Arithmetic Circuits — Digital Logic -

🧮 Combinational Logic Circuits (Arithmetic Circuits)

Let’s start with something familiar — doing math.
Whenever you add, subtract, or multiply numbers, you’re performing arithmetic operations.
Now imagine your calculator, computer, or even your phone doing the same thing — but using only 0s and 1s.
That’s exactly what arithmetic circuits do in digital systems.

💡 What Are Arithmetic Circuits?

Arithmetic circuits are combinational logic circuits that perform mathematical operations like:

Addition
Subtraction
Multiplication
Division

These circuits don’t “remember” anything (no memory elements).
They simply take input binary numbers, perform the operation, and instantly give the output.

So, you can think of them as tiny math machines inside a computer’s brain that calculate results at lightning speed.

🧱 The Basic Building Blocks

Before we move to big arithmetic circuits, let’s understand their smallest pieces — the logic gates.

Logic gates like AND, OR, and XOR act as the “workers” in these circuits.
Each gate performs a small part of the job, and when combined properly, they can perform complex math operations.

For example, just by connecting a few AND, OR, and XOR gates, you can build an adder circuit — a device that adds two binary numbers!

⚙️ Types of Arithmetic Circuits

Let’s explore the most common types of arithmetic circuits one by one.

1️⃣ Half Adder

The half adder is the simplest arithmetic circuit.
It adds two single-bit binary numbers (say J and K).

But here’s the catch — it doesn’t handle any carry from previous additions.
That’s why it’s called “half” adder — it does only half the job.

Inputs	Output (Sum)	Output (Carry)
J	K	S	C
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

From this truth table:

Sum (S) = J ⊕ K (XOR gate)
Carry (C) = J ⋅ K (AND gate)

So, a half adder needs just two gates — one XOR and one AND.

Think of it as a tiny “addition helper” that can only add two bits at a time.

2️⃣ Full Adder

The full adder takes things one step further.
It adds three bits — two input bits (J and K) and a carry-in (Cin) from a previous addition.

Now it’s a “complete” adder since it can handle carry as well.

| J | K | Cin | Sum | Carry |
| – | – | — | — | —– |
| 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 |

From this:

Sum = J ⊕ K ⊕ Cin
Carry = (J ⋅ K) + (Cin ⋅ (J ⊕ K))

A full adder can be built using two half adders and an OR gate.
Pretty neat, right?

If you connect several full adders side by side, you can add multi-bit numbers — like 4-bit, 8-bit, or even 32-bit numbers.
This is how computers add large binary values so quickly.

3️⃣ Half Subtractor

Now that we can add, let’s learn how to subtract.

A half subtractor is similar to a half adder but performs subtraction instead.
It works with two bits — a minuend (J) and a subtrahend (K).

| J | K | Difference (D) | Borrow (B) |
| – | – | ————– | ———- |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |

From this:

Difference = J ⊕ K
Borrow = K ⋅ J′

So, it needs an XOR gate and an AND + NOT combination.

It’s like saying, “Can I subtract K from J? If not, I’ll borrow 1!”

4️⃣ Full Subtractor

A full subtractor handles three bits — J, K, and a borrow-in (Bin) from a previous operation.

| J | K | Bin | Difference | Borrow |
| – | – | — | ———- | —— |
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 |

Expressions:

Difference = J ⊕ K ⊕ Bin
Borrow = K ⋅ J′ + Bin ⋅ (K′ ⊕ J)

Like the full adder, a full subtractor can also be made using two half subtractors and an OR gate.

5️⃣ Parallel Adders and Subtractors

When we want to add or subtract multiple-bit numbers, we connect many full adders or full subtractors in a chain.
This setup is known as a parallel adder or parallel subtractor.

For example, a 4-bit parallel adder adds two 4-bit numbers (say 1010 and 0111) at once by using four full adders connected together.

It’s like having a team of full adders working side-by-side — each handling one pair of bits.

⚡ Other Arithmetic Circuits

Apart from adders and subtractors, there are a few more:

Binary Incrementer – adds 1 to a binary number.
Binary Decrementer – subtracts 1 from a binary number.
Multipliers – perform binary multiplication using AND gates and adders.
Arithmetic Logic Unit (ALU) – a powerful circuit that combines arithmetic and logic operations (used inside every CPU!).