Natural Numbers and Whole Numbers
Interactive, step-by-step solutions for Exercises 3(A) to 3(E), including number properties, operations, patterns, magic squares and objective questions.
Exercise 3(A)
5 fully explained questions
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
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(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).
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(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
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(i) By the Commutative Property of Addition: a + b = b + a.
(ii) By the Commutative Property of Addition: a + b = b + a.
(iii) By the Additive Identity property, adding 0 to a whole number gives the number itself.
(iv) By the Multiplicative Identity property, multiplying a whole number by 1 gives the number itself.
(v) By the Associative Property of Addition: 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
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(i) 3 + (5 + 4) = (3 + 5) + 4
Since L.H.S. = R.H.S., the result is verified.
(ii) 8 × (8 + 0) = 8 × 8 + 8 × 0
Since L.H.S. = R.H.S., the result is verified.
(iii) (7 + 6) × 10 = 7 × 10 + 6 × 10
Since L.H.S. = R.H.S., the result is verified.
(iv) (15 - 12) × 18 = 15 × 18 - 12 × 18
Since L.H.S. = R.H.S., the result is verified.
(v) 16 + 0 = 16
By the Additive Identity property, the result is verified.
(vi) 23 + (-23) = 0
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.
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(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.
Exercise 3(B)
6 fully explained questions
Consider two whole numbers a and b such that a is greater than b.
(i) Is a - b a whole number? Is this result always true?
(ii) Is b - a a whole number? Is this result always true?
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(i) Since a is greater than b, the difference a - b is a positive number.
So, a - b is a whole number, and yes, this result is always true (whenever a is greater than b).
(ii) Since a is greater than b, the difference b - a is a negative number, and negative numbers are not whole numbers.
So, b - a is not a whole number, and yes, this result is always true (whenever a is greater than b, b - a is always negative).
Write the identity number, if possible for subtraction of whole numbers.
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For any whole number x, we have x - 0 = x, but 0 - x ≠ x.
An identity element must give the same number when applied from both sides, but for subtraction 0 works only from the right side and not from the left.
Hence, the identity number for subtraction of whole numbers is not possible (it does not exist).
Fill in the blanks:
(i) 12 × (9 - 6) = =
(ii) 12 × 9 - 12 × 6 = =
(iii) Is 12 × (9 - 6) = 12 × 9 - 12 × 6?
(iv) Is this type of result always true?
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(i) 12 × (9 - 6) = 12 × 3 = 36.
(ii) 12 × 9 - 12 × 6 = 108 - 72 = 36.
(iii) Both results are equal to 36, so yes, 12 × (9 - 6) = 12 × 9 - 12 × 6.
(iv) Yes. By the Distributive Law of Multiplication over Subtraction, this type of result is always true.
Fill in the blanks:
(i) (16 - 8) × 24 = =
(ii) 16 × 24 - 8 × 24 = - =
(iii) Is (16 - 8) × 24 = 16 × 24 - 8 × 24?
(iv) Is this type of result always true?
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(i) (16 - 8) × 24 = 8 × 24 = 192.
(ii) 16 × 24 - 8 × 24 = 384 - 192 = 192.
(iii) Both results are equal to 192, so yes, (16 - 8) × 24 = 16 × 24 - 8 × 24.
(iv) Yes. By the Distributive Law of Multiplication over Subtraction, this type of result is always true.
Find the difference between the largest number of four digits and the smallest number of six digits.
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The largest number of four digits = 9999.
The smallest number of six digits = 100000.
Hence, the required difference is 90001.
Find the difference between the smallest number of eight digits and the largest number of five digits.
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The smallest number of eight digits = 10000000.
The largest number of five digits = 99999.
Hence, the required difference is 9900001.
Exercise 3(C)
6 fully explained questions
Fill in the blanks:
(i) 42 × 0 =
(ii) 592 × 1 =
(iii) 328 × 573 = × 328
(iv) 229 × = 578 × 229
(v) 32 × 15 = 32 × 6 + 32 × 7 + 32 ×
(vi) 23 × 56 = 20 × 56 + × 56
(vii) 83 × 54 + 83 × 16 = 83 × () = 83 × =
(viii) 98 × 273 - 75 × 273 = () × 273 = × 273
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(i) Any number multiplied by 0 gives 0.
(ii) Any number multiplied by 1 gives the number itself.
(iii) By the Commutative Property of Multiplication: a × b = b × a.
(iv) By the Commutative Property of Multiplication: a × b = b × a.
(v) Since 6 + 7 + 2 = 15, we have:
(vi) Since 23 = 20 + 3, we have:
(vii) Using the Distributive Law of Multiplication over Addition:
(viii) Using the Distributive Law of Multiplication over Subtraction:
Evaluate (using distributive property):
(i) 984 × 102
(ii) 385 × 1004
(iii) 446 × 10002
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(i) 984 × 102
Using the Distributive Law of Multiplication over Addition:
Hence, 984 × 102 = 100368.
(ii) 385 × 1004
Using the Distributive Law of Multiplication over Addition:
Hence, 385 × 1004 = 386540.
(iii) 446 × 10002
Using the Distributive Law of Multiplication over Addition:
Hence, 446 × 10002 = 4460892.
Evaluate using properties:
(i) 548 × 98
(ii) 924 × 988
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(i) 548 × 98
Using the Distributive Law of Multiplication over Subtraction:
Hence, 548 × 98 = 53704.
(ii) 924 × 988
Using the Distributive Law of Multiplication over Subtraction:
Hence, 924 × 988 = 912912.
Evaluate using properties:
(i) 679 × 8 + 679 × 2
(ii) 284 × 12 - 284 × 2
(iii) 55873 × 94 + 55873 × 6
(iv) 7984 × 15 - 7984 × 5
(v) 8324 × 1945 - 8324 × 945
(vi) 3333 × 987 + 13 × 3333
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(i) 679 × 8 + 679 × 2
Using the Distributive Law of Multiplication over Addition: a × b + a × c = a × (b + c).
Hence, 679 × 8 + 679 × 2 = 6790.
(ii) 284 × 12 - 284 × 2
Using the Distributive Law of Multiplication over Subtraction: a × b - a × c = a × (b - c).
Hence, 284 × 12 - 284 × 2 = 2840.
(iii) 55873 × 94 + 55873 × 6
Using the Distributive Law of Multiplication over Addition:
Hence, 55873 × 94 + 55873 × 6 = 5587300.
(iv) 7984 × 15 - 7984 × 5
Using the Distributive Law of Multiplication over Subtraction:
Hence, 7984 × 15 - 7984 × 5 = 79840.
(v) 8324 × 1945 - 8324 × 945
Using the Distributive Law of Multiplication over Subtraction:
Hence, 8324 × 1945 - 8324 × 945 = 8324000.
(vi) 3333 × 987 + 13 × 3333
Using the Distributive Law of Multiplication over Addition:
Hence, 3333 × 987 + 13 × 3333 = 3333000.
Find the product of the:
(i) greatest number of three digits and smallest number of five digits.
(ii) greatest number of four digits and the greatest number of three digits.
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(i) The greatest number of three digits = 999.
The smallest number of five digits = 10000.
Hence, the required product is 9990000.
(ii) The greatest number of four digits = 9999.
The greatest number of three digits = 999.
Hence, the required product is 9989001.
Fill in the blanks:
(i) (437 + 3) × (400 - 3) = 397 × =
(ii) 66 + 44 + 22 = 11 × () = 11 × =
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(i) Here, 437 + 3 = 440 and 400 - 3 = 397.
So, (437 + 3) × (400 - 3) = 440 × 397 = 397 × 440 [by commutative property]
(ii) Since 66 = 11 × 6, 44 = 11 × 4 and 22 = 11 × 2, using the distributive property:
Exercise 3(D)
4 fully explained questions
Show that:
(i) division of whole numbers is not closed.
(ii) any whole number divided by 1, always gives the number itself.
(iii) every non-zero whole number divided by itself gives 1 (one).
(iv) zero divided by any non-zero number is zero only.
(v) a whole number divided by 0 is not defined.
For each part, given above, give two suitable examples.
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(i) Division of whole numbers is not closed because the result of dividing one whole number by another is not always a whole number.
(ii) Any whole number divided by 1, always gives the number itself.
(iii) Every non-zero whole number divided by itself gives 1.
(iv) Zero divided by any non-zero whole number gives zero only.
(v) A whole number divided by 0 is not defined.
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For every non-zero whole number, dividing it by itself gives 1, i.e. x ÷ x = 1.
So, x ÷ x = x means x = 1.
Hence, the value of x is 1.
Fill in the blanks:
(i) 987 ÷ 1 =
(ii) 0 ÷ 987 =
(iii) 336 - (888 ÷ 888) =
(iv) (23 ÷ 23) - (437 ÷ 437) =
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(i) Any whole number divided by 1 gives the number itself.
(ii) Zero divided by any non-zero whole number gives 0.
(iii) Any non-zero whole number divided by itself gives 1.
(iv) Any non-zero whole number divided by itself gives 1.
Which of the following statements are true?
(i) 12 ÷ (6 × 2) = (12 ÷ 6) × (12 ÷ 2)
(ii) a ÷ (b - c) = ab - ac
(iii) (a - b) ÷ c = ac - bc
(iv) (15 - 13) ÷ 8 = (15 ÷ 8) - (13 ÷ 8)
(v) 8 ÷ (15 - 13) = 815 - 813
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(i)
Since L.H.S. ≠ R.H.S., statement (i) is false.
(ii) Division is not distributive over subtraction when the divisor is a difference. In general, a ÷ (b - c) ≠ ab - ac.
Statement (ii) is false.
(iii) When a difference is divided by the same non-zero number:
Statement (iii) is true.
(iv)
Since L.H.S. = R.H.S., statement (iv) is true.
(v)
Since L.H.S. ≠ R.H.S., statement (v) is false.
Exercise 3(E)
4 fully explained questions
For each pattern, given below, write the next three steps:
(i) 1 × 9 + 1 = 10
(ii) 9 × 9 + 7 = 88
(iii) 1 × 8 + 1 = 9
(iv) 111 ÷ 3 = 37
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(i) The next three steps are:
(ii) The next three steps are:
(iii) The next three steps are:
(iv) The next three steps are:
Complete each of the following magic squares.
| 6 | 7 | |
| 5 | 9 | |
| 8 | 4 |
| 4 | 8 | |
| 7 | ||
| 10 |
| 16 | 2 | |
| 10 | ||
| 4 |
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In a magic square, every row, column and diagonal has the same total. For a 3 by 3 magic square, the magic sum is three times the centre number.
(i) Magic sum = 3 × 5 = 15
| 6 | 7 | 2 |
| 1 | 5 | 9 |
| 8 | 3 | 4 |
(ii) Magic sum = 3 × 7 = 21
| 4 | 9 | 8 |
| 11 | 7 | 3 |
| 6 | 5 | 10 |
(iii) Magic sum = 3 × 10 = 30
| 16 | 2 | 12 |
| 6 | 10 | 14 |
| 8 | 18 | 4 |
Study the matchstick pattern carefully.
(i) If n denotes the figure number and S denotes the number of matchsticks, find S in terms of n.
(ii) Find the number of matchsticks required for the 15th figure and the 40th figure.
(iii) Describe the pattern in words.
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Each new figure adds one square that shares one side with the existing pattern. Therefore, every new figure needs 3 additional matchsticks.
| Figure number (n) | 1 | 2 | 3 | 4 |
| Matchsticks (S) | 7 | 10 | 13 | 16 |
In words, the number of matchsticks is 4 more than three times the figure number.
Observe the matchstick pattern and draw the next two figures.
(i) Draw figures 4 and 5.
(ii) Construct a table for the pattern.
(iii) If n denotes the figure number and L denotes the number of matchsticks, find L in terms of n.
(iv) Find the number of matchsticks in the 12th and 20th figures.
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The next two figures are:
| Figure number (n) | 1 | 2 | 3 | 4 | 5 |
| Matchsticks (L) | 2 | 4 | 6 | 8 | 10 |
Each figure contributes 2 matchsticks, so the rule is:
Multiple Choice Question
10 fully explained questions
The whole number that does not have its predecessor is:
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The predecessor of a whole number is obtained by subtracting 1 from it.
The predecessor of 0 would be 0 - 1 = -1, which is not a whole number.
So, 0 is the only whole number that does not have a predecessor.
Hence, option 2 is the correct option.
The predecessor of ten thousand is:
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Hence, option 2 is the correct option.
The value of 43 × 27 + 43 × 73 is equal to:
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Using the Distributive Law of Multiplication over Addition:
Hence, option 1 is the correct option.
484 × 101 is equal to:
View step-by-step answerOpen
Using the Distributive Law of Multiplication over Addition:
Hence, option 3 is the correct option.
57 × 13 - 57 × 3 is equal to:
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Using the Distributive Law of Multiplication over Subtraction:
Hence, option 2 is the correct option.
(17 ÷ 17) - (97 ÷ 97) is equal to:
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Every non-zero whole number divided by itself gives 1.
Hence, option 3 is the correct option.
The number line for the whole numbers between 3 and 7 is:
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The whole numbers between 3 and 7 are 4, 5 and 6.
The correct number line is the one in which only the points 4, 5 and 6 are marked between 3 and 7.
Hence, option 2 is the correct option.
The product of the greatest number of two digits and the smallest two digit number is:
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The greatest number of two digits = 99.
The smallest number of two digits = 10.
Hence, option 2 is the correct option.
If a ÷ a = a, then a is:
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For every non-zero whole number, a ÷ a = 1.
So, a ÷ a = a means a = 1.
Hence, option 3 is the correct option.
If 8 - (a - 7) = 8 + (4 - b), then:
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So, 15 - a = 12 - b, which gives b - a = -3.
Substituting values from option 2:
Hence, option 2 is the correct option.
Statement I-II Type Questions
2 fully explained questions
Which of the following options is correct?
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Statement 1: x - 0 = x shows that 0 is a right identity for subtraction. But 0 is an identity element only if it works from both the right and the left, i.e. 0 - x = x must also hold. Since 0 - x ≠ x, the number 0 is not the identity element for subtraction.
∴ Statement 1 is false.
∴ Statement 2 is false.
Hence, option 2 is the correct option.
Which of the following options is correct?
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Statement 1: Subtracting one whole number from another does not always give a natural number. For example, 5 - 5 = 0 (which is not a natural number) and 5 - 8 = -3 (which is not even a whole number).
∴ Statement 1 is false.
∴ Statement 2 is true.
Hence, option 4 is the correct option.
Assertion Reason Type Questions
2 fully explained questions
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Reason (R): The relation a + (b + c) = (a + b) + c is exactly the statement of the associative property of addition of natural numbers. So, the reason is true and it correctly explains the assertion.
Both A and R are true.
Hence, option 3 is the correct option.
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Reason (R): For any three numbers x, y and z, x × (y - z) = x × y - x × z, which is the distributive property of multiplication over subtraction. So, the reason is true.
A is false and R is true.
Hence, option 2 is the correct option.
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What is the difference between natural numbers and whole numbers?
Natural numbers begin with 1, while whole numbers begin with 0. Every natural number is a whole number, but 0 is a whole number that is not a natural number.
What is the smallest natural number?
The smallest natural number is 1.
What is the smallest whole number?
The smallest whole number is 0.
Do natural numbers and whole numbers have a largest number?
No. Both sets continue without end, so neither has a largest number.
How do we find the successor and predecessor of a number?
Add 1 to find the successor and subtract 1 to find the predecessor.
What is the distributive property of multiplication?
Multiplication distributes over addition and subtraction: a times the sum or difference of two numbers equals the sum or difference of the two separate products.
Why is division by zero not defined?
There is no number that can be multiplied by zero to produce a non-zero dividend, so division by zero is not defined.
