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HSSLIVE Plus One Computer Science Chapter 2: Data Representation and Boolean Algebra Notes

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At the core of computing lies the representation of all information in binary form. This chapter delves into number systems (binary, octal, hexadecimal) and explains how computers encode different types of data—from numbers and text to images and sound—using only 0s and 1s. Boolean algebra, developed by George Boole, provides the mathematical foundation for digital logic that enables computation through logical operations (AND, OR, NOT), which are implemented using electronic circuits to perform the decision-making operations that drive computing systems.

Chapter 2: Data Representation and Boolean Algebra

Number Systems in Computing

Computers process data in binary (base-2) format, but humans typically work with decimal (base-10) numbers. Understanding different number systems is crucial:

  1. Decimal System (Base-10):
    • Uses digits 0-9
    • Each position represents a power of 10
    • Example: 253₁₀ = 2×10² + 5×10¹ + 3×10⁰
  2. Binary System (Base-2):
    • Uses only 0 and 1 (bits)
    • Each position represents a power of 2
    • Example: 1011₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 11₁₀
  3. Octal System (Base-8):
    • Uses digits 0-7
    • Each position represents a power of 8
    • Example: 52₈ = 5×8¹ + 2×8⁰ = 42₁₀
  4. Hexadecimal System (Base-16):
    • Uses digits 0-9 and letters A-F (where A=10, B=11, …, F=15)
    • Each position represents a power of 16
    • Example: 2F₁₆ = 2×16¹ + 15×16⁰ = 47₁₀

Conversion Between Number Systems

Converting between different number systems:

Decimal to Binary Conversion: Divide by 2 repeatedly and record remainders in reverse order. Example: Convert 25₁₀ to binary

  • 25 ÷ 2 = 12 remainder 1
  • 12 ÷ 2 = 6 remainder 0
  • 6 ÷ 2 = 3 remainder 0
  • 3 ÷ 2 = 1 remainder 1
  • 1 ÷ 2 = 0 remainder 1 Result: 25₁₀ = 11001₂

Binary to Decimal Conversion: Multiply each bit by its positional value (power of 2) and sum. Example: Convert 10110₂ to decimal

  • 1×2⁴ + 0×2³ + 1×2² + 1×2¹ + 0×2⁰
  • 16 + 0 + 4 + 2 + 0 = 22₁₀

Data Representation in Computers

  1. Character Representation:
    • ASCII (American Standard Code for Information Interchange): 7-bit encoding (128 characters)
    • Extended ASCII: 8-bit encoding (256 characters)
    • Unicode: Modern standard supporting multiple languages (UTF-8, UTF-16)
  2. Integer Representation:
    • Unsigned integers: Only positive values
    • Signed integers: Positive and negative values using two’s complement
  3. Floating Point Representation:
    • Used for real numbers with decimal points
    • IEEE 754 standard: Sign bit, exponent, and mantissa

Boolean Algebra

Boolean algebra, developed by George Boole, forms the mathematical foundation for digital electronics and computer logic.

  1. Basic Operators:
    • AND (conjunction, ∧): Output is true only when both inputs are true
    • OR (disjunction, ∨): Output is true when at least one input is true
    • NOT (negation, ¬): Inverts the input value
  2. Truth Tables: AND Operation:
    A B A AND B
    0 0 0
    0 1 0
    1 0 0
    1 1 1

    OR Operation:

    A B A OR B
    0 0 0
    0 1 1
    1 0 1
    1 1 1

    NOT Operation:

    A NOT A
    0 1
    1 0
  3. Boolean Laws and Properties:
    • Commutative Law: A + B = B + A, A · B = B · A
    • Associative Law: (A + B) + C = A + (B + C), (A · B) · C = A · (B · C)
    • Distributive Law: A · (B + C) = A · B + A · C
    • Identity Law: A + 0 = A, A · 1 = A
    • Complement Law: A + Ā = 1, A · Ā = 0
  4. Logic Gates:
    • AND gate
    • OR gate
    • NOT gate
    • NAND gate (NOT AND)
    • NOR gate (NOT OR)
    • XOR gate (Exclusive OR)
    • XNOR gate (Exclusive NOR)
  5. Applications of Boolean Algebra:
    • Digital circuit design
    • Programming logic
    • Database queries
    • Search algorithms

Binary Arithmetic

Performing arithmetic operations with binary numbers:

  1. Binary Addition:
    • 0 + 0 = 0
    • 0 + 1 = 1
    • 1 + 0 = 1
    • 1 + 1 = 10 (carry the 1)
  2. Binary Subtraction:
    • 0 – 0 = 0
    • 1 – 0 = 1
    • 1 – 1 = 0
    • 0 – 1 = 1 (borrow 1)

Understanding data representation and Boolean algebra is fundamental to computer science as it provides the mathematical foundation upon which all computing systems are built. These concepts directly influence how computers store, process, and manipulate data at the most basic level.

Complete Chapter-wise Hsslive Plus One Computer Science Notes

Our HSSLive Plus One Computer Science Notes cover all chapters with key focus areas to help you organize your study effectively:

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