Radix Number Systems — Digital Logic

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🔢 Basics of Digital Logic (Radix Number Systems)

Imagine you’re counting apples. You start with one, then two, three, and so on — that’s simple enough, right?
But have you ever wondered why we stop at 9 before jumping to 10? Or how computers, which only understand 0s and 1s, manage to represent huge numbers?
That’s where Radix Number Systems come in — they tell us how numbers are written and understood in different forms.

💡 What Is a Radix (or Base)?

The word radix simply means base, and it tells us how many unique digits a number system uses before it rolls over to the next place value.

For example:

  • In our everyday life, we use the Decimal System — its base (or radix) is 10 because it uses 10 digits: 0 to 9.
  • A computer, on the other hand, uses the Binary System — its base is 2, since it only uses two digits: 0 and 1.

So, the radix defines the “alphabet” of a number system — how many symbols it has for counting.

🧮 Common Number Systems in Digital Logic

  1. Binary System (Base 2)
    • Digits used: 0, 1
    • Example: (1011)₂ = 11 in decimal
    • Used by computers because electronic circuits can easily represent two states — ON (1) and OFF (0).
  2. Octal System (Base 8)
    • Digits used: 0 to 7
    • Example: (17)₈ = 15 in decimal
    • It’s often used as a shorthand for binary since 3 binary digits (bits) make one octal digit.
  3. Decimal System (Base 10)
    • Digits used: 0 to 9
    • Example: (245)₁₀ = 2×10² + 4×10¹ + 5×10⁰ = 245
    • This is the system humans are most comfortable with — we have ten fingers, after all!
  4. Hexadecimal System (Base 16)
    • Digits used: 0–9 and A–F (where A=10, B=11, …, F=15)
    • Example: (1A)₁₆ = 1×16¹ + 10×16⁰ = 26 in decimal
    • Used in computer science because one hexadecimal digit equals four binary bits — it’s compact and easy to read.

🧠 How Place Values Work

In any number system, the position of each digit determines its value.
Each position represents a power of the base (radix).

For example, in the decimal number 345,
→ 3 represents 3×10²,
→ 4 represents 4×10¹,
→ 5 represents 5×10⁰.

In binary (1011)₂,
→ 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11 (decimal).

So, changing the base changes the weight of each position.

🪄 Real-Life Analogy

Think of it like clocks in different countries showing different time zones.
The concept of time is the same everywhere, but how we read it (AM/PM or 24-hour format) depends on the system we use.
Similarly, numbers represent the same quantity, but their symbols and place values depend on the radix system.

⚙️ Why Radix Systems Matter in Digital Logic

Digital circuits communicate using binary, but humans find binary too long and confusing.
That’s why engineers use octal or hexadecimal as shortcuts.
For instance, instead of writing 101011110000 (binary), we can simply write AF0 (hexadecimal).
It saves time, space, and reduces errors.

A Radix Number System defines how numbers are represented using different sets of symbols and place values.
Binary, Octal, Decimal, and Hexadecimal are the most common systems in digital logic.
Understanding them helps bridge the gap between how humans think in decimal and how machines operate in binary — forming the foundation of all digital communication.