Half Adders and Full Adders

Adder circuits are fundamental building blocks of computer arithmetic. Every processor uses adders to perform addition, and understanding how they work from basic logic gates is a core part of the OCR H446 specification.

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Binary Addition Recap

Binary addition follows the same rules as decimal addition, but with only two digits (0 and 1):

A	B	Sum	Carry
0 + 0		0	0
0 + 1		1	0
1 + 0		1	0
1 + 1		0	1

When both bits are 1, the sum is 0 with a carry of 1 (just like 5 + 5 = 10 in decimal — the "0" stays and the "1" carries to the next column).

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The Half Adder

A half adder adds two single-bit inputs (A and B) and produces a sum (S) and a carry (C).

Truth table:

A	B	Sum (S)	Carry (C)
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

Observations:

• The Sum column is exactly the XOR function: S = A XOR B
• The Carry column is exactly the AND function: C = A AND B

Circuit:

A ---+--- XOR --- S (Sum)
     |
B ---+--- AND --- C (Carry)

The half adder uses just two gates: one XOR and one AND.

Why is it called "half"? Because it can only add two bits — it has no input for a carry-in from a previous column. When adding multi-bit numbers, each column (except the least significant) must also handle a carry from the previous column.

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The Full Adder

A full adder adds three single-bit inputs: A, B, and a carry-in (Cin). It produces a sum (S) and a carry-out (Cout).

Truth table:

A	B	Cin	Sum (S)	Cout
0	0	0	0	0
0	0	1	1	0
0	1	0	1	0
0	1	1	0	1
1	0	0	1	0
1	0	1	0	1
1	1	0	0	1
1	1	1	1	1

Boolean expressions:

• S = A XOR B XOR Cin
• Cout = (A AND B) OR (Cin AND (A XOR B))

Explanation of Cout: There is a carry-out if:

• Both A and B are 1 (regardless of Cin), OR
• One of A and B is 1 (A XOR B = 1) AND the carry-in is 1

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Full Adder from Two Half Adders

A full adder can be constructed from two half adders and an OR gate:

Half Adder 1: Inputs A and B

• S1 = A XOR B
• C1 = A AND B

Half Adder 2: Inputs S1 and Cin

• S = S1 XOR Cin = (A XOR B) XOR Cin
• C2 = S1 AND Cin = (A XOR B) AND Cin

Final carry: Cout = C1 OR C2 = (A AND B) OR ((A XOR B) AND Cin)

A ---+                          +--- S
     |--- Half Adder 1 ---S1---|
B ---+           |              +--- Half Adder 2 ---S
                 C1---------+        |
                             |  Cin--+
                             |       C2
                             +--OR------ Cout

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The Ripple Carry Adder