NCERT 6TH CLASS MATHEMATICS CHAPTER WHOLE NUMBERS
When we count a set of objects, we start counting from one then go on to two, three, four, etc.
This is the natural way of counting any set of objects.
Hence, 1, 2, 3, 4,... are called counting numbers or natural numbers. The number of students in a class, the number of days in a week, and the number of trees in a garden are all examples of natural numbers.
Let us consider an example to understand the concept of zero. If we want to divide 7 sweets equally among 3 children, 1 sweet will be left. But if we were to divide 6 sweets equally among 3 children, we are left with no sweets.
Zero means absence of the item (or no item)
The numbers 1, 2, 3,...., are called natural number or counting numbers. Let us add one more number, i.e., zero (0), to the collection of natural numbers. Now the numbers are 0, 1, 2, 3,.... . These numbers are called whole numbers.
PROPERTIES OF ADDITION
· Closure Property : If ‘a’ and ‘b’ are two whole numbers and their sum is c, i.e., a + b = c, then c will always be a whole number. This property of a addition is called the closure property of addition.
For Example. : 3 + 4 = 7
2 + 8 = 10 i.e., whole number + whole number = whole number
· Commutative Property : If a and b are two whole numbers a + b = a. This property of addition, where the order of addition does not alter the sum, is called the commutative property of addition.
For Example. : 3 + 4 = 7
Also, 4 + 3 = 7
i.e., 3 + 4 = 4 + 3
· Associative Property : If a, b and c three whole numbers then, then a + (b + c) = (a + b) + c.
ln other words, in the addition of whole numbers, the sum does not change even if the grouping is changed. This property is called the associative property of addition.
For Example. : 2 + (3 + 4) = (2 + 3) + 4
2 + 7 = 5 + 4
9 = 9
· Additive ldentity : If a is a whole number, then a + 0 = 0 + a = a.
Hence, zero is called the additive identity of the whole numbers because it maintains (or does not change) the identity (value) of the numbers during the operation of addition.
For Example. : 7 + 0 = 7 = 0 + 7
PROPERTIES OF SUBTRACTION
· Closure Property : If a and b are two whole numbers, then a – b will be a whole number only if a is greater than b or a is equal to b. If a is smaller than b, than the answer will not be a whole number. Hence subtraction is not closed under whole numbers.
For Example. : 7 – 2 = 5 is whole number
but 3 – 8 is not a whole number
· Commutative Property : If a and b are two distinct whole numbers, then a – b is not equal to b – a. Hence, the commutative property is not true for subtraction of whole numbers.
For Example. : a – b ¹ b – a
7 – 2 ¹ 2 – 7
· Associative Property : If a, b and c are whole numbers, then (a – b) – c is not equal to a – (b – c). So, the associative property also not hold true for subtraction of whole numbers
For Example. (12 – 4) – 3 = 8 – 3 = 5
12 – (4 – 3) = 12 – 1 = 11
\ (12 – 4) – 3 ¹ 12 – (4 – 3)