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Chapter 1: Number Systems and Logic

Introduction

This chapter introduces the foundational building blocks of digital logic design, focusing on number systems, Boolean algebra, and logical representation techniques. Understanding these concepts is essential for analyzing and building complex digital systems.

1.1 Number Systems and Representations

Digital systems use various number systems to represent and process data:

  • Binary: Base-2, uses digits 0 and 1.
  • Octal: Base-8, uses digits 0 to 7.
  • Decimal: Base-10, common human-readable format.
  • Hexadecimal: Base-16, uses digits 0-9 and letters A-F.

Conversions between these systems are critical for interfacing between human-readable formats and hardware-level data representation.

1.2 Boolean Algebra and Logic Gates

Boolean algebra provides the theoretical basis for digital logic. Common operations include:

  • AND (A · B): True only when both A and B are true.
  • OR (A + B): True if either A or B is true.
  • NOT A: Inverts the input.

Important laws include identity, null, idempotent, and De Morgan’s Theorems. Logic gates implement these operations in hardware.

1.3 Truth Tables and Logic Expressions

Truth tables map all possible input combinations to corresponding outputs. They help define logic circuits precisely and support:

  • Verifying expressions
  • Simplifying logic
  • Designing from requirements

1.4 Sum of Products (SOP) and Product of Sums (POS)

Boolean expressions can be standardized as:

  • SOP: A sum (OR) of product (AND) terms, e.g., (AB + AC)
  • POS: A product (AND) of sum (OR) terms, e.g., (A + B + C + D)

These forms are used for simplifying and implementing logic using gates.

1.5 Karnaugh Maps and Simplification Techniques

Karnaugh Maps (K-maps) are visual tools for minimizing Boolean expressions by grouping adjacent cells representing 1s in the truth table.

  • 2, 3, or 4-variable maps are common
  • Minimizes gate count and circuit complexity
  • Groups must contain 1, 2, 4, 8... cells in powers of two

1.6 Binary Arithmetic and Two’s Complement

Binary arithmetic forms the basis of all numeric operations in digital systems.

  • Addition: Follows base-2 rules (1+1 = 10)
  • Subtraction: Uses two’s complement for negative numbers
  • Two’s complement: Invert all bits and add 1 to represent negative numbers

Summary

  • Binary and hexadecimal systems are fundamental to hardware
  • Boolean algebra enables logic simplification and expression
  • Truth tables and SOP/POS forms assist in accurate circuit design
  • Karnaugh Maps provide an intuitive way to minimize logic
  • Binary arithmetic and two’s complement support signed number operations

🧪 MicroSim

Link to Simulation Demo

✅ Quiz: Check Your Understanding

1. What is the binary representation of decimal 13?

  • A) 1010
  • B) 1101
  • C) 1110
  • D) 1001
Show Answer

Correct answer: B) 1101

2. What is the result of A + A?

  • A) A
  • B) 1
  • C) 0
  • D) A
Show Answer

Correct answer: B) 1

3. What is the two’s complement of 0101?

  • A) 1011
  • B) 1101
  • C) 1110
  • D) 1010
Show Answer

Correct answer: B) 1011