Half Adder Calculator
Calculate the Sum and Carry outputs of a half adder from two binary input bits. See the XOR sum logic, AND carry logic, Boolean equations, truth-table result and step-by-step binary addition.
What Is a Half Adder?
A half adder is a combinational digital logic circuit that adds two one-bit binary inputs. The inputs are normally labeled A and B, and the circuit produces two outputs: Sum and Carry.
The Sum output represents the least significant bit of the addition. The Carry output represents a value that would be passed to the next higher binary position when the half adder forms part of a larger arithmetic circuit.
A half adder is one of the basic building blocks used to understand binary addition, Boolean algebra and digital arithmetic circuits.
Half Adder Boolean Equations
Sum Equation
The Sum output is 1 only when the two inputs are different. This behavior is exactly the XOR operation.
Carry Equation
The Carry output becomes 1 only when both A and B are 1. This is the AND operation.
Half Adder Truth Table
| Input A | Input B | Sum | Carry | Binary Result | Decimal Total |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 00 | 0 |
| 0 | 1 | 1 | 0 | 01 | 1 |
| 1 | 0 | 1 | 0 | 01 | 1 |
| 1 | 1 | 0 | 1 | 10 | 2 |
Worked Example 1: A = 0 and B = 0
B = 0
Sum = 0 XOR 0 = 0
Carry = 0 AND 0 = 0
Result = 00
Decimal total = 0
Worked Example 2: A = 0 and B = 1
B = 1
Sum = 0 XOR 1 = 1
Carry = 0 AND 1 = 0
Result = 01
Decimal total = 1
Worked Example 3: A = 1 and B = 0
B = 0
Sum = 1 XOR 0 = 1
Carry = 1 AND 0 = 0
Result = 01
Decimal total = 1
Worked Example 4: A = 1 and B = 1
B = 1
Binary addition:
1 + 1 = 10
Sum = 1 XOR 1 = 0
Carry = 1 AND 1 = 1
Result = 10
Decimal total = 2
How Half Adder Logic Gates Work
XOR Gate for Sum
The XOR gate outputs 1 when exactly one of the two inputs is 1. Therefore it produces the correct low-order Sum bit for one-bit binary addition.
AND Gate for Carry
A carry is required only when both input bits equal 1. An AND gate provides exactly this condition.
Half Adder vs Full Adder
| Feature | Half Adder | Full Adder |
|---|---|---|
| Input A | Yes | Yes |
| Input B | Yes | Yes |
| Carry-In | No | Yes |
| Sum Output | Yes | Yes |
| Carry-Out | Yes | Yes |
| Main purpose | Add two single bits | Add two bits plus incoming carry |
| Multi-bit addition | Insufficient alone | Can be chained |
Why Is It Called a Half Adder?
A half adder can add two binary operand bits but cannot accept a carry from a previous binary position. This limitation prevents a half adder alone from handling the middle stages of general multi-bit addition.
A full adder adds a third input called Carry-In, allowing multiple stages to be connected together for wider binary addition.
Where Half Adders Are Used
Digital Logic Education
Half adders provide a simple introduction to combinational logic and demonstrate how Boolean gates perform arithmetic.
Adder Circuit Design
Half-adder logic can form part of larger arithmetic circuit structures.
Boolean Algebra
The circuit gives a practical example of XOR and AND expressions operating together.
Computer Architecture
Understanding half adders provides a foundation for studying full adders and wider arithmetic logic units.
Important Half Adder Notes
It does not have a Carry-In input.
The Sum output is calculated with XOR, while the Carry output is calculated with AND.
For A = 1 and B = 1, the binary result is 10: Sum = 0 and Carry = 1.