A half adder (HA), is an electronic component used to add two bits. It is distinguished from the full adder by the lack of a carry in input. A half adder does, however, have a carry out output.

Typically a half adder would be used to add the least significant bits in a ripple adder, as this addition can have no carry in. It is also significantly less complex than a full adder, and saves on hardware in the situation where a carry in is not needed.

The HA might be implemented as follows:

SUM = A xor B

Cout = A · B

Where A and B are the two bits to be added, and Cout is the bit carried out.

A possible symbol:

     |
     |
  +--+-------+
  |  Cout    |
--|A      Sum|--
--|B  HA     |
  |          |
  +----------+

A half adder is an important part of of what may be called a full adder. The full adder has the ability to add any two binary digits together, whether the output has a carry or not.

The half adder can only add two binary digits together and output one digit. The carry is not taken into account.

		Full adder				Half Adder
			 1					1
		+	 1				+	1
			10					0

With the half adder we only get a one digit output.

Well how do we make one of these 'half adders'?
We use things called logic gates.

Note: Before you continue reading, please ensure you have read the logic gates node, I say this so you will have a basic understanding of what is to follow.
First of all we'll need to draw up a truth table of our desired outputs. We will use this truth table to test our logic gate diagram. Having two inputs you will have only 4 possible binary combinations.
I1  I2     O
____________
0   0   |  0
0   1   |  1
1   0   |  1
1   1   |  0

Looking at our output column we can see that the pattern of outputs is somewhat similiar to the output of an OR gate. Shown below.

OR    O
_______
0  |  0
1  |  1
1  |  1
1  |  0

The only output that is different is when both inputs are 1. We'll start the diagram with an OR gate, this will take care of the first three inputs. But we also need something to take care of the last output.

Note: I'd probably defeat my own purpose if I tried to explain the complete logic gate diagram in words so I'll make a bad attempt at ASCII art.
First we need an OR gate.
           ___
I1  -------\   `-._
            \      `.______
            /     _,'
I2  -------/___,-'
We now take the output of the OR and feed it into an AND gate:
            ___
I1  --------\   `-._            ___
             \      `._________|   `-.
             /     _,'         |      \_______
I2  --------/___,-'            |      /
                               |___,-'

As we look at the output of the AND gate we can see that if one input is 0 then the output will also be 0. This is a very important discovery!

  AND Gate
I1  I2     O
____________
0   0   |  0
0   1   |  0
1   0   |  0
1   1   |  1
Using this important information, we now know that we can use the AND gate as a switch of sorts. How? Well, consider this: If we were to make one input of an AND gate a 1, then the output would be the same as the second input. Take this for example:
  AND Gate
I1  I2     O
____________
1   0   |  0
1   1   |  1

So, when we know the first input of an AND gate will be 1, then we know that the output of that AND gate will be the same as the second input.

In saying that, when I1 is 1 then the switch is open and I2 will go through as the output. This is why we have an AND gate after the OR. The OR will give us an output of 1 if it has an input of 1. So the AND gate/switch will be open whenever there is a 1 as an input.

And having established that, we can now carry on with the logic gate diagram.

So far we have:
            ___
I1  --------\   `-._            ___
             \      `._________|   `-.
             /     _,'         |      \_______
I2  --------/___,-'            |      /
                               |___,-'
We now need to have a look at our AND gate's second input. We know that the first two inputs of 0 and 0 will give us a 0 as the AND gates first input. This means that the output will, of course, be a 0. So all we need to look at is the next three inputs. We want them to be as follows.
When...      We want...
I1  I2
_________________
0   1        |  1
1   0        |  1
1   1        |  0
Looking at these three outputs we look and see if there is a pattern... and is there? We already know that a normal AND gate's output is:
  AND Gate
I1  I2     O
____________
0   0   |  0
0   1   |  0
1   0   |  0
1   1   |  1
We're only interested in the last three outputs though. Lets compare them with our desired outputs.
Desired      AND Gate's
Output Output
_____________ 1 | 0 1 | 0 0 | 1

Well, there you go! They're opposite as we can quite plainly see. So the second output of our AND gate is the output of another AND gate. Only this AND gate's output will be reversed.

Well, how do I reverse an output?
To reverse an output you need a NOT gate. Basically a NOT gate is just a negating gate. It has one input and one output. The output is opposite to the input.

The final diagram:
            ___
I1  --+-----\   `-._            ___
      |      \      `._________|   `-.
      |      /     _,'         |      \_______ Output
I2  --|--+--/___,-'            |      /
      |  |               +-----|___,-'
      |  |               o
      |  |               |
      |  |              / \
      |  |   ___       /___\
      +--|--|   `-.      |
         |  |      \_____|
         |  |      /
         +--|___,-'
The final output of our wonderful diagram is:
I1  I2     O
____________
0   0   |  0
0   1   |  1
1   0   |  1
1   1   |  0
Many thanks to TenMinJoe who has kindly pointed out to me that this truth table is recognisable as the XOR truth table. XOR is basically a half adder.
References:
David Hartley - Verdon College, Invercargill, Southland, New Zealand. (My computer teacher)
Many thanks also to dinkypoo's logic gates node - I kinda copied his ASCII Art.

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