# Properties of Whole Numbers

The properties of whole numbers are as follows:

• The number 0 is the first and the smallest whole numbers.

• All natural numbers along with zero are called whole numbers

• There is no last or greatest whole number.

• There is no largest whole number since they are infinite.

• There is infinitely many or uncountable number of whole numbers.

• All natural numbers are whole numbers.

• Each number is 1 more than its previous number.

• All whole numbers are not natural numbers.

For example: 0 is a whole number but it is not a natural number.

• Whole numbers are denoted by 'W' normally.

Note:

The system has infinite numbers.

Thus, W = {0, 1, 2, 3, 4, ……….}

Even Whole Numbers (E):

A system of whole numbers, which are divisible by 2 or are multiples of 2, is called a set of even numbers. It is denoted by 'E'.

Thus, E = {2, 4, 6, 8, 10, 12, .....}

There are infinite even numbers.

Odd Whole Numbers (O):

A system of whole numbers, which are not divisible by 2 or are not multiples of 2, is called a set of odd numbers. It is denoted by 'O'.

Thus, O = {1, 3, 5, 7, 9, 11, .....}

There are infinite odd numbers.

Note:

0’ (Zero) is neither a negative number nor a positive number, it’s a natural number.

Having learnt about four basic operations are addition, subtraction, multiplication and division on whole numbers. We shall now study the properties of these operations on whole numbers. Set of whole numbers: w = {0, 1, 2, 3 ,............}

## I: Properties of Addition of Whole Numbers:

A: Closure Property of Addition of Whole Numbers:

The sum of any two whole numbers is a whole number.
Hence, whole numbers are closed under addition. If x and y are any two whole numbers, then x + y is also a whole number.

For Example:

(i) 3 + 8 = 11 (a whole number)

(ii) 25 + 42 = 67 (a whole number)

B: Commutative Property of Addition of Whole Numbers:

Two whole numbers can be added in any order.

Hence, addition is commutative for whole numbers.

If a and b are any whole numbers, then

x + y = y + x

For Example:

(i) 0 + 7 = 7 and 7 + 0 = 7

(ii) 85 + 73 = 158 and 73 + 85 = 158

REMEMBER

In whatever order two whole numbers are added, their sum always remains the same.

C: Additive Property Zero of Whole Numbers:

The sum of a whole number and 0 is the whole number itself. Zero is called the identity element under addition or additive identity for whole numbers.

If x is any whole number, then

x + 0 = 0 + x = x

For Example:

(i) 752 + 0 = 0 + 752 = 752

(ii) 2565 + 0 = 0 + 2565 = 2565

D: Associative Property Zero of Whole Numbers:

Three or more whole numbers can be grouped in any order to find their sum.

Hence, addition is associative for whole numbers.

If x, y and z are any three whole numbers, then

(x + y) + z = x + (y + z)

For Example:

(i) (645 + 707) + 883 = 1,352 + 883 = 2,235

and 645 + (707 + 883) = 645 + 1590 = 2,235

Therefore, (645 + 707) + 883 = 645 + (707 + 883)

(ii) (888 + 2,330) + 755 = 3218 + 755 = 3,973

and 888 + (2,330 + 755) = 888 + 3,085 = 3,973

Therefore, (888 + 2,330) + 755 = 888 + (2,330 + 755)

By using the commutative and associative properties of addition of whole numbers, often it becomes convenient to add two or more whole numbers as shown in the following examples.

Solved Examples Properties of Whole Numbers

1. Find the sum by suitable rearrangement.

(i) 943 + 508 + 557

(ii) 2920 + 532 + 2580 + 468

Solution:

(i) 943 + 508 + 557

(943 + 557) + 508

1500 + 508 = 2008

(ii) 2920 + 532 + 2580 + 468

(2920 + 2580) + (532 + 468)

= 5500 + 1000

= 6500

2. Find the sum of the following.

(i) 12873 + 9999

(ii) 99999 + 289326

Solution:

(i) 12873 + 9999

= 12873 + (10000 - 1)

= (12873 + 10000) - 1

= 22873 - 1

= 22872

(ii) 99999 + 289326

= (100000 - 1) + 289326

= (289326 + 100000) - 1

= 389326 - 1

= 389325

## II: Properties of Subtraction of Whole Numbers:

A: Closure Property of Subtraction of Whole Numbers:

The difference of two whole numbers is not always a whole number. Subtraction results in a whole number only when the number from which the other is subtracted is greater than or equal to the other number.

Hence, whole numbers are not closed under subtraction.

If x and y are two whole numbers such that x > y or x = y, then x - y is whole number but if x < y, then subtraction x - y is not possible in whole numbers.

For Example:

(i) Subtraction of two equal whole numbers results in whole number 0.

For Example:

8 - 8 = 0,

57 - 57 = 0,

549 - 549 = 0, etc.

(ii) Subtraction of a smaller whole number from a larger whole number results in a whole number,

For Example:

48 - 15 = 33,

249 - 64 = 185,

100000 - 1 = 99,999, etc.

(iii) Subtraction of a larger whole number from a smaller one does not result in a whole number.

For Example:

45 - 84 is not equal to a whole number.

235 - 460 is not equal to a whole number.

B: Commutative Property of Subtraction of Whole Numbers:

Two whole numbers cannot be subtracted in any order. Hence, subtraction is not commutative for whole numbers. If x and y are any two whole numbers, then

x - y ≠ y - x

For Example:

(i) 42 - 12 = 30, a whole number but 12 - 42 is not defined in whole numbers.

(ii) 850 - 170 = 680, a whole number but 170 - 850 is not defined in whole numbers.

C: Property of Zero of Subtraction of Whole Numbers:

If zero is subtracted from any whole number, the result is the number itself. Thus, the identity property of a zero holds good when 0 is subtracted from a whole number. When a whole number other than 0 is subtracted from 0, the result is not a whole number.

If x is any whole number, then x - 0 = x but 0 - x is not defined in whole numbers.

For Example:

(i) 39 - 0 = 39, a whole number but 0 - 39 is not defined in whole numbers.

(ii) 559 - 0 = 559, a whole number but 0 - 559 is not defined in whole numbers.

C: Associative Property of Subtraction of Whole Numbers:

Subtraction is not associative for whole numbers. If x, y, z are any three whole numbers, then

(x - y)- z ≠ x - (y - z)

For Example:

(8 - 4) - 2 = 4 - 2 = 2

and 8 - (4 - 2) = 8 - 2 = 6

Therefore, (8 - 4) - 2 ≠ 8 - (4 - 2)

D: Inverse operations of Subtraction of Whole Numbers:

The operations of addition and subtraction are inverse of each other. Thus, subtracting a whole number, say, 18 from the whole number 60 can be considered as finding a whole number which when added to the whole number 18 results in 60.

60 - 18 = 42 ⟹ 18 + 42 = 60

If x, y, z are any three whole numbers such that

x - y = z, then y + z = x.

For Example:

(i) 30 - 17 = 13 ⟹ 17 + 13 = 30

(ii) 23 - 8 = 15 ⟹ 8 + 15 = 23

## II: Properties of Multiplication of Whole Numbers:

A: Closure Property of Multiplication of Whole Numbers:

The product of two whole numbers is always a whole number.

Hence, whole numbers are closed under multiplication.

If m and n are any two whole numbers, then their product m × n is also a whole number.

For Example:

(i) 8 × 6 = 54 a whole number

(ii) 42 × 4 = 168 a whole number

B: Commutative Property of Multiplication of Whole Numbers:

Two whole numbers can be multiplied in any order. Hence, multiplication is commutative for whole numbers.

If y and z are any two whole numbers, then

y × z = z × y

For Example:

(i) 8 × 4 = 32 and 4 × 8 = 32

Therefore, 8 × 4 = 4 × 8

(ii) 15 × 20 = 300 and 20 × 15 = 300

Therefore, 15 × 20 = 20 × 15

C: Multiplicative Property of Zero of Whole Numbers:

The product of any whole number and zero is always zero.

If z is any whole number, then z × 0 = 0 × z = 0

For Example:

(i) 6 × 0 = 0 × 6 = 0

(ii) 365 × 0 = 0 × 365 = 0

D: Existence of Multiplicative Identity Property of Whole Numbers:

The product of any whole number and 1 is the number itself.

1 is called the multiplicative identity or identity element for whole numbers under multiplication.

If z is any whole number, then

× 1 = 1 × z = z

For Example:

(i) 72 × 1 = 1 × 72 = 72

(ii) 245 × 1 = 1 × 245 = 245

E: Associative Property Multiplication of Whole Numbers:

Three or more whole numbers can be grouped in any order to find their product. Hence, multiplication is associative for whole numbers.

If a, b, c are any three whole numbers, then

(a × b) × c = a × (b × c)

For Example:

(3 × 5) × 4 = 15 × 4 = 60

and 3 × (5 × 4) = 3 × 20 = 60

Therefore, (3 × 5) × 4 = 3 × (5 × 4)

F: Distributive Property Multiplication Over Addition of Whole Numbers:

The multiplication of whole numbers distributes over their addition.

If a, b, c are any three whole numbers, then

a × (b + c) = a × b + a × c

For Example:

3 × (5 + 7) = 3 x (12) = 36

and 3 × 5 + 3 × 7 = 15 + 21 = 36

Therefore, 3 × (5 + 7) = 3 × 5 + 3 × 7

F: Distributive Property Multiplication Over Subtraction of Whole Numbers:

The multiplication of whole numbers distributes over their subtraction.

If a, b, c are whole numbers such that b > c, then

a × (b - c) = a × b - a × c

For Example:

18 × (8 - 5) = 18 × 3 = 54

and 18 × 8 - 18 × 5 = 144 - 90 = 54

Therefore, 18 × (8 - 5) = 18 × 8 - 18 × 5

Solved Examples on Properties of Multiplication of Whole Numbers:

1. Determine each of the following products by suitable rearrangement.

(i) 4 × 6798 × 250

(ii) 2388 × 25 × 2 × 40

Solution:

(i) 4 × 6798 × 250

= 6798 × (4 × 250)

= 6798 x 1000

= 6798000

(ii) 2388 × 25 × 2 × 40

= (2388 × 2) × (25 × 40)

= 4776 × 10000

= 4776000

2. Find the value of each of the following using various properties.

(i) 638 × 7 + 638 × 3

(ii) 997 × 10 × 982 - 882 × 9970

Solution:

(i) 638 × 7 + 638 × 3

= 638 × (7 + 3)

= 638 × 10

6380

(ii) 997 × 10 × 982 - 882 × 9970

= (997 × 10 × 982) - (9970 × 882)

= (9970 × 982) - (9970 × 882)

= 9970 × (982 - 882)

= 9970 × 100

= 997000

3. Using distributive property of multiplication over addition/subtraction in whole numbers, find the product of each of the following.

(i) 498 × 102

(ii) 736 ×1003

(iii) 482 × 64

Solution:

(i) 498 × 102

= 498 × (100 + 2)

= 498 × 100 + 498 × 2

= 49800 + 996

= 50796

(ii) 736 × 1003

= 736 × (1000 + 3)

= 736 × 1000 + 736 × 3

= 736000 + 2208

= 738208

(iii) 482 × 64

= (400 + 80 + 2) × 64

= 400 × 64 + 80 × 64 + 2 × 64

= 25600 + 5120 + 128

= 30848

4. Find the product of the greatest number of five digits and the greatest number of four digits.

Solution:

Greatest five digit number = 99999

Greatest four digit number = 9999

Now find the product of greatest 5-digit and greatest 4-digit numbers

Product = 99999 × 9999

= 99999 × (10000 - 1)

= 99999 x 10000 - 99999 × 1 [Using distributive property]

= 999990000 - 99999

= 999990000 - (100000 - 1)

= 999990000 - 100000 + 1

= 999890000 + 1

= 999890001

5. 29 laptops and 29 tables were purchased for a new office. If each laptop costs $2,632 and each table costs$368, find the total amount spend.

Solution:

Cost of 29 laptops = $2,632 × 29 Cost of 29 chairs =$368 × 29

Total cost = $(2,632 × 29 + 368 × 29) =$ 29 (2,632 + 368); [Using Distributive Property]

= $29 (3,000) =$ (30 - 1) (3,000)

= $(30 × 3,000 - 1 × 3,000); [Using Distributive Property] =$ (90,000 - 3,000)

= $87,000 Hence, the total amount spend =$ 87,000

## II: Properties of Division of Whole Numbers:

A: Closure Property of Division of Whole Numbers:

The quotient of two whole numbers when one is divided by the other is not always a whole number.

Hence, whole numbers are not closed under division. If x and y (y ≠ 0) are whole numbers, then x ÷ y not always a whole number.

For Example:

(i) 18 ÷ 3 = 6, a whole number

(ii) 23 ÷ 7, is not a whole number

B: Commutative Property of Division of Whole Numbers:

Division is not commutative for (non-zero) whole numbers.

If x and y are whole numbers, then

÷ y y ÷ x (x y, x 0, y 0)

For Example:

24 ÷ 6 = 4, a whole number and 6 ÷ 24 = $$\frac{6}{24}$$ = 1/4 not a whole number.

Clearly, 24 ÷ 6 6 ÷ 24

C: Associative Property of Division of Whole Numbers:

Division is not associative for whole numbers.

If x, y, z are whole numbers, then

(x ÷ y) ÷ z ≠ x ÷ (y ÷ z); (y ≠ 0, z ≠ 0)

For Example:

(30 ÷ 5) ÷ 6 = 6 ÷ 6 = 1

and 30 ÷ (5 ÷ 6) = 30 ÷ $$\frac{5}{6}$$ = 30 × $$\frac{6}{5}$$ = 36

Therefore, (30 ÷ 5) ÷ 6 30 ÷ (5 ÷ 6)

D: Property of Division by 1 of Whole Numbers:

Any whole number divided by 1 gives the quotient as the number itself.

If x is any whole number, then x ÷ 1 = x.

For Example:

(i) Since,    1 × 5 =  5;      Therefore, 5 ÷ 1   =  5

(ii) Since, 85 × 1 = 85;      Therefore, 85 ÷ 1 = 85

(iii) Since,  0 × 1 =  0;       Therefore, 0 ÷ 1   =  0

(iv) Since,  1 × 1 =  1;       Therefore, 1 ÷ 1   =  1

E: Property of Division by a Number itself of Whole Numbers:

Any whole number (other than zero) divided by itself gives 1 as the quotient.

If x is a whole number (other than zero), then x ÷ x = 1.

For Example:

(i) Since,    6 =   6 × 1;        Therefore,     6 ÷ 6  =  1

(ii) Since,  43 = 43 × 1;        Therefore,  43 ÷ 43 = 1

(iii) Since, 85 = 85 × 1;        Therefore, 85 ÷ 85  =  85

(iv) Since,  1 =   1 × 1;        Therefore,    1 ÷ 1   =  1

F: Property of Zero Divided by a Whole Numbers:

Zero divided by any whole number (other than 0) gives the quotient as 0.

If x is a whole number other than 0, then 0 ÷ x = 0.

For Example:

(i) Since,       0 × 8 = 0;       Therefore, 0 ÷ 8    = 0

(ii) Since,      0 × 5 = 0;       Therefore,  0 ÷ 5   = 0

(iii) Since,    0 × 49 = 0;      Therefore, 0 ÷ 49  = 0

(iv) Since,  0 × 845 = 0;      Therefore, 0 ÷ 845 = 0

G: Property of Divided by Zero of a Whole Numbers:

Division by zero is not defined. To evaluate 4 ÷ 0, we need to find a whole number which when multiplied by zero results in 4. No such number can be obtained as the product of any whole number and zero is always zero.

Hence, division of a whole number by zero is meaningless.

H. Division Algorithm of a Whole Numbers:

Dividend = Divisor × Quotient + Remainder,

where dividend is the number to be divided, divisor is the number by which the dividend is divided, quotient is the number which is left over after division.

In general, if a and b are whole numbers such that a > b and b ≠ 0 and on dividing a by b the quotient and the remainder obtained are q and r respectively, then a = bq + r, where 0 ≤ r < b.

For Example:

Divide 69 by 5 and verify the division algorithm.

Clearly, 69 = 5 × 13 + 4

I. Even and Odd Whole Numbers:

Even whole numbers are the whole numbers which are divisible by 2.

For example, 0, 2, 4, 6, 8, 10, 12, etc. are all even numbers.

Odd whole numbers are the whole numbers which are not divisible by 2.

For example, 1, 3, 5, 7, 9, 11, 13, etc. are all odd numbers.

I. Worksheet on Properties of Addition of Whole Numbers:

1. Fill in the blanks.

(i) If two whole numbers are added, the sum is always a _____ number.

(ii) Whole numbers are _____ under addition.

(iii) _____ is called the identity element for whole numbers under addition.

(iv) In whatever order two whole numbers are added, their sum always remains _____.

(v) If a is a whole number such that a + a = a, then a = _____.

1. (i) whole number

(ii) Identity element

(iii) zero

(iv) the same

(v) 0

2. Fill in the blanks to make each of the following a true statement.

(i) 6728 + 0 = __________

(ii) __________ + 125831 = 125831

(iii) 5239 + 928 = 928 +

(iv) 9325 + __________ = 998 + 9325

(ν) 337 + (892 + 617) = (337 + 892) + __________

2. (i) 6728

(ii) 0

(iii) 5239

(iv) 998

(ν) 617

3. Find the sum by using a short method.

(i) 312875 + 9999

(ii) 1289345 + 99999

3. (i) 312875 + 9999

= 312875 + (10000 - 1)

= (312875 + 10000) - 1

= 322875 - 1

= 322874

(ii) 1289345 + 99999

= 1289345 + (100000 - 1)

= (1289345 + 100000) -1

= 1389345 -1

= 1389344

4. Determine each of the following sums using suitable rearrangement.

(i) 647 + 142 + 858 + 253

(ii) 211 + 684 + 389 + 5816

(iii) 15209 + 378 + 791 + 122

(iv) 496 + 497 + 498 + 499 + 1 + 2 + 3 + 4

4. (i) 647 + 142 + 858 + 253

= (647 + 858) + (253 + 142)

= 1505 + 395

= 1900

(ii) 211 + 684 + 389 + 5816

= (211 + 389) + (684 + 5816)

= 600 - 6500

= 7100

(iii) 15209 + 378 + 791 + 122

(15209 + 791) + (378 + 122)

= 16000 + 500

= 16500

(iv) 496 + 497 + 498 + 499 + 1 + 2 + 3 + 4

= (496 + 4) + (497 + 3) + (497 + 2) + (499 + 1)

= 500 + 500 + 500 + 500

= 2000

II. Worksheet on Properties of Subtraction of Whole Numbers:

1. For each of the following addition sentence write two subtraction sentences.

(i) 23 + 12 = 35

(ii) 326 + 422 = 748

(iii) 135 + 92 = 227

1. (i) 23 + 12 = 35

⟹ 23 = 35 - 12

⟹ 12 = 35 - 23

(ii) 326 + 422 = 748

⟹ 326 = 748 - 422

⟹ 422 = 748 - 326

(iii) 135 + 92 = 227

⟹ 135 = 227 - 92

⟹ 92 = 227 - 135

2. Perform the following subtractions and check your results by corresponding additions.

(i) 9,328 - 427

(ii) 41,000 - 10,999

(iii) 100,000 - 75,652

(iv) 6,050,501 - 787,879

2. (i) 9328 - 427 = 8901

Check: 8901 + 427 = 9328

(ii) 41000 - 10999 = 30,001

Check: 30,001+ 10,999=41000

(iii) 100,000 - 75,652 = 24,348

Check: 24,348 + 75,652 = 100,000

(iv) 6,050,501 - 787,879 = 5,262,622

Check: 5,262,622 + 787,879 = 6,050,501

3. Fill in the blanks.

(i) 100,000 - __________ = 8,019

(ii) 759,999 - 83,599 = __________

3. (i) 91,981

(ii) 676,400

4. Find the difference between the smallest number of 5 digits and the largest number of 4 digits.

4. 1

5. Find the difference between the smallest number of 7 digits and the largest number of 4 digits.

5. 990,001

6. Find the whole number x when

(i) x + 7 = 20

(ii) x + 43 = 215

(iii) x - 17 = 41

(iv) x - 30,298 = 32,307

6. (i) x = 13

(ii) x = 172

(iii) x = 58

(iv) x = 62,605

7. The population of a town was 150800. In one year it increased by 4290 due to new births. In the increased population if the number of men is 61296, determine the number of women.

7. 93,794

III. Worksheet on Properties of Multiplication of Whole Numbers:

1. Fill in the blanks.

(i) 992 × 0 = __________

(ii) 6675 × 1 = __________

(ii) 2198 × 557 = 557 × __________

(iv) 88 × (125 × 45) = (88 × 125) × __________

(v) 90 × 100 × __________ = 900000

(vi) 125 × (86 + 95) = 125 × 86 + __________ × 95

1. (i) 0

(ii) 6675

(ii) 2198

(iv) 45

(v) 100

(vi) 125

2. Determine each of the following products by suitable rearrangements.

(i) 25 × 928 × 4

(ii) 50 × 6793 × 2

(iii) 25 × 8 × 5432

(iv) 874 × 625 × 16

(v) 40 × 1345 × 25 × 2

(vi) 20 × 625 × 8 × 50

2. (i) 25 × 928 × 4

= 928 (25 × 4)

= 928 × 100

= 92800

(ii) 50 × 6793 × 2

= 6793 (50 × 2)

= 6793 × 100

= 679300

(iii) 25 × 8 × 5432

= (25 × 8) × 5437

= 200 × 5432

= 1,086,400

(iv) 874 × 625 × 16

= 874(625 × 16)

= 874 × 10000

= 8,740,000

(v) 40 × 1345 × 25 × 2

= (1345 × 2)(40 × 25)

= 2690 × 1000

= 2,690,000

(vi) 20 × 625 × 8 × 50

= (625 × 8)(50 × 20)

= 5000 * 1000

= 5,000,000

3. Find the value of each of the following using various properties.

(i) 199 × 29 + 199 × 71

(ii) 1679 × 999 + 1679

(iii) 389 × 17 + 389 × 23 + 389 × 60

(iv) 683 × 36 + 683 × 17 - (683 × 48) - (5 × 683)

(v) 2398 × 761 - 2398 × 661

(vi) 12345 × 12345 - 12345 × 2345

3. (i) 199 × 29 + 199 × 71

= 199 (29 + 71)

= 199 × 100

= 19,900

(ii) 1679 × 999 + 1679

= 1679 × (999 + 1)

= 1679 × 1000

= 1,679,000

(iii) 389 × 17 + 389 × 23 + 389 × 60

= 389 (17 + 23 + 60)

= 389 × 100

= 38,900

(iv) 683 × 36 + 683 × 17 - (683 × 48) - (5 × 683)

= 683 (36 + 17) - 683 × 48 - 5 × 683

= 683 (53) - 683 (48 + 5)

= 683 (53) - 683 (53)

= 683(53 - 53)

= 683 × 0

= 0

(v) 2398 × 761 - 2398 × 661

= 2,398 (761 - 661)

= 2,398 × 100

= 239,800

(vi) 12345 × 12345 - 12345 × 12345

= 12345 (12345 - 12345)

= 12345 (0)

= 0

4. Use the distributive property of multiplication over addition, subtraction to find the following.

(i) 932 × 103

(ii) 345 × 1008

(iii) 3847 × 97

(iv) 350 × 64

4. (i) 932 × 103

= 932 (100 + 3)

= 932 × 100 + 932 × 3

= 93200 + 2796

= 95,996

(ii) 345 × 1,008

= 345 (1,000 + 8)

= 345 × 1,000 + 345 ×

= 345,000 + 2,760

= 347,760

(iii) 3847 × 97

= 3847 (100 - 3)

= 3847 × 100 - 3847 × 3

= 384,700 - 11,541

= 373,159

(iv) 350 × 64

= (300 + 50) × 64

= 300 × 64 + 50 × 64

= 19,200 + 3,200

= 22,400

Worksheet on Properties of Division of Whole Numbers:

1. Find the value of the following

(i) 923287 ÷ 1

(ii) 0 ÷ 6993

(iii) 990 ÷ (640 ÷ 64)

(iv) 7896 ÷ (2347 ÷ 2347)

(v) 999 + (2975 ÷ 2975)

(vi) (6208 ÷ 6208) - (2358 ÷ 2358)

1. (i) 923287

(ii) 0

(iii) 99

(iv) 7896

(v) 1000

(vi) 0

2. Divide and check the result by division algorithm in each of the following.

(i) 2873 ÷ 35

(ii) 93875 ÷ 651

(iii) 121878 ÷ 88

(iv) 254254 ÷ 675

2. (i) 2873 = 35 × 82 + 3

(ii) 93875 651 × 128 + 447

(iii) 121878 88 × 1384 + 86

(iv) 254254 675 × 376 + 454

3. Divide and find out the quotient and remainder. Check your answer.

(i) 94335 ÷ 93

(ii) 10000 ÷ 125

(ⅲ) 66087 ÷ 285

(iv) 99999 ÷ 423

3. (i) Quotient = 1014; Remainder = 33

94335 = 93 × 1014 + 33

(ii) Quotient = 80; Remainder = 0

10000 125 × 80 + 0

(ⅲ) Quotient = 231; Remainder = 252

66087 285 × 231 + 252

(iν) Quotient = 263; Remainder = 171

99999 423 × 263 + 171

● Whole Numbers

The Number Zero

Properties of Whole Numbers

Successor and Predecessor

Representation of Whole Numbers on Number Line

Properties of Subtraction

Properties of Multiplication

Properties of Division

Division as The Inverse of Multiplication

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• ### Numbers | Notation | Numeration | Numeral | Estimation | Examples

Numbers are used for calculating and counting. These counting numbers 1, 2, 3, 4, 5, .......... are called natural numbers. In order to describe the number of elements in a collection with no objects

The properties of addition whole numbers are as follows: Closure property: If a and b are two whole numbers, then a + b is also a whole number. In other words, the sum of any two whole numbers i

• ### Properties of Multiplication | Multiplicative Identity | Whole Numbers

There are six properties of multiplication of whole numbers that will help to solve the problems easily. The six properties of multiplication are Closure Property, Commutative Property, Zero Property, Identity Property, Associativity Property and Distributive Property.

• ### Estimating Sum and Difference | Reasonable Estimate | Procedure | Math

The procedure of estimating sum and difference are in the following examples. Example 1: Estimate the sum 5290 + 17986 by estimating the numbers to their nearest (i) hundreds (ii) thousands.

• ### Estimating Product and Quotient |Estimated Product |Estimated Quotient

The procedure of estimating product and quotient are in the following examples. Example 1: Estimate the product 958 × 387 by rounding off each factor to its greatest place.

• ### Representation of Whole Numbers on Number Line | Compare Whole Numbers

Numbers on a line is called the representation of whole numbers on number line. The number line also helps us to compare two whole numbers, i.e., to decide which of the two given whole numbers

• ### Properties of Subtraction |Whole Numbers |Subtraction of Whole Numbers

1. When zero is subtracted from the number, the difference is the number itself. For example, 8931 – 0 = 8931, 5649 – 0 = 5649 2. When a number is subtracted from itself the difference is zero. For example, 5485 – 5485 = 0 3. When 1 is subtracted from a number, we get its

• ### Whole Numbers | Definition of Whole Numbers | Smallest Whole Number

The whole numbers are the counting numbers including 0. We have seen that the numbers 1, 2, 3, 4, 5, 6……. etc. are natural numbers. These natural numbers along with the number zero

• ### Addition of Numbers using Number Line | Addition Rules on Number Line

Addition of numbers using number line will help us to learn how a number line can be used for addition. Addition of numbers can be well understood with the help of the number line.

• ### Subtraction of Numbers using Number Line |Subtracting with Number Line

Subtraction of numbers using number line will help us to learn how a number line can be used for subtracting one number from the another number.

• ### Worksheet on Reading and Writing Large Numbers|Writing Numbers in Word

Practice the questions given in the worksheet on reading and writing large numbers to group place values into periods in hundred, thousand, million and billion. The questions are related to writing

• ### Worksheet on Estimation | Estimate the Product |Nearest Tens, Hundreds

Practice the questions given in the worksheet on estimation. The questions are based on estimating the sum, difference, product and quotient to the nearest tens, hundreds and thousands.

• ### Properties of Adding Integers | Closure |Commutative | Associative ...

The properties of adding integers are discussed here along with the examples. 1. The addition (sum) of any two integers is always an integer. For example: (i) 5 + 9 = 14 ∈ Z (ii) (-5) + 9 = 4 ∈ Z

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## Recent Articles

1. ### Subtracting Integers | Subtraction of Integers |Fundamental Operations

Jun 13, 24 02:51 AM

Subtracting integers is the second operations on integers, among the four fundamental operations on integers. Change the sign of the integer to be subtracted and then add.

2. ### Properties of Subtracting Integers | Subtraction of Integers |Examples

Jun 13, 24 02:28 AM

The properties of subtracting integers are explained here along with the examples. 1. The difference (subtraction) of any two integers is always an integer. Examples: (a) (+7) – (+4) = 7 - 4 = 3

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4. ### Addition of Integers | Adding Integers on a Number Line | Examples

Jun 12, 24 01:11 PM

We will learn addition of integers using number line. We know that counting forward means addition. When we add positive integers, we move to the right on the number line. For example to add +2 and +4…