IntroEdit
When adding one number A with another number B the operation will produce a Sum S and a Carry C.The Operation of an adder is Shown below.
Digital AdderEdit
Half adderEdit
If there are two binary number A and B . The operation of adding two numbers can be shown below
 A + B = S C
 0 + 0 = Sum 0 Carry 0
 0 + 1 = Sum 1 Carry 0
 1 + 0 = Sum 1 Carry 0
 1 + 1 = Sum 0 Carry 1
The operation of Half Adder can be summarized in the Truth Table below

0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0
From above, In term of logic gate XOR will produce a sum of two input . logic gate AND produce carry
Half Adder can constructed from AND gate and XOR gate as shown below
___________ A    Half   Adder    B ___________
Full adderEdit
A full adder is a logical circuit that performs an addition operation on three onebit binary numbers. The full adder produces a sum of the two inputs and carry value. It can be combined with other full adders (see below) or work on its own.

Input Output 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1
Note that the final OR gate before the carryout output may be replaced by an XOR gate without altering the resulting logic. This is because the only difference between OR and XOR gates occurs when both inputs are 1; for the adder shown here, this is never possible. Using only two types of gates is convenient if one desires to implement the adder directly using common IC chips.
A full adder can be constructed from two half adders by connecting A and B to the input of one half adder, connecting the sum from that to an input to the second adder, connecting C_{i} to the other input and OR the two carry outputs. Equivalently, S could be made the threebit XOR of A, B, and C_{i}, and C_{o} could be made the threebit majority function of A, B, and C_{i}.
Multiplebit addersEdit
Ripple carry adderEdit
It is possible to create a logical circuit using multiple full adders to add Nbit numbers. Each full adder inputs a C_{in}, which is the C_{out} of the previous adder. This kind of adder is 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.
The layout of ripple carry adder is simple, which allows for fast design time; however, the ripple carry adder is relatively slow, since each full adder must wait for the carry bit to be calculated from the previous full adder. The gate delay can easily be calculated by inspection of the full adder circuit. Each full adder requires three levels of logic. In a 32bit [ripple carry] adder, there 32 full adders,so the critical path (worst case) delay is gate delays.
Carry lookahead addersEdit
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'), a carry is generated in that bit position (both inputs are '1'), or if a carry is killed in that bit position (both inputs are '0'). In most cases, P is simply the sum output of a halfadder 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 look ahead architectures are the Manchester carry chain, BrentKung adder, and the KoggeStone adder.
Some other multibit 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 bypass adder which will determine P and G values for each block rather than each bit, and the carry select adder which pregenerates sum and carry values for either possible carry input to the block.
Other adder designs include the conditional sum adder, carry skip adder, and carry complete adder.
Lookahead Carry UnitEdit
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 64bit adder that uses four 16bit CLAs with two levels of LCUs.