Adders and the Arithmetic Logic Unit | OCR H446

Learning Objectives

By the end of this topic you will be able to:

Describe the half adder: its inputs, outputs and Boolean expressions

Describe the full adder and how it extends the half adder

Explain how multiple adders are chained to add multi-bit numbers

Describe the role and components of the Arithmetic Logic Unit (ALU)

Half Adder

The Half Adder

A half adder adds two single bits (A and B) and produces two outputs: Sum (S) and Carry (C). It cannot accept a carry-in from a previous bit position.

Boolean Expressions

S = A ⊕ B    (XOR gate)

C = A · B    (AND gate)

Truth Table

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

Full Adder

The Full Adder

A full adder adds three bits: A, B and a Carry-In (Cin) from the previous bit position. It produces Sum (S) and Carry-Out (Cout). Built from two half adders plus an OR gate.

Boolean Expressions

S = A ⊕ B ⊕ Cin

Cout = (A·B) + (Cin·(A⊕B))

Key rows from truth table

A	B	Cin	S	Cout
0	0	0	0	0
1	1	0	0	1
1	1	1	1	1

Ripple Adder

Multi-bit Addition: Ripple Carry Adder

To add multi-bit binary numbers, multiple full adders are chained. The Carry-Out of each full adder becomes the Carry-In of the next more significant bit position. For the least significant bit, Cin = 0 (or a half adder is used).

Adding 0111 (7) + 0101 (5) using 4 full adders:

Bit 0: 1+1+Cin=0 → S=0, Cout=1

Bit 1: 1+0+Cin=1 → S=0, Cout=1

Bit 2: 1+1+Cin=1 → S=1, Cout=1

Bit 3: 0+0+Cin=1 → S=1, Cout=0

Result: 1100 = 12 ✓

The delay as carry ripples from LSB to MSB is called propagation delay — a limitation of the ripple carry adder. More advanced look-ahead carry adders avoid this.

ALU

The Arithmetic Logic Unit (ALU)

The ALU is the component within the CPU that performs arithmetic operations (addition, subtraction) and logical operations (AND, OR, NOT, XOR, bit shifts). It is a core part of the CPU's datapath.

ALU Inputs

• Two operands (from registers or memory)
• Operation select (from control unit)
• Carry-in flag (from status register)

ALU Outputs

• Result (to register or memory)
• Status flags: Zero (Z), Carry (C), Negative (N), Overflow (V) — stored in status/flags register, used for branching decisions

Exam Practice

OCR H446 Style · 4 marks

Explain the difference between a half adder and a full adder. State the Boolean expressions for both outputs of a full adder and explain why a full adder is needed to add multi-bit numbers.

[4 marks]

A half adder has two inputs (A, B) and two outputs (Sum, Carry). It cannot accept a carry-in from a previous bit.

A full adder has three inputs (A, B, Carry-in) allowing it to incorporate the carry produced by a less significant bit position.

Full adder expressions: S = A ⊕ B ⊕ Cin; Cout = (A·B) + (Cin·(A⊕B))

For multi-bit addition, each bit position may receive a carry from the position to its right. Only a full adder can accept this carry-in — a half adder would discard it, giving incorrect results for all bits except the least significant.

Common Mistakes

Don't Lose Marks

Saying the Sum output of a half adder uses an AND gate — the Sum output uses an XOR gate (A ⊕ B). The AND gate produces the Carry output. Swapping these is a very common error that fails the Boolean expression question.

Thinking the ALU only does arithmetic — the ALU performs both arithmetic (add, subtract, increment) and logical operations (AND, OR, NOT, XOR, bit shifts). The 'L' in ALU stands for Logic — always mention both types when describing the ALU.

Not explaining why carry-in matters in multi-bit adders — when adding multi-bit numbers, each bit position can receive a carry from the right. Without carry-in capability, carries would be lost, producing wrong results. This is the key reason full adders are needed.

Adders & theArithmetic Logic Unit