The Half Adder

The half adder is the simplest arithmetic circuit. It adds two single-bit binary numbers, producing a sum and a carry output. Despite its simplicity, the half adder introduces the fundamental concept of binary addition hardware and serves as a building block for the full adder.

Objectives

• Write the truth table for a half adder
• Derive the Boolean expressions: Sum = A⊕B, Carry = A·B
• Build a half adder circuit from XOR and AND gates
• Understand why it's called "half" - it has no carry input

Key Takeaways

• Half adder adds two single bits: Sum = A⊕B, Carry = A·B
• Only 2 gates needed: one XOR + one AND
• Cannot handle carry input - only works for the least significant bit
• Building block for the full adder
• The XOR function is the fundamental "addition without carry" operation

Applications

• LSB Addition: The first bit position in any multi-bit adder.
• Parity Generation: XOR chain for parity bit calculation.
• Increment Circuits: Adding 1 to a binary number.

Practice Problems

Problem 1: What are the Sum and Carry outputs when A=1 and B=1?

Problem 2: Can a half adder produce Sum=1 and Carry=1 simultaneously?

Problem 3: Build a half adder using only NAND gates. How many do you need?

Problem 4: Why can't you simply chain half adders for multi-bit addition?