Integers Class 6 Notes: Chapter 6

Introduction

Whole Numbers

  • Whole numbers include zero and all natural numbers i.e. 0, 1, 2, 3, 4, and so on.

Integers-1

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Negative Numbers

  • The numbers with a negative sign and which lies to the left of zero on the number line are called negative numbers.

Integers-2

 

Introduction to Zero

The number Zero

  • The number zero means an absence of value.

The Number Line

Integers

  • Collection of all positive and negative numbers including zero are called integers. ⇒ Numbers …, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, … are integers.

Representing Integers on the Number Line

Integers-3

 

  • Draw a line and mark a point as 0 on it
  • Points marked to the left (-1, -2, -3, -4, -5, -6) are called negative integers.
  • Points marked to the right (1, 2, 3, 4, 5, 6) or (+1, +2, +3, +4, +5, +6) are called positive integers.

Absolute value of an integer

  • Absolute value of an integer is the numerical value of the integer without considering its sign.
  • Example: Absolute value of -7 is 7 and of +7 is 7.

Ordering Integers

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

Addition of Integers

 Positive integer + Negative integer

  • Example: (+5) + (-2) Subtract: 5 – 2 = 3 Sign of bigger integer (5): + Answer: +3
  • Example: (-5) + (2) Subtract: 5-2 = 3 Sign of the bigger integer (-5): – Answer: -3

Positive integer + Positive integer

  • Example: (+5) + (+2) = +7
  • Add the 2 integers and add the positive sign.

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

Operations that can be performed on integers:

  • Addition
  • Subtraction
  • Multiplication
  • Division.

Subtraction of Integers

  • The subtraction of an integer from another integer is same as the addition of the integer and its additive inverse.
  • Example: 56 – (–73) = 56 + 73 = 129 and 14 – (8) = 14 – 8 = 6

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

Multiplication of Integers

  • Product of a negative integer and a positive integer is always a negative integer. 10×−2=−20
  • Product of two negative integers is a positive integer. −10×−2=20
  • Product of even number of negative integers is positive. (−2)×(−5)=10
  • Product of an odd number of negative integers is negative. (−2)×(−5)×(6)=−60

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Properties of Multiplication of Integers

Properties of Multiplication of Integers

Closure under Multiplication

  • Integer * Integer = Integer

Commutativity of Multiplication

  • For any two integers a and b, a × b = b × a.

Associativity of Multiplication

  • For any three integers a, b and c, (a × b) × c = a × (b × c).

Distributive Property of Integers

  • Under addition and multiplication, integers show the distributive property.
  • For any integers a, b and c, a × (b + c) = a × b + a × c.

Multiplication by Zero

  • For any integer a, a × 0 = 0 × a = 0.

Multiplicative Identity

  • 1 is the multiplicative identity for integers.
  • a × 1 = 1 × a = a

Dividing Integers

Division of 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

Properties of Division of Integers

For any integer a,

  • a/0 is not defined
  • a/1=a

Integers are not closed under division.

Example: (–9)÷(–3)=3 result is an integer but (−3)÷(−9)=−3−9=13=0.33 which is not an integer.

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