# Properties of Adding Integers

The properties of adding integers are discussed here along with the examples.

1. Closure Property: The addition (sum) of any two integers is always an integer.

i.e.,

The sum of integers is always an integer.

Hence, integers are closed under addition. If x and y are two integers, then x + y is always an integer.

For example:

(i) 16 + 48 = 64, which is an integer.

(ii) 12 + (-38) = -26, which is an integer.

(iii) - 24 + (- 14) = - 38, which is an integer.

(iv) 42 + (- 10) = 32, which is an integer.

(v) 5 + 9 = 14 ∈ Z

(vi) (-5) + 9 = 4 ∈ Z

(vii) (-5) + (-9) = -14 ∈ Z

(viii) 5 + (-9) = -4 ∈ Z                    and so on.

2. Commutative Property: Two integers can be added in any order.

Hence, addition is commutative for integers.

For any two integers ‘x’ and ‘y’;

x + y = y + x

For example:

(i) (-7) + 18 = 11 and 18 + (-7) = 11

Therefore, (-7) + 18 = 18 + (-7)

(ii) (-28) + (-5) = - 33 and (-5) + (-28) = -33

Therefore, (-28) + (-5) = (-5) + (-28)

(iii) (+3) + (+8) = (+8) + (+3)

(iv) (-7) + (+3) = (+3) + (-7)

(v) (-9) + (-3) = (-3) + (-9)

(vi) (+5) + (-3) = (+5) + (-3)                    and so on.

3. Associative Property: Three or more integers can be grouped in any order to find their sum. Hence, addition is associative for integers.

For any three integers ‘x’ ‘y’ and ‘z’;

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

For example:

(i) [(-5) + (-3)] + 10 = (-8) + 10 = 2 and (-5) + [(-3) + 10] = (-5) + (7) = 2

Therefore, [(-5) + (-3)] + 10 = (-5) + [(-3) + 10]

(ii) [(- 24) + 12] + 6 = (-12) + 6 = -6 and (- 24) + (12 + 6) = - 24 + 18 = -6

Therefore, [(- 24) + 12] + 6 = (- 24) + (12 + 6)

(iii) (+5) + [(-2) + (+3)] = [(+5) + (-2)] + (+3)

(iv) (-3) + [(-4) + (-5)] = [(-3) + (-4)] + (-5)

(v) (+4) + [(+2) + (+3)] = [(+4) + (+2)] + (+3)

(vi) (-2) + [(+3) + (-4)] = [(-2) + (+3)] + (-4)

(vii) (-4) + [(-3) + (+5)] = [(-4) + (-3)] + (+5)

(viii) (+3) + [(+4) + (-2)] = [(+3) + (+4)] + (-2)

(ix) (-3) + [(2) + (7)] = [(-3) + (2)] + (7)

(x) 9 + [(-4) + (-2)] = [9 + (-4)] + (-2)                    and so on.

4. Existence of Additive Identity: The sum of any integer and 0 is the integer itself, 0 is the additive identity for integers.

For any integer ‘x’;

x + 0 = 0 + x = x

For example:

(i) 100 + 0 = 0 + 100 = 100

(ii) (-45) + 0 = 0 + (-45) = -45

(iii) (+7) + 0 = 0 + (+7) = +7

(iv) (-11) + 0 = 0 + (-11) = -11

(v) 0 + (+9) = (+9) + 0 = +9

(vi) 0 + (-5) = (-5) + 0 = -5                    and so on.

5. Existence of Additive Inverse: For any integer x, there exists its opposite -x such that their sum is zero, i.e.,

x + (-x) = (-x) + x = 0

Integers x and -x are called opposites or negatives or additive inverses of each other.

For example:

(i) 15 + (-15) = (-15) + 15 = 0.

Thus, the additive inverse of 15 is -15 and

the additive inverse of -15 is 15.

(ii) 56 + (-56) = (-56) + 56 = 0.

Thus, the additive inverse of 56 is -56 and

the additive inverse of -56 is 56.

(iii) 5 + (-5) = 0

(iv) (-7) + 7 = 0                    and so on.

6. Successor and Predecessor of an Integers: If x is any integer, then (x + 1) is called the successor of x and x - 1 is called the predecessor of x.

For example:

(i) Successor of 6 is 6 + 1 = 7;     Predecessor of 6 is 6 - 1 = 5

(ii) Successor of -5 is -5 + 1 = -4;     Predecessor of -5 is -5 - 1 = -6

Solved Examples on Properties of Adding Integers:

1. Fill in the blanks and make each of the following a true statement.

(i) The additive inverse of 17 is __________.

(ii) The additive inverse of -48 is __________.

(iii) The successor of 12 is __________.

(iv) The successor of -90 is __________.

(v) The predecessor of 1000 is __________.

(vi) The predecessor of -10000 is __________.

Solution:

(i) The additive inverse of 17 is -17; [Since, 17 + (-17) = 0]

(ii) The additive inverse of -48 is 48; [Since, (-48) + 48 = 0]

(iii) The successor of 12 is 13; [Since, 12 + 1 = 13]

(iv) The successor of -90 is -89; [Since, -90 + 1 = -89]

(v) The predecessor of 1000 is 999; [Since, 1000 - 1 = 999]

(vi) The predecessor of -10000 is -10001; [Since, -10000 - 1 = -10001]

2. Example in Find the sum of the following.

(i) (- 15) + (- 18) + 26 + 45

(ii) 42 + (- 4) + (- 78) + (- 7)

Solution:

(i) (- 15) + (- 18) + 26 + 45

= (- 33) + (71)

= + (71 - 33)

= +38

= 38

(ii) 42 + (- 4) + (- 78) + (- 7)

= 42 + (-89)

= - (89 - 42)

= - (47)

= - 47

3. Find an integer 'n" such that

(i) 10 + n = 0

(ii) n + (- 7) = 0

Solution:

(i) 10 + n = 0

⟹ (- 10) + 10 + n = (- 10) + 0; [Adding (-10) on both sides]

⟹ [(- 10) + 10] + n = - 10; [Using associative property and property of 0]

⟹ 0 + n = - 10

Hence, n = - 10.

(ii) n + (- 7) = 0

⟹ n + (- 7) + 7 = 0 + 7; [Adding 7 on both sides]

⟹ n + [(- 7) + 7] = 7; [Using associative property and property of 0]

⟹ n + 0 = 7

Hence, n = 7

## You might like these

• ### Counting Natural Numbers | Definition of Natural Numbers | Counting

Natural numbers are all the numbers from 1 onwards, i.e., 1, 2, 3, 4, 5, …... and are used for counting. We know since our childhood we are using numbers 1, 2, 3, 4, 5, 6, ………..

• ### Reading and Writing Large Numbers | Large Numbers in Words in Billion

In reading and writing large numbers we group place values into periods ‘ones or unit’, ‘tens’, ‘hundred’, ‘thousand’, ‘10 thousand’, ‘100 thousand’, ‘million’, ’10 million’, ‘100 million

• ### 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

• ### Properties of Addition | Identity, Commutative, Associative, Additive

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.

• ### Properties of Whole Numbers | Closure Property | Commutative Property

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.

• ### 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.

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

## Recent Articles

1. ### 2nd Grade Place Value | Definition | Explanation | Examples |Worksheet

Sep 14, 24 04:31 PM

The value of a digit in a given number depends on its place or position in the number. This value is called its place value.

2. ### Three Digit Numbers | What is Spike Abacus? | Abacus for Kids|3 Digits

Sep 14, 24 03:39 PM

Three digit numbers are from 100 to 999. We know that there are nine one-digit numbers, i.e., 1, 2, 3, 4, 5, 6, 7, 8 and 9. There are 90 two digit numbers i.e., from 10 to 99. One digit numbers are ma

3. ### Worksheet on Three-digit Numbers | Write the Missing Numbers | Pattern

Sep 14, 24 02:12 PM

Practice the questions given in worksheet on three-digit numbers. The questions are based on writing the missing number in the correct order, patterns, 3-digit number in words, number names in figures…

4. ### Comparison of Three-digit Numbers | Arrange 3-digit Numbers |Questions

Sep 13, 24 02:48 AM

What are the rules for the comparison of three-digit numbers? (i) The numbers having less than three digits are always smaller than the numbers having three digits as: