An adder, or summer, 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 (ALUs). They are also used 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.

George Stibitz invented the 2-bit binary adder (the Model K) in 1937.

The half adder adds two single binary digits ${\displaystyle A}$ and ${\displaystyle B}$. It has two outputs, sum (${\displaystyle S}$) and carry (${\displaystyle C}$). The carry signal represents an overflow into the next digit of a multi-digit addition. The value of the sum is ${\displaystyle 2C+S}$. The simplest half-adder design, pictured on the right, incorporates an XOR gate for ${\displaystyle S}$ and an AND gate for ${\displaystyle C}$. The Boolean logic for the sum (in this case ${\displaystyle S}$) will be ${\displaystyle A\oplus B}$ whereas for the carry (${\displaystyle C}$) will be ${\displaystyle A\cdot B}$. With the addition of an OR gate to combine their carry outputs, two half adders can be combined to make a full adder.

The truth table for the half adder is:

Various half adder digital logic circuits:

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 ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C_{in}}$; ${\displaystyle A}$ and ${\displaystyle B}$ are the operands, and ${\displaystyle C_{in}}$ is a bit carried in from the previous less-significant stage. The circuit produces a two-bit output. Output carry and sum are typically represented by the signals ${\displaystyle C_{out}}$ and ${\displaystyle S}$, where the sum equals ${\displaystyle 2C_{out}+S}$. The full adder is usually a component in a cascade of adders, which add 8, 16, 32, etc. bit binary numbers.

A full adder can be implemented in many different ways such as with a custom transistor-level circuit or composed of other gates. The most common implementation is with: