Interactive 1-bit Full Adder
Addition is the fundamental arithmetic operation in digital computers. Just like elementary school decimal addition—where adding 9 + 3 produces a sum of 2 and a carry-out of 1 —computers add digits column by column, passing carries to the next position.
While humans calculate in base-10, computers operate in binary (0 and 1). Check out the Binary Number Explorer for some interactive fun. The standard building block for multi-bit addition is the 1-bit Full Adder. It takes three 1-bit inputs:
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A and B (the bits to add)
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Carry_in (the Carry_out bit received from the previous column)
The adder processes these inputs to produce two outputs:
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Sum (the result bit for the current column)
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Carry_out (the carry bit passed to the next column as Carry_in)
Expressed as an equation:
A + B + Carry_in = 2 x Carry_out + Sum
The classic adder truth table below shows all possible input and output values. With 3 inputs there are 8 possible sets of outputs. The "Dec" column is the decimal equivalent of the 2 binary outputs. See for example the last row where all the inputs add to decimal 3.
The adder below is the classic 1-bit form made of AND, OR and Exclusive OR gates. Learn more about these gates here. This basic block can be replicated and chained to create any length of adder (8, 16, 32, 64,...bits). The R30 relay computer uses a 2-relay, 1-bit adder, six times to create a 6-bit adder. The two-relay adder is described elsewhere on this site and is equivalent to the diagram below in logic operation.
Importantly, there are many variations of this circuit so don't get confused if you see something called a "Full Binary Adder" and the design looks different. All methods are equivalent logically with 3 inputs and 2 outputs. To see the adder in operation, experiment with the demo below. Try different inputs and see the resultant outputs.