If you are conversant with the working of computers, you understand they use a binary digit system. Also, in a digital circuit, there's a binary digit system made of adders. Therefore, if you are keen on understanding the arithmetic circuits, you should know the working of adders. Typically, we have two main types of adders that combine binary inputs, including the full adder vs half adder. Figure 1: An illustration of a Digital Processor

It is a logic circuit with three inputs: an OR gate, 2 AND logic gates, and 2 EX-OR logic gates. It is a combinational circuit which means it does not feature a storage property. Nonetheless, it features additional logic gates making it fit for complex arithmetic operations.

It is also a multibit adder capable of adding two one-bit binary numbers. The multi-bit operation enables the addition of several input digits to create a series. Hence, it is also a ripple-carry adder.

In the adder module, you will find the first two inputs as A and B in most circuits. The third input is an additional carry-In (Cin) input. For the outputs, C-out represents the output carry while S or SUM represents the standard output.

In this case, the input carry represents what the circuit had in the previous carry. Therefore, the complete adder circuit combines the three input bits to give two outputs, and one of them is the standard output, while the other is the output carry.

You can also create a full adder using a single OR gate and two half adders.

Note that when you add the binary digits, you will obtain the equation at the EX-OR gate. On the other hand, the output from the AND gate is the effect of the addition of the carry.

Here is the Boolean expression of the C-out:  AB + BC + AC.

The C in this equation represents the Carry-in input, ideally the previous carry input. Figure 2: A Man Using a Calculator

A half adder like the full adder is also a combinational logic circuit. However, unlike the full adder with three inputs, a half adder only comprises an EX-OR gate and an AND gate. It is a 1-bit adder that will add the two input bits.

It consists of two input terminals A and B, and two output terminals. One gate output is the half-adder sum output, while the other is the half-adder carry output. The EX-OR gate will provide an output which is a sum of two input values.

Nonetheless, for the half adder, the carry-in one addition does not reflect in the next addition. Ideally, the cause of this is because there lacks a logic gate to facilitate the operation. Hence, that is where it derives the name half-adder circuit as it lacks the full-adder characteristics.

Here is the equation for the logical operation of the two-bit output circuit.

S = A + B. The sum is the addition of the two inputs.

On the other hand, the Carry (C) for the half adder is A x B.

## Truth Table of Full Adder Figure 3: A Full Adder Truth Table

From the above truth table, we can create the logical expression for sum and carry output. We have already explained the two equations above.

Consequently, the final Boolean expression for the above truth table is

C (out) = AB + A(Cin) + B(Cin)

Here are the necessary deductions from the truth table:

• The combinations that result in ones will give a high on the sum output. Such varieties include 010, 001, and 100. When there is only one high at the second XOR gate, the sum output is high.
• The result is a low sum and high carry out for the combinations that lead to a two. They include 101, 110, and 011.

## Truth Table of Half Adder Figure 4: A Half Adder Truth Table Figure 5: An Illustration of the Binary System

They are fast adders that improve the time of deciphering the carry bits in digital electronics. A carry-lookahead adder is different from the typical ripple carry adder.  In logic operations of the latter, the sum bit and carry bits require simultaneous calculation.

However, each calculation has to wait for the other to end so as the other one commences. The Carry lookahead adders seek to solve this lengthy process.

Carry lookahead adders lead to a significant aggressive reduction of the wait time of calculations. The reduction process is possible as these adders calculate more than one carry bit before the sum. Also, the bits to be added are available immediately.

Ideally, these kinds of adders do not have to await the arrival of the carry from the former black. Hence, the carry-lookahead adders can handle 4-bit blocks faster than the typical one-bit full adder. Figure 6: An Illustration of a Complex Mathematical Expressions

### Similarities

1. Both the full adder and half adder are combinational digital circuits. Unlike sequential circuits, the two lack a memory perspective.
2. They both provide a means of performing arithmetic operations. Any arithmetic circuit will therefore feature one of them.

### Differences

1. The half adder will add two binary inputs to give out a carry and a sum. On the other hand, the full adder will add three binary inputs also to generate the carry and sum. Hence, there is a significant difference in their hardware architectures.
2. In half adder, the carry from the previous addition is inconsequential as it does not add to the next step. On the other hand, in the full adder digital logic, the carry from the previous step is helpful in the subsequent addition.
3. There are two logic gates in the half adder that include the EX-OR gate and the AND gate. The full adder features three logic gates: an OR gate, three AND gates, and two EX-OR gates.
4. The half adder consists of two input bits, A and B, while the full adder features three input bits. In addition to the A and B, there is an additional C input bit.
5. The carry equations of the two adders are also different. For the half adder, the Sum = A + B, and C = A x B. For the full adder, the Sum = A + B + Cin and C= A x B + Cin  (A + B). Figure 7: A Calculator

1. The adder facilitates the addition of two single-bit numbers in digital arithmetic operations.
2. It is also easy to design.

1. They facilitate adding the carry from the previous operation, and half adders cannot perform such an action.
2. The full adder is essential in the creation of critical circuits such as multiplexers.
3. The power consumption by the full adder is relatively less than for the half adder circuits.
4. Adding an inverter to the full adder will facilitate the creation of a half subtractor.
5. A full adder will also give a higher output than the half adder, allowing more combinations that facilitate the increased work.
6. The speed of the full adder during applications is also relatively higher than that of the half adder.
7. The full adder is also strong enough to support voltage scaling.

1. The half adder is not fit for carrying the addition of the previous operation since it has two inputs. Hence, it will not apply in conducting multibit operations.
2. When the full adder is in the form of a ripple adder, the output drive capability diminishes. Figure 8: A Binary Numbers Illustration

1. The half adder can be applicable in the creation of complete adder combinations.
2. Full adders are useful in Arithmetic Logic Unit (ALU) systems.
3. The binary addition property of half adders is applicable in the working of calculators.
4. Full adders are helpful in various forms of digital circuits and digital electronics.
6. Full adders are applicable in the generation of memory addresses and the creation of program counterpoints.
7. The Full adders are essential in the creation of complex circuits capable of adding numerous bits simultaneously.
8. Full adders are critical components in the creation of the graphical processing unit (GPU).