### Mathematics Class VI

Knowing Our Numbers
Whole Numbers
Playing With Numbers
Basic Geometrical Ideas
Fractions
Integers
Decimals
Data Handling
Mensuration
Algebra
Ratio and Proportion
Practical Geometry
Symmetry

# 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…

• 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

Properties of Addition and Subtraction of Integers

Operations on Integers

• Subtraction
• Multiplication
• Division.

Properties of Addition and Subtraction of Integers

• 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.
• (a + b) + c = a + (b + c) for all integers a, b and c.
• 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.
• Product of two negative integers is a positive integer.
• Product of even number of negative integers is positive.
• Product of an odd number of negative integers is negative.

Properties of Multiplication of Integers

• Closure under Multiplication
• Integer Integer = Integer
• Commutativity of Multiplication
• For any two integers a and b,
• Associativity of Multiplication
• For any three integers a, b and c,
• Distributive Property of Integers
• Under addition and multiplication, integers show the distributive property.
• For any integers a, b and c, .
• Multiplication by Zero
• For any integer a,
• Multiplicative Identity
• 1 is the multiplicative identity for integers.

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,

• is not defined

Integers are not closed under division.

Example: result is an integer but

which is not an integer.

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