Conversion Among Radices — Digital Logic -

🔄 Basics of Digital Logic (Conversion Among Radices)

When we deal with digital systems, numbers don’t always appear in the form we’re used to — like 10, 25, or 99.
Computers speak in binary, engineers sometimes use octal or hexadecimal, and we humans prefer decimal.
So, to understand or design digital circuits, we must know how to convert numbers from one system (radix) to another.

Don’t worry — it’s easier than it sounds once you know the trick behind it.

💡 What Does “Conversion Among Radices” Mean?

Every number system has its own base (or radix) — that’s the count of unique digits it uses.
When we convert among radices, we simply translate a number from one base to another — like translating a word from English to another language.
The number’s value doesn’t change; only the way it’s written does.

🧮 The Four Main Radices in Digital Logic

Binary (Base 2) → Uses 0 and 1
Octal (Base 8) → Uses 0 to 7
Decimal (Base 10) → Uses 0 to 9
Hexadecimal (Base 16) → Uses 0–9 and A–F (A=10, B=11, …, F=15)

Now let’s see how we can move between these systems step by step.

🔹 1. Converting from Any Base to Decimal

This is the easiest and most important step because decimal acts like a “bridge” for other conversions.
To convert, multiply each digit by its base raised to its position power, starting from right to left.

Example:
Convert (1011)₂ to decimal.
= 1×2³ + 0×2² + 1×2¹ + 1×2⁰
= 8 + 0 + 2 + 1 = 11₁₀

Another one: (1A)₁₆ to decimal
A = 10
= 1×16¹ + 10×16⁰ = 16 + 10 = 26₁₀

🔹 2. Converting from Decimal to Any Other Base

Here we use division and remainder.
Keep dividing the decimal number by the new base until you get 0.
Then, read the remainders backward — that’s your new number.

Example: Convert (25)₁₀ to binary.
25 ÷ 2 = 12 R1
12 ÷ 2 = 6 R0
6 ÷ 2 = 3 R0
3 ÷ 2 = 1 R1
1 ÷ 2 = 0 R1
Now read remainders upward → 11001₂

🔹 3. Binary ↔ Octal Conversion

Binary and octal are closely related because 3 binary bits = 1 octal digit.
Just group binary digits in sets of 3 from right to left and convert.

Example: (101011)₂ → Group as 101 011
101 = 5, 011 = 3 → (53)₈

Reverse is just as simple — replace each octal digit with its 3-bit binary equivalent.

Example: (47)₈ → 4 = 100, 7 = 111 → (100111)₂

🔹 4. Binary ↔ Hexadecimal Conversion

Here, 4 binary bits = 1 hexadecimal digit.

Example: (11011010)₂ → Group as 1101 1010
1101 = D, 1010 = A → (DA)₁₆

Reverse works the same way — each hexadecimal digit turns into a 4-bit binary block.

Example: (1F)₁₆ → 1 = 0001, F = 1111 → (00011111)₂

🔹 5. Octal ↔ Hexadecimal Conversion (Using Binary as a Bridge)

There’s no direct shortcut here, but you can convert:
Octal → Binary → Hexadecimal (or vice versa).

For example:
(57)₈ → Binary = 101111 → Group into 4 bits → 0010 1111 → (2F)₁₆

🎯 A Simple Analogy

Think of these number systems like currencies — dollars, euros, yen.
The value of your money doesn’t change; only the representation does.
Conversion among radices is the same — the number means the same thing, but we write it differently depending on the system.

Radix conversion means changing the form of a number without changing its value.
Decimal acts as a helpful middle step between systems.
Binary pairs neatly with octal (3 bits) and hexadecimal (4 bits).
Once you practice a few conversions, it starts to feel as natural as counting.

Digital systems depend on these conversions — they help humans, engineers, and computers “speak the same language” in their own ways.