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↪ We know that Predecessor of any natural number is the number that comes before it, & we get it by subtracting one from the given number.
One has no predecessor in natural numbers because when we subtract one from itself, we get zero which is not a natural number.
↪ Also, when we subtract any number from itself, we get 0.
↪ 0 (zero) is not a part of counting number as we can't count 0. It was also not thought as a actual number for many years. But now, we know its significance.
"When we add 0 to the set of natural numbers, we get a new set of numbers called Whole numbers."
W={0,1,2,3...}
Time to Think
1) Are all natural number also whole numbers?2) Are all whole numbers also natural number?
3) Has the natural number 1 no predecessor?
4) Has the whole number 1 no predecessor?
5) Has the whole number 0 no predecessor?
6) Which is smallest whole number?
7) Which is smallest whole number?
Answer
1) Yes.2) No, 0 is not a natural number.
3) Yes.
4) No, predecessor of whole number 1 is 0.
5) Yes.
6) 0.
7) none.
Properties of Whole numbers
Closure property
➤ For addition- closed - sum of any two whole number is whole number,e.g.,0+2 = 2
5+6 = 11
➤ For multiplication- closed -product of any two whole number is whole number,e.g.,
0×2 = 0
5×6 = 30
➤ For subtraction- not closed -difference of any two whole number is not always whole number,e.g.,
0-2 = ?, not a whole number
6-5 = 1
5-6 = ?, not a whole number
➤ For division - not closed - division of any two whole number is not always whole number,e.g.,
4÷2 = 2
2÷4 = ?, not a whole number
6÷5 = ?, not a whole number
Commutative property
➤ For addition- commutative - we can add two whole numbers in any order & the result will be same, e.g.,0+2 = 2 ⇔ 2+0 =2
5+6 =11 ⇔ 6+5 =11
➤ For multiplication- commutative - we can multiply two whole numbers in any order & the result will be same, e.g.,
0×2 = 0 ⇔ 2×0 = 0
5×6 =30 ⇔ 6×5 =30
➤ For subtraction- not commutative - we can not subtract two whole numbers in any order, the result will get changed if we reverse the order, e.g.,
2-0 =2 ⇎ 0-2 =?
6-5 =1 ⇎ 5-6 =?
➤ For division - not commutative - we can not divide two whole numbers in any order, the result will get changed if we reverse the order, e.g.,
4÷2 =2 ⇎ 2÷4 =?
9÷3 =3 ⇎ 3÷9 =?
Associative property
➤For addition- associative- sum of natural numbers will give same result when the grouping of numbers is changed, e.g.,(0+2)+5 =2+5 =7 ⇔ 0+(2+5) =2+5 =7
(3+6)+7 =9+7 =16 ⇔ 3+(6+7) =3+13 =16
➤For multiplication- associative- product of natural numbers will give same result when the grouping of numbers is changed, e.g.,
(0×2)×4 =0×4 =0 ⇔ 0×(2×4)= 0×8 =0
(3×4)×7 =12×7 =84 ⇔ 3×(4×7) =3×28 =84
➤For subtraction- not associative- difference of natural numbers may not give same result when the grouping of numbers is changed, e.g.,
(4-2)-1=2-1=1 ⇎ 4-(2-1)= 4-1= 3 , results are not same
(6-3)-1=3-1=2 ⇎ 6-(3-1)= 6-2 =4 , results are not same
➤For division - not associative- division of natural numbers may not give same result when the grouping of numbers is changed, e.g.,
(6÷3)÷1=2÷1 =2 ⇎ 6÷(3÷1) =6÷3 =2, results are same
(16÷4)÷2=4÷2 =2 ⇎ 16÷(4÷2) =16÷2 =8, results are not same
Distributive Property/ Distribution of multiplication over addition :
Multiplying a number with sum of two numbers will give same result when the number is first multiplied with each addend & then adding the products.2×(0+5) = 2×5 = 10
2×(0+5) =(2×0)+(2×5) =0+10 =10 (using distributive property)
Additive Identity
When we add 0 to any whole number, we get the same number as a result and we can say that the number has kept its identity.0 is called additive identity for whole numbers (or any number).
Multiplicative Identity
When we multiply 1 to any whole number, we get the same number as a result and we can say that the number has kept its identity.1 is called multiplicative identity for whole numbers (or any number).
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