Definition

  • Natural numbers are the set of positive integers starting from 1.
  • They are used for counting and ordering.
  • N = {1, 2, 3, 4, 5, ...}
  • The symbol used to denote natural numbers is N.

Basic Properties

  • The smallest natural number is 1.
  • Zero (0) is not a natural number.
  • Natural numbers are infinite.
  • They are a subset of whole numbers and integers.
  • Natural numbers do not include negative numbers, fractions, or decimals.

Operations and Properties

  • Closed under addition and multiplication.
  • Not closed under subtraction and division.
  • Commutative property holds for addition and multiplication.
  • Associative property also holds for both operations.
  • Distributive property of multiplication over addition is valid.

Identity Elements

  • No identity element for addition within natural numbers.
  • 1 is the multiplicative identity.

Examples of Operations

  • 3 + 2 = 5 (natural number)
  • 4 × 5 = 20 (natural number)
  • 3 - 5 = -2 (not a natural number)
  • 5 ÷ 2 = 2.5 (not a natural number)

Types and Categories

  • Even and odd numbers are subsets of natural numbers.
  • Prime and composite numbers are types of natural numbers.
  • 1 is neither prime nor composite.

Number Line Representation

  • Natural numbers start from 1 and go infinitely to the right.
  • Successor of a natural number is obtained by adding 1.
  • Predecessor of 1 does not exist in natural numbers.

Applications in Mathematics

  • Used in LCM, HCF, factors, and multiples.
  • Form the base of arithmetic operations.
  • Every natural number is a multiple of itself.
  • Natural numbers are used in divisibility rules.
  • Factorial (n!) is defined only for natural numbers.

Sequences and Patterns

  • Used in arithmetic progression (AP).
  • Often used to generate number patterns.
  • Squares and cubes of natural numbers are also natural.

Real-Life Use

  • Used for counting objects (e.g., 1 pen, 2 books).
  • Used in ranking, ordering, and measuring quantities (except decimals).

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