The properties of adding whole numbers are as follows:

Closure of addition:

If a and b are two integers, then a + b is also an integer. In other words, the sum of two whole numbers is a whole number or whole numbers are closed to addition.

Verification: To check this property, we take any two integers and add them up. We find that the sum is always an integer as shown below


7 + 3 = 10 (10 is also an integer)

0 + 8 = 8 (8 is also an integer)

29 + 37 = 66 (66 is also an integer)

Commutative addition property / order addition property:

If a and b are any two integers, then a + b = b + a.

In other words, the sum of two whole numbers remains the same even if the order of whole numbers (called addends) is changed.

The numbers can be added in any order. The sum of two numbers remains the same even if the order of the numbers is changed.

For example:

I. 4313 + 3142 = 7455

3142 + 4313 = 7455

Changing the order of addends 4313 and 3142 does not change the total.

II. 133 + 142 = 275

142 + 133 = 275

Changing the order of addends 133 and 142 does not change the total.

Verification:
To check this property, let’s look at a few pairs of integers and add them in two different sequences. We notice that the sum stays the same as shown below:

9 + 3 = 3 + 9

13 + 25 = 25 + 13

0 + 32 = 32 + 0

Existence of the additive identity of addition / identity property of addition / zero property of addition:

If a is an integer then

a + 0 = a = 0 + a

In other words, the sum of an integer and zero is the number itself. That is, zero is the only integer that doesn’t change the value (identity) of the number it is added to.

The integer 0 (zero) is known as additive identity or an identity element for adding whole numbers.

The number stays the same if zero is added to the number.

For example:

I. 5918 + 0 = 5918

The identity of 5918 stays the same when added to zero.

II. 45 + 0 = 45

The identity of 45 stays the same when added to zero.

Verification:
To check this property, we take any integer and add it to zero. We find that the sum is the integer itself as shown below:

5 + 0 = 5 = 0 + 5

27 + 0 = 27 = 0 + 27

137 + 0 = 137 = 0 + 137

Note:

Zero is called additive identity because it maintains or does not change the identity (value) of the numbers during the addition process.

Associativity of addition / Associativity of addition:

If a, b, c are any three integers, then

(a + b) + c = a + (b + c)

In other words, integer addition is associative.

If three or more numbers are added, the total will stay the same regardless of their group or place.

For example:

I.4610 + 1129 + 2382 = 5739 + 2382 = 8121

4610 + 1129 + 2382 = 4610 + 3511 = 8121

4610 + 2382 + 1129 = 6992 + 1129 = 8121

The grouping of the summands does not change the sum.

II. 23 + 45 + 16 = 68 + 16 = 84

23 + 45 + 16 = 23 + 61 = 84

23 + 16 + 45 = 39 + 45 = 84

The grouping of the summands does not change the sum.

Verification:
To check this property, let’s take three integers, say a, b, c, and find the values ​​of the expression (a + b) + c and a + (b + c). We notice that the values ​​of these expressions stay the same, as shown below;

(i) (2 + 5) + 7 = 2 + (5 + 7)

then 7 + 7 = 2 + 12

14 = 14

(ii) (5 + 10) + 13 = 5 + (10 + 13)

then 15 + 13 = 5 + 23

28 = 28

(iii) (9 + 0) + 11 = 9 + (0 + 11)

then 9 + 11 = 9 + 11

20 = 20

Let us consider any three integers a, b, c.

We have (a + b) + c

= (b + a) + c [By using commutativity of addition we have a + b = b + a]

= b + (a + c) [By using associativity of addition]

= b + (c + a) [By using commutativity of addition]

= (b + c) + a [By using associativity of addition]

= (c + b) + a [By using commutativity of addition]

Property of the opposites of addition:

For every real number a there is a unique real number -a such that

a + (-a) = 0 and (-a) + a = 0

The sum of the real number (a) and its opposite real number (-a) is zero, then they are called the additive inverses of each other.

Verification:

5 + (-5) = 0 and (-5) + 5 = 0

or, 5 – 5 = 0 and -5 + 5 = 0

Here 5 is the real number and (-5) is the opposite real number. The sum of 5 and (-5) is zero.

Hence, (-5) is the additive inverse of 5

or 5 is an additive inverse of (-5).

Property of the opposite of an addition sum:

If a and b are any two integers, then

– (a + b) = (-a) + (-b)

The opposite of the sum of whole numbers is equal to the sum of the opposite whole numbers.

Verification:

– (3 + 4) = (-3) + (-4)

or – (7) = -3 -4

or -7 = -7

Here the opposite of the sum of 3 and 4 is -7.

The opposites of 3 and 4 are (-3) and (-4), respectively.

The sum of the opposites (-3) and (-4) is equal to -7.

Successor to a total / successor to the addition:

If a is an integer then

a + 1 = (a + 1), which is a successor to “a”.

If we add 1 to the sum of a number, we have a descendant of the number.

If we add 1 to any number, we get the number immediately after it.

For example:

I. 26519 + 1 = 26520

26520 is the successor to 26519

II. 276 + 1 = 277

277 is the successor to 276

Verification:

2420 + 1 = 2421

2421 is the successor to 2420.

Similar, 1 + 2542 = 2543

2543 is the successor to 2542.

Questions and answers about addition properties:

1. Fill in the indicated gaps with the properties of addition.

(i) 19.94.450 + 3.07.689 = __________ + 19.94.450

(ii) 18,47,336 + __________ = 18,47,336

(iii) 11,300,999 + 1 = __________

(iv) __________ + 0 = 18.95,72,025

(v) (84.32.583 + 22.68.592) + 90.81.225 = 84.32.583 + (__________ + 90.81.225)

(vi) 37.46,442 + 20,000 = __________

(vii) 209,718,660 + 1,000,000 = __________

(i) 674 + 0 = ………….

(ii) 0 + …………. = 174

(iii) 723 + 122 = …………. + 723

(iv) 118 + 687 = 687 + ………….

(v) 250 + 211 + …………. = 211 + 134 + 250

(vi) 433 + …………. = 123 + 433

(vii) 102 + …………. = 326 + 102

(viii) 361 + …………. = 361

(ix) …………. + 537 + 216 = 909 + 537 + 216

(x) …………. + 773 = 773 + 612

Reply:

(i) 3.07.689

(ii) 0

(iii) 11,301,000

(iv) 18,95,72,025

(v) 22.68.592

(vi) 37,66,442

(vii) 210.718.660

(i) 674

(ii) 174

(iii) 122

(iv) 118

(v) 134

(vi) 123

(vii) 326

(viii) 0

(ix) 909

(x) 612

2. Fill in the specified gaps with the properties of addition:

(i) 9508 + 8857 = ……………. + 9508

(ii) 6698 + ……………. = 6698

(iii) 7397 + 1 = …………….

(iv) 8647 + ……………. = 8648

(v) 7498 + ……………. = 5096 + 7498

(vi) ……………. + 0 = 2985

(vii) (6654 + 3011) + 8010 = 6654 + (……………. + 8010)

(viii) 3997 + 2000 = …………….

(ix) ……………. To 50 = 150. added

(x) 1 more than 999 = …………….

Reply:

(i) 8857

(ii) 0

(iii) 7398

(iv) 1

(v) 5096

(vi) 2985

(vii) 3011

(viii) 5997

(ix) 100

(x) 1000

3. Write the successor to the following numbers:

(i) 433

(ii) 127

(iii) 484

(iv) 579

(v) 397

(vi) 625

(vii) 650

(viii) 823

(ix) 34

(x) 0

Reply:

(i) 434

(ii) 128

(iii) 485

(iv) 580

(v) 398

(vi) 626

(vii) 651

(viii) 824

(ix) 35

(x) 1

Whole numbers

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