Definition

  • Whole numbers are the set of numbers starting from 0 and increasing positively without end.
  • The set of whole numbers is represented by the symbol W.
  • W = {0, 1, 2, 3, 4, 5, ...}
  • All natural numbers are whole numbers.
  • 0 is a whole number but not a natural number.

Basic Properties

  • Whole numbers do not include negative numbers, fractions, or decimals.
  • They are a subset of integers and real numbers.
  • They are infinite in count.
  • They are used in counting, ordering, and labeling.

Closure Properties

  • Closed under addition: Sum of any two whole numbers is a whole number.
  • Closed under multiplication: Product of any two whole numbers is a whole number.
  • Not closed under subtraction: Subtracting two whole numbers may not result in a whole number.
  • Not closed under division: Division may result in a fraction.

Properties of Operations

  • Commutative for addition: a + b = b + a
  • Commutative for multiplication: a × b = b × a
  • Associative for addition: (a + b) + c = a + (b + c)
  • Associative for multiplication: (a × b) × c = a × (b × c)
  • Distributive of multiplication over addition: a × (b + c) = a×b + a×c

Identity Elements

  • 0 is the identity for addition: a + 0 = a
  • 1 is the identity for multiplication: a × 1 = a

Representation and Examples

  • 0, 5, 23, 100 are all whole numbers.
  • Numbers like −3, 1.5, ¾ are not whole numbers.
  • Whole numbers can be plotted on a number line starting from 0.
  • They are used in real-life counting where negative or fractional values are not practical (e.g., counting books, balls).

Number Classification

  • All natural numbers are whole numbers.
  • Whole numbers are a subset of integers.
  • Natural numbers start from 1, while whole numbers start from 0.

Mathematical Applications

  • Used in basic arithmetic (addition, subtraction, multiplication).
  • Used in finding LCM, HCF, and performing factorization.
  • Helps in forming algebraic expressions and equations.
  • Useful in data analysis where whole units are measured.

Real-Life Use

  • Used for measuring objects (lengths in cm, counts of items).
  • Used in time representation in full hours or days.
  • Used in digital systems for counting binary values starting from 0.

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