Binary Number System: Definition, Types & Conversions (Complete Guide for BCA Students)

Introduction

The binary number system is the foundation of all digital electronics and computing. Unlike the decimal system (base-10), binary uses only two digits: 0 and 1, making it ideal for computers that operate using ON/OFF states. This post covers:

✔ Definition of Binary System
✔ Types of Binary Systems
✔ Binary → Decimal, Octal, Hexadecimal Conversion
✔ Exam Tricks & Practice Problems

1. What is the Binary Number System?

Definition: A base-2 number system using only 0 (OFF) and 1 (ON).
Used in: All digital systems (CPUs, memory, networking).
Example:
- Decimal 5 = Binary 101
- Decimal 10 = Binary 1010

Why Binary in Computers?

Transistors (hardware) work in two states:
- 0 = Low voltage (~0V)
- 1 = High voltage (~3.3V or 5V)

2. Types of Binary Systems

(A) Pure Binary (Standard Base-2)

Represents numbers using 0s and 1s.
Example: 1101 = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 13 (Decimal)

(B) Binary-Coded Decimal (BCD)

Each decimal digit (0-9) is represented by a 4-bit binary code.
Example:
- Decimal 12 = BCD 0001 0010
- (Not the same as pure binary 1100)

(C) Gray Code (Reflected Binary)

Only one bit changes between consecutive numbers.
Used in encoders, Karnaugh maps.
Example:

3. Binary Conversions (Step-by-Step)

(A) Binary → Decimal

Method: Multiply each bit by 2ⁿ (where n = position from right, starting at 0).

Example: Convert 1011 to decimal.

(B) Binary → Octal (Base-8)

Method: Group binary digits into sets of 3 (from right), then convert each group to octal.

Example: Convert 110101 to octal.

(C) Binary → Hexadecimal (Base-16)

Method: Group binary digits into sets of 4, then convert each group to hex.

Example: Convert 11011010 to hex.

4. Conversion Table (Quick Reference)

Binary	Decimal	Octal	Hexadecimal
`0000`	0	0	0
`0001`	1	1	1
`0010`	2	2	2
...	...	...	...
`1010`	10	12	A
`1111`	15	17	F

5. Exam-Focused FAQs

❓ Q1: Why is hexadecimal used instead of binary in programming?
✅ A: Hex shortens long binary strings (e.g., FF = 11111111) for readability.

❓ Q2: How to convert fractional binary (e.g., 101.101) to decimal?
✅ A: Use negative powers for bits after the point:
1×2² + 0×2¹ + 1×2⁰ + 1×2⁻¹ + 0×2⁻² + 1×2⁻³ = 5.625

❓ Q3: What is the largest decimal number representable by 8-bit binary?
✅ A: 11111111 = 255 (2⁸ - 1).

6. Practice Problems

Convert 10011 to decimal.
Convert 1100110 to octal.
Convert 10101111 to hexadecimal.

(Answers: 19, 146, AF)

7. Summary

Binary = Base-2 (0,1), used in all digital systems.
Types: Pure binary, BCD, Gray code.
Conversions:
- Binary → Decimal (Multiply by 2ⁿ).
- Binary → Octal (Group in 3s).
- Binary → Hex (Group in 4s).

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