Definition

  • Integers are the set of whole numbers and their negative counterparts.
  • The set of integers is represented by the symbol Z (from the German word "Zahlen").
  • Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
  • Integers include positive numbers, negative numbers, and zero.
  • Fractions and decimals are not integers.

Types of Integers

  • Positive integers: {1, 2, 3, 4, ...}
  • Negative integers: {..., -4, -3, -2, -1}
  • Zero is an integer but neither positive nor negative.

Number Line Representation

  • Integers can be represented on a number line extending on both sides of 0.
  • On the number line, positive integers are to the right of 0.
  • Negative integers are to the left of 0.

Basic Properties

  • Integers are closed under addition, subtraction, and multiplication.
  • They are not closed under division.
  • Addition of two integers always results in an integer.
  • Subtraction of two integers is also an integer.
  • Multiplication of two integers gives an integer.
  • Division may not result in an integer. (e.g., 3 ÷ 2 = 1.5)

Properties of Operations

  • Commutative property holds for addition and multiplication.
  • Associative property also holds for addition and multiplication.
  • Distributive property of multiplication over addition: a × (b + c) = a×b + a×c

Identity Elements

  • 0 is the additive identity: a + 0 = a
  • 1 is the multiplicative identity: a × 1 = a

Rules of Signs

  • Positive × Positive = Positive
  • Negative × Negative = Positive
  • Positive × Negative = Negative
  • Negative × Positive = Negative
  • Negative + Positive = depends on which number is larger (in magnitude)
  • Negative − Positive = More negative

Comparison

  • All positive integers are greater than 0.
  • All negative integers are less than 0.
  • Among any two integers, the one farther to the right on the number line is greater.

Set Relationships

  • Natural numbers ⊂ Whole numbers ⊂ Integers
  • Every natural number and whole number is also an integer.
  • Integers form a subset of rational numbers.

Examples

  • −5, 0, 12 are all integers.
  • 3.5, ½, √2 are not integers.

Applications in Mathematics

  • Used to express temperature (e.g., −10°C), bank balances (debit = negative), and altitudes.
  • Essential in algebra, number theory, and coordinate geometry.
  • Important in competitive exams for simplification and number system questions.

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Questions