Number system

  🔢 Numbering Systems in Digital Electronics

Published on: Jan 7, 2025

Author: Vijaya Bala V – ASIC Simplified

✨️Introduction 

In digital electronics, the way we represent information is crucial. Unlike the natural decimal system we use in daily life, digital circuits use different numbering systems to process and interpret data efficiently. Let's simplify four core numbering systems.

Decimal (Base-10)

Binary (Base-2)

Octal (Base-8)

Hexadecimal (Base-16)

Each system has unique features that support digital logic, memory addressing, and data encoding.

🔟 Decimal System (Base-10)

The decimal system uses digits from 0 to 9 and is based on powers of 10. It's the most intuitive system for humans but not ideal for computers.This is the number system we use every day — it has ten digits (0 to 9). Each digit’s position represents a power of 10. It's intuitive to us because it's built into our counting habits and daily life.Digits used: 0 to 9 (10 symbols)Most common system we use daily.Each digit's place represents a power of 10.

Example:

321 in decimal =

3 × 10² + 2 × 10¹ + 1 × 10⁰ = 300 + 20 + 1 = 321

💡 Binary System (Base-2)

Binary is the heart of digital systems. It uses only two digits: 0 and 1, which correspond to OFF and ON states in circuits.Binary is the native language of computers. It uses only two digits: 0 and 1. Each binary digit (bit) represents a switch being off (0) or on (1). All software, images, videos, and commands you see on a computer are ultimately made of binary data.

Example:

🔹 Convert Binary 1101 to Decimal

Step-by-step:

Binary:

1 1 0 1  

↑ ↑ ↑ ↑

2³ 2² 2¹ 2⁰

Now multiply each bit by its place value:

= (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1)  

= 8 + 4 + 0 + 1  

= 13

✅ Binary 1101 = Decimal 13 

Binary Voltage Logic:

1 = High voltage (e.g., 3.3V or 5V)

0 = Low voltage (e.g., 0V)

🧮 Octal System (Base-8)

The octal system simplifies binary by grouping 3 bits together. It includes digits 0 through 7.Octal simplifies binary into shorter, readable chunks. It uses digits from 0 to 7. Every 3 binary bits form one octal digit, making it useful in microprocessor programming and low-level computing where memory optimization matters.

Example:

 🔹 Convert Octal 237 to Decimal

Step-by-step:

Octal:

2 3 7  

↑ ↑ ↑

8² 8¹ 8⁰

Now multiply each digit:

= (2 × 64) + (3 × 8) + (7 × 1)  

= 128 + 24 + 7  

= 159

✅ Octal 237 = Decimal 159

🔄 Binary to Octal

101101 = (101)(101) → 5 5 → Octal 55

🧪 Hexadecimal System (Base-16)

Hexadecimal offers a compact representation of binary data using 4-bit groups. It includes digits 0-9 and letters A-F (for values 10-15).Hex uses 16 symbols: 0–9 and A–F. It's widely used in programming, especially in defining colors in web design (#FF5733), memory addresses, and machine-level debugging. Every 4 binary bits form one hex digit, offering compactness and clarity.

Example:

🔹 Convert Hexadecimal 2F to Decimal

Step-by-step:

Hex Digits:

2 F  

↑ ↑

16¹ 16⁰

Remember: F = 15 in decimal.

Now multiply:

= (2 × 16) + (15 × 1)  

= 32 + 15  

= 47

✅ Hexadecimal 2F = Decimal 47

🔄 Binary to Hex

11011010 = (1101)(1010) → D A → Hex DA

🧠 Why Learn Number Systems?

Understanding number systems is foundational to:

Writing Verilog/SystemVerilog code

Interpreting signal values in simulation

Working with memory addresses and register maps

Debugging and interpreting waveform outputs

🧩 Interactive Activity

Can you convert the following?

1. Decimal 75 to Binary

Step-by-step: Divide 75 by 2 and keep track of remainders:

75 ÷ 2 = 37 remainder 1  

37 ÷ 2 = 18 remainder 1  

18 ÷ 2 = 9 remainder 0  

9 ÷ 2 = 4 remainder 1  

4 ÷ 2 = 2 remainder 0  

2 ÷ 2 = 1 remainder 0  

1 ÷ 2 = 0 remainder 1 ← stop here

Now reverse the remainders:

→ 1 0 0 1 0 1 1

So, 75₁₀ = 1001011₂

✅ Answer: 1001011

2. Binary 101011 to Hex

Group bits in 4s from right to left (add leading zeros if needed):

Binary: 0010 1011  

              ↑ ↑

            (2) (B)

0010 = 2

1011 = B

✅ Answer: 2B₁₆

Explanation: Binary groups of 4 bits map directly to hexadecimal digits.

3. Hex 1A3 to Decimal

Step-by-step: Use positional weights of 16:

1A3₁₆ = (1 × 16²) + (A × 16¹) + (3 × 16⁰)  

           = (1 × 256) + (10 × 16) + (3 × 1)  

           = 256 + 160 + 3 = **419**

✅ Answer: 419₁₀

4. Octal 172 to Binary

Each octal digit → 3-bit binary group:

1 = 001  

7 = 111  

2 = 010

So, 172₈ = 001 111 010 = 001111010₂

✅ Answer: 001111010

Tip: Don’t drop leading zeros if alignment matters in hardware design (like address decoding).