Fill in the blanks:
(i) Smallest natural number is ...............
(ii) Smallest whole number is ...............
(iii) Largest natural number is ...............
(iv) Largest whole number is ...............
(v) All natural numbers are ...............
(vi) All whole numbers are not ...............
(vii) Successor of 4099 is ...............
(viii) Predecessor of 4330 is ...............
Answer
(i) The smallest natural number is 1.
(ii) The smallest whole number is 0.
(iii) The largest natural number is not possible (natural numbers are infinite).
(iv) The largest whole number is not possible (whole numbers are infinite).
(v) All natural numbers are whole numbers.
(vi) All whole numbers are not natural numbers.
(vii) Successor of 4099 = 4099 + 1 = 4100.
(viii) Predecessor of 4330 = 4330 - 1 = 4329.
State true or false:
(i) Whole numbers are closed for addition.
(ii) If a and b are any two whole numbers, then a + b is not a whole number.
(iii) If a and b are any two whole numbers, then a + b = b + a.
(iv) 0 + 18 = 18 + 0.
(v) Addition of whole numbers is associative.
(vi) 10 + 12 + 16 = (10 + 12) + 16 = 10 + (12 + 16).
Answer
(i) True. The sum of two whole numbers is always a whole number, so whole numbers are closed for addition.
(ii) False. By the closure property of addition, a + b is always a whole number.
(iii) True. Addition of whole numbers is commutative, so a + b = b + a.
(iv) True. By the commutative property, 0 + 18 = 18 + 0 = 18.
(v) True. Addition of whole numbers is associative.
(vi) True. By the associative property, (10 + 12) + 16 = 10 + (12 + 16) = 38.
Fill in the blanks:
(i) 54 + 234 = 234 + ...............
(ii) 332 + 497 = ............... + 332
(iii) 286 + 0 = ...............
(iv) 286 × 1 = ...............
(v) a + (b + c) = (a + ...............) + c
Answer
(i) By the Commutative Property of Addition: a + b = b + a.
54 + 234 = 234 + 54.
(ii) By the Commutative Property of Addition: a + b = b + a.
332 + 497 = 497 + 332.
(iii) By the Additive Identity property, adding 0 to a whole number gives the number itself.
286 + 0 = 286.
(iv) By the Multiplicative Identity property, multiplying a whole number by 1 gives the number itself.
286 × 1 = 286.
(v) By the Associative Property of Addition: a + (b + c) = (a + b) + c.
a + (b + c) = (a + b) + c.
Verify that:
(i) 3 + (5 + 4) = (3 + 5) + 4
(ii) 8 × (8 + 0) = 8 × 8 + 8 × 0
(iii) (7 + 6) × 10 = 7 × 10 + 6 × 10
(iv) (15 - 12) × 18 = 15 × 18 - 12 × 18
(v) 16 + 0 = 16
(vi) 23 + (-23) = 0
Answer
(i) 3 + (5 + 4) = (3 + 5) + 4
L.H.S. = 3 + (5 + 4) = 3 + 9 = 12.
R.H.S. = (3 + 5) + 4 = 8 + 4 = 12.
Since L.H.S. = R.H.S., the result is verified.
(ii) 8 × (8 + 0) = 8 × 8 + 8 × 0
L.H.S. = 8 × (8 + 0) = 8 × 8 = 64.
R.H.S. = 8 × 8 + 8 × 0 = 64 + 0 = 64.
Since L.H.S. = R.H.S., the result is verified.
(iii) (7 + 6) × 10 = 7 × 10 + 6 × 10
L.H.S. = (7 + 6) × 10 = 13 × 10 = 130.
R.H.S. = 7 × 10 + 6 × 10 = 70 + 60 = 130.
Since L.H.S. = R.H.S., the result is verified.
(iv) (15 - 12) × 18 = 15 × 18 - 12 × 18
L.H.S. = (15 - 12) × 18 = 3 × 18 = 54.
R.H.S. = 15 × 18 - 12 × 18 = 270 - 216 = 54.
Since L.H.S. = R.H.S., the result is verified.
(v) 16 + 0 = 16
L.H.S. = 16 + 0 = 16 = R.H.S.
By the Additive Identity property, the result is verified.
(vi) 23 + (-23) = 0
L.H.S. = 23 + (-23) = 23 - 23 = 0 = R.H.S.
The result is verified.
State true or false:
(i) The sum of two odd numbers is an odd number.
(ii) The sum of two odd numbers is an even number.
(iii) The sum of two even numbers is an even number.
(iv) The sum of two even numbers is an odd number.
(v) The sum of an even number and an odd number is odd number.
(vi) Every whole number is a natural number.
(vii) Every natural number is a whole number.
(viii) Every whole number + 0 = The whole number itself.
(ix) Every whole number × 1 = The whole number itself.
(x) Commutativity and associativity are properties of natural numbers and whole numbers both.
(xi) Commutativity and associativity are properties of addition for natural numbers and whole numbers both.
(xii) If x is a whole number then -x is also a whole number.
Answer
(i) False. The sum of two odd numbers is an even number. For example, 3 + 5 = 8.
(ii) True. The sum of two odd numbers is an even number. For example, 7 + 9 = 16.
(iii) True. The sum of two even numbers is an even number. For example, 4 + 6 = 10.
(iv) False. The sum of two even numbers is an even number, not an odd number.
(v) True. The sum of an even number and an odd number is an odd number. For example, 4 + 3 = 7.
(vi) False. 0 is a whole number but it is not a natural number, so every whole number is not a natural number.
(vii) True. Every natural number is a whole number.
(viii) True. Adding 0 to any whole number gives the number itself (additive identity).
(ix) True. Multiplying any whole number by 1 gives the number itself (multiplicative identity).
(x) False. Commutativity and associativity are not properties of every operation (for example, they do not hold for subtraction and division). The statement is incomplete as it does not mention the operation.
(xi) True. For the operation of addition, both commutativity and associativity hold for natural numbers as well as whole numbers.
(xii) False. If x is a whole number, then -x is a negative number, and negative numbers are not whole numbers.