Binary, Octal, Decimal & Hexadecimal Number Systems

Understanding Number Systems | CS Study Guide
CS Fundamentals Series

Binary, Octal, Decimal & Hexadecimal Number Systems

Bridging human logic and machine execution: A comprehensive guide for Computer Science students.

Introduction

In computer science, data represents electronic states. Number systems bridge human logic and machine execution, crucial for understanding digital logic and computer architecture. A mathematical method for representing numbers using a set of symbols (digits). Most are positional, where a digit's value depends on its position relative to the decimal point. The number of unique symbols is the Base or Radix.

Positional Notation Breakdown showing digit weights for decimal and binary
Figure 1: Visual breakdown of positional notation and digit weights.

1 Core Number Systems

Decimal (Base 10)

  • Base (r): 10
  • Digits: 0 through 9
  • Weight: Powers of 10 (10⁰, 10¹, 10²...)
523 = (5×10²) + (2×10¹) + (3×10⁰)

Binary (Base 2)

  • Base (r): 2
  • Digits: 0 and 1 ("bits")
  • Importance: Fundamental to electronic logic.
ON = 1 | OFF = 0

Octal (Base 8)

  • Base (r): 8
  • Digits: 0 through 7
  • Usage: Groups binary digits into 3s.

Hexadecimal (Base 16)

  • Base (r): 16
  • Digits: 0-9 and A-F (A=10...F=15)
  • Usage: Memory addresses, CSS colors.

The 'Four-Bit' Relationship Chart

Number Systems Conversion Table (0-15)
Figure 2: Comparing structural relationships between bases (0-15).

Master the Conversions

A. Any Base to Decimal

Method: Sum of Weights

Multiply each digit by its positional weight and sum the results.

Binary 1011 → (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11₁₀

B. Decimal to Any Base

Method: Successive Division

Repeatedly divide the decimal number by the target base, recording remainders from bottom to top.

Successive Division Method flowchart for Decimal 25 to Binary
Figure 3: Step-by-step conversion of decimal 25 to binary.

C. Binary to Hexadecimal

Method: Grouping (Nibbles)

Group binary digits into sets of four from right to left (LSB).

Binary to Hexadecimal Grouping visualization
Figure 4: Direct mapping between binary nibbles and hex characters.

Summary for Exams

System Base Digits Common Application
Decimal 10 0–9 Human calculations
Binary 2 0, 1 Machine logic / CPU
Octal 8 0–7 Legacy compact binary
Hexadecimal 16 0–F Memory addresses, Debugging

★ Why Base 2?

Hardware Compatibility: Bistable electronic circuits simplify design, reduce noise sensitivity, and increase reliability. Two states are easier to maintain than ten.

★ Why Base 16?

Efficiency: Since 16 is 2⁴, four binary bits (a "nibble") map perfectly to one hex digit, making it an efficient shorthand for long binary strings.

Final Takeaway

Mastering number systems is essential for understanding computer architecture and for skills like debugging and low-level code optimization. Fluency between Binary, Decimal, and Hexadecimal is a fundamental computer science skill that will serve you throughout your career.