Integers

Introduction

Whole Numbers

Whole numbers include all-natural numbers and zero

 i.e., 0, 1, 2, 3, 4, and so on.

Negative Numbers

  • The numbers lying to the left of the zero with negative sign on the numbers line are called negative numbers.

The Number Line

Integers

  • Set of all positive and negative numbers including zero are called integers.

…, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, … are integers.

Representing Integers on the Number Line

Absolute value of an integer

  • When we consider only the numerical value of the integer without considering its sign it is called as the absolute value of an integer.
  • Example: Absolute value of -7 is 7 and of +7 is 7.

Ordering Integers

  • The number increases as we move towards right and decreases as we move towards left on a number line.
  • Hence, the order of integers is …, –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5…

Addition of Integers

  • Positive integer + Negative integer
  • Example: (+5) + (-2) Subtract: 5 – 2 = +3 (Sign of bigger integer)
  • Example: (-5) + (2) Subtract: 5-2 = -3 (Sign of bigger integer)
  • Positive integer + Positive integer
  • Example: (+5) + (+2) = +7
  • Negative integer + Negative integer
  • Example: (-5) + (-2) = -7
  • Add the two integers and add the negative sign.

Properties of Addition and Subtraction of Integers

Operations on Integers

  • Addition
  • Subtraction
  • Multiplication
  • Division.

Properties of Addition and Subtraction of Integers

  • Closure under Addition
  • a + b and a – b are integers, where a and b are any integers.
  • Commutativity Property
  • a + b = b + a for all integers a and b.
  • Associativity of Addition
  • (a + b) + c = a + (b + c) for all integers a, b and c.
  • Additive Identity
  • Additive Identity is 0, because adding 0 to a number leaves it unchanged.
  • a + 0 = 0 + a = a for every integer a.

Multiplication of Integers

  • Product of a negative integer and a positive integer is always a negative integer.
    •  10 \times -2 = -20
  • Product of two negative integers is a positive integer.
    •  -10 \times -2 = 20
  • Product of even number of negative integers is positive.
    • \left( { - 2} \right) \times \left( { - 5} \right) = 10
  • Product of an odd number of negative integers is negative.
    •  \left( {-2} \right) \times \left( { -5} \right) \times \left( 6 \right) = -60

Properties of Multiplication of Integers

  • Closure under Multiplication
    • Integer \times Integer = Integer
  • Commutativity of Multiplication
    • For any two integers a and b, a\; \times \;b{\rm{ }} = {\rm{ }}b{\rm{ }} \times {\rm{ }}a.
  • Associativity of Multiplication
    • For any three integers a, b and c, \left( {a{\rm{ }} \times {\rm{ }}b} \right){\rm{ }} \times {\rm{ }}c{\rm{ }} = {\rm{ }}a{\rm{ }} \times {\rm{ }}\left( {b{\rm{ }} \times {\rm{ }}c} \right).
  • Distributive Property of Integers
    • Under addition and multiplication, integers show the distributive property.
    • For any integers a, b and c, a{\rm{ }} \times {\rm{ }}\left( {b{\rm{ }} + {\rm{ }}c} \right){\rm{ }} = {\rm{ }}a\; \times {\rm{ }}b{\rm{ }} + {\rm{ }}a{\rm{ }} \times {\rm{ }}c.
  • Multiplication by Zero
    • For any integer a, a{\rm{ }} \times {\rm{ }}0{\rm{ }} = {\rm{ }}0{\rm{ }} \times {\rm{ }}a{\rm{ }} = {\rm{ }}0.
  • Multiplicative Identity
    • 1 is the multiplicative identity for integers.
    • a{\rm{ }} \times {\rm{ }}1{\rm{ }} = {\rm{ }}1{\rm{ }} \times {\rm{ }}a{\rm{ }} = {\rm{ }}a

Dividing Integers

  • (positive integer/negative integer) or (negative integer/positive integer)
    ⇒ The quotient obtained is a negative integer.
  • (positive integer/positive integer) or (negative integer/negative integer)
    ⇒ The quotient obtained is a positive integer.

Properties of Division of Integers

For any integer a,

  • \dfrac{a}{0} is not defined
  • \dfrac{a}{1} = a

Integers are not closed under division.

Example: \left( {-9} \right){\rm{ }} \div {\rm{ }}\left( {-3} \right){\rm{ }} = 3 result is an integer but

 \left( {-3} \right) \div \left( { -9} \right) = \dfrac{3}{9} = 0.33 which is not an integer.

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