR-gate in adder and through 2 gates (AND and OR) in carry-block and therefore, if AND- or OR-gates takes 1 delay to complete, has a delay of

${\displaystyle T_{c}=T_{\text{XOR}}+T_{\text{AND}}+T_{\text{OR}}=D+D+D=3D.}$

The truth table for the full adder is:

Inputs Outputs
A B Cin Cout S
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1

4-bit adder with logical block diagram shown

It is possible to create a logical circuit using multiple full adders to add N-bit numbers. Each full adder inputs a Cin, which is the Cout of the previous adder. This kind of adder is called a ripple-carry adder, since each carry bit "ripples" to the next full adder. Note that the first (and only the first) full adder may be replaced by a half adder (under the assumption that Cin = 0).

${\displaystyle T_{\text{CRA}}(n)=(n-1)\cdot T_{c}+T_{s}=(n-1)\cdot 3D+2D=(3n-1)D.}$

The delay from bit position 0 to the carry-out is a little different:

${\displaystyle T_{{\text{CRA}}_{[0:c_{\text{out}}]}}=T_{{\text{CRA}}_{[c_{0}:c_{n}]}}(n)=n\cdot T_{c}=n\cdot 3D=3nD.}$

The carry-in must travel through n XOR-gates in adders and n carry-generator blocks to have an effect on the carry-out.

A design with alternating carry polarities and optimized AND-OR-Invert gates can be about twice as fast.[4]

To reduce the computation time, engineers devised faster ways to add two binary numbers by using carry-lookahead adders. They work by creating two signals (P and G) for each bit position, based on whether a carry is propagated through from a less significant bit position (at least one input is a 1), generated in that bit position (both inputs are 1), or killed in that bit position (both inputs are 0). In most cases, P is simply the sum output of a half adder and G is the carry output of the same adder. After P and G are generated, the carries for every bit position are created. Some advanced carry-lookahead architectures are the Manchester carry chain, Brent–Kung adder, and the Kogge–Stone adder.

Some other multi-bit adder architectures break the adder into blocks. It is possible to vary the length of these blocks based on the propagation delay of the circuits to optimize computation time. These block based adders include the carry-skip (or carry-bypass) adder which will determine P and G values for each block rather than each bit, and the carry select adder which pre-generates the sum and carry values for either possible carry input (0 or 1) to the block, using multiplexers to select the appropriate result when the carry bit is known.

By combining multiple carry-lookahead adders, even larger adders can be created. This can be used at multiple levels to make even larger adders. For example, the following adder is a 64-bit adder that uses four 16-bit CLAs with two levels of LCUs.