CLASS 6TH | UNIT 1: NUMBERS | NEGATIVE NUMBERS AND INTEGERS | REVISION NOTES

Negative Numbers

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

To know more about Application of Negative Numbers in Daily Life

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

• 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

• 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

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

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

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.
To know more about Number Lines