Properties of Addition and Subtraction of Integers: This article explains the properties of addition and subtraction of integers for students. The set of numbers, including natural numbers, additive inverse, and zero, constitutes the set of integers. The numbers \(\left\{ { \ldots ..,~ – 3,~ – 2,~ – 1,~0,~1,~2,~3,~ \ldots ..}\right\}\) are called integers. If we subtract a whole number from another whole number, the difference may not always be a whole number.

When a small number is subtracted from a larger number, we get a whole number. But, we have no whole numbers to represent 3−8. So, to describe such differences, we have created integers. This article will see the addition and subtraction of these integers and the properties of addition and subtraction of integers.

Integers

Corresponding to natural numbers \(1,2,3,4,5, \ldots \ldots\) etc., we have their additive inverses as negative numbers like \(-1,-2,-3,-4,-5, \ldots \ldots\). etc., which is called minus one, minus two, minus three, minus four, minus five, etc. respectively, such that

\(1+(-1)=0\)

\(2+(-2)=0\) and so on.

\(-1\) is also called as negative of one, and \(-1\) and \(1\) are called opposites.

Merging these new numbers with whole numbers, we obtain a new set of numbers which is written as …….. \(-3,-2,-1,0,1,2,3, \ldots..\) These are called integers. They are represented as

\(Z = \left\{ {…….. – 3,{\text{ }} – 2,{\text{ }} – 1,{\text{ }}0,{\text{ }}1,{\text{ }}2,{\text{ }}3,{\text{ }} \ldots ..}\right\}\)

Integers are of three types: Positive, negative and zero.

Positive Integers: The numbers \(1,2,3,4,5,6 \ldots..\) are called positive integers. They are nothing but natural numbers.

Negative Integers: The numbers \(-1,-2,-3,-4,-5,-6 \ldots \ldots\) are called negative integers. They are the additive inverse of natural numbers.

Zero: Zero lies in the middle of positive and negative integers. Zero is neither positive nor negative.

Representation of Integers on Number Line

To represent whole numbers on a number line, we draw a straight line and point \(O\) on it, as shown below.

To represent integers on a number line, we draw a straight line and a point \(O\) in the middle, as shown below. We set off equal distances on the right side as well as the left side of \(O\). On the right side of \(O\), we label the divisions as \(1,2,3,4, \ldots\). Since negative integers are opposite of positive integers, we represent negative integers in the opposite direction, i.e., on the left side of \(O\). We label these divisions as \(-1,-2,-3,-4, \ldots \ldots\) as shown below.

We can arrive at some conclusions regarding the integers from this representation of integers on the number line.

1. Every positive integer is greater than every negative integer since every positive integer is to the right side of every negative integer.
2. Zero is less than every positive integer since \(0\) is to the left side of every positive integer.
3. Zero is greater than every negative integer since \(0\) is to the right of every negative integer. 
4. The greater the number, the lesser is its opposite.
5. The largest negative integer is \(-1\), and the smallest negative integer cannot be determined.
6. The smallest positive integer is \(1\), and the greatest positive integer cannot be determined.

All mathematical operations like addition, subtraction, multiplication and division can be done with integers. Let us discuss in detail the addition and subtraction of integers.

Addition of Integers

The addition of integers is done in a similar way we do the addition of whole numbers or natural numbers. The difference comes only when negative integers are involved.

Case 1: When two positive or two negative integers are added, we add their absolute values and assign the sign of the addends to the sum.

Example:

(i) Add \(243\) and \(144\)

Solution: Here, both integers are positive. So, \(243+144=387\)

(ii) Add \(-320\) and \(-180\)

Solution: Here, both integers are negative. So, we add their absolute values and assign the sign of the addends to the sum.

\(\Rightarrow(320+180)=500\)

Therefore, \((-320)+(-180)=-500\)

Case 2: When we add a positive and a negative integer, we determine the difference of their absolute values and assign the sign of addend having a greater absolute value.

Example: Add \(-523\) and \(937\)

Solution: Here, one integer is negative, and one is positive. So, we determine the difference of their absolute values and assign the sign of addend having a greater absolute value.

\(\Rightarrow-523+937=414\)

Properties of Addition of Integers

All properties of operations on whole numbers are also true for Integers. The properties are:

1. Closure Property
2. Commutative Property of Addition
3. Associative Property of Addition
4. Additive Identity
5. Additive inverse

Closure Property: The sum of two integers is always an integer. In other words, if \(a\) and \(b\) are any two integers, \(a+b\) is always an integer.

Example: \(3+(-6)=-3\). Here, \(-3\) is an integer.

Commutative Property: The addition of integers is commutative, i.e., for any two integers \(a\) and \(b\), we have \(a+b=b+a\).

Example: \(4+(-8)=(-8)+4=-4\)

Associative Property: The addition of integers is associate, i.e., for any three integers \(a, b, c\), we have \((a+b)+c=a+(b+c)\)

Example: \((-7)+(2+6)=(-7+2)+6=1\)

Additive Identity: There is integer \(0\) such that for any integer \(a\), we have \(a+0=0+a=a\)

Example: \(-6+0=0+(-6)=-6\)

Additive Inverse: For any integer \(a\) there exists its opposite \(-a\) such that \(a+(-a)=0\)

Example: \(-7+7=7-7=0\)

Existence of Successor and Predecessor: Every integer has a successor and a predecessor, i.e., for every integer \(a, a+1\) is the successor and \(a-1\) is its predecessor.

Example: \(5\) is the successor of \(4\), and \(3\) is the predecessor of \(4\).

Also, If \(a, b, c\) are integers, then

(i) \(a>b \Rightarrow a+c>b+c\)
(ii) \(a<b \Rightarrow a+c<b+c\)

Subtraction of Integers

Subtraction is an inverse process to that of addition. 

For example: subtracting \(3\) from \(6\) means finding a number which, when added to \(3\), gives \(6\). Similarly, subtracting \(-2\) from \(5\) means finding a number that, when added to \(-2\), gives \(5\). If \(a\) and \(b\) are two integers, then to subtract \(b\) from \(a\), we change the sign of \(b\) and add it to \(a\). That is, \(a-b=a+-b\)

Properties of Subtraction of Integers

1. The subtraction of any two integers is an integer, i.e., for any two integers \(a\) and \(b\), \(a-b\) is an integer.
   Example: \(3-6=-3\). Here,\(-3\) is an integer.
   Hence, subtraction is closed under integers.
2. Subtraction of integers is not commutative, i.e., for any two integers \(a\) and \(b\), \(a-b \neq b-a\)
   Example: \(4-2 \neq 2-4\)
3. The subtraction of integers is not associative.
4. For any integer \(a\), we have \(a-0=a\). Here \(0\) is the right identity for subtraction.
   Also, \(0-a \neq a\).
5. For any integer \(a\) there exists its opposite \(a\) such that \(a+(-a)=0\).
6. If \(a, b, c\) are three integers and \(a>b\), then \(a-c>b-c\)

We can consolidate the properties of addition and subtraction of integers in the following table.

Properties	Addition of Integers	Subtraction of Integers
Closure property	✔️	✔️
Commutative property	✔️	✖️
Associative property	✔️	✖️
Existence of Identity	✔️	✔️
Existence of Inverse	✔️	✔️

Solved Examples – Properties of Addition and Subtraction of Integers Class 7

Q.1. Find the additive inverse of the following integers:
a. \(52\)
b. \(-176\)
Ans: We know that for any integer \(a\) there exists its opposite \(-a\) such that \(a+(-a)=0\). For any integer \(-a\) there exists its opposite \(a\) such that \(a+(-a)=0\).
Therefore, the additive inverse of \(52\) is \(-52\), and the additive inverse of \(-176\) is \(176\).

Q.2. Subtract \(-1235\) from \(4532\).
Ans: To subtract \(-1235\) from \(4532\), we change the sign of \(-1235\) and add it to \(4532\). Thus, we have
\(4532-(-1235)=4532+1235=5767\)

Q.3. Find the value of \(412+(-98)+(-84)+(-7)+35\).
Ans: We have, \(412+(-98)+(-84)+(-7)+35\)
Separating the positive and negative integers, we get
\(412+(-98)+(-84)+(-7)+35=(412+35)+[(-98)+(-84)+(-7)]\)
\(=447+(-189)\)
\(=447-189\)
\(=258\)
Therefore, \(412+(-98)+(-84)+(-7)+35=258\)

Q.4. On a particular day, the temperature of Delhi at \(10\,\text {a.m.}\) was \(13^{\circ} \mathrm{C}\), but by midnight, it fell to \(6^{\circ} \mathrm{C}\). The temperature of Chennai at \(10\,\text {a.m.}\) the same day was \(18^{\circ} \mathrm{C}\) but fell to \(10^{\circ} \mathrm{C}\) by midnight. Which temperature drop is greater?
Ans: We have, fall in Delhi’s temperature \(=13^{\circ} \mathrm{C}-6^{\circ} \mathrm{C}=7^{\circ} \mathrm{C}\)
Also, fall in Chennai’s temperature \(=18^{\circ} \mathrm{C}-10^{\circ} \mathrm{C}=8^{\circ} \mathrm{C}\)
So, \(8^{\circ} \mathrm{C}>7^{\circ} \mathrm{C}\)
Hence, the fall in temperature of Chennai is greater.

Q.5. Subtract the sum of \(-5020\) and \(2320\) from \(-709\).
Ans: Sum of \(-5020\) and \(2320=-5020+2320=-2700\)
Also, \((-709)-(-2700)=-709+2700=1991\)
Therefore, the required answer is \(1991\).

Summary

In this article, we have learnt the definition of integers, the representation of integers on a number line, meaning of addition and subtraction of integers. Also, we have learnt several properties of the addition of integers like closure property, commutative property, associative property, additive identity and additive inverse. Also, we have learnt the properties of subtraction of integers and solved some example problems on the same.

FAQs on Properties of Addition and Subtraction of Integers Class 7

Here are some most frequently asked questions on the properties of addition and subtraction of integers:

Q.1: What are the properties of integers in addition?
Ans: 1. Closure property
2. Commutative property of addition
3. Associative property of addition
4. Additive identity
5. Additive inverse

Q.2: What is the commutative property of addition?
Ans: The addition of integers is commutative, i.e., for any two integers \(a\) and \(b\), we have \(a+b=b+a\).
Example: \(5+(-8)=(-8)+5=-3\)

Q.3: What is the commutative property of subtraction?
Ans: Subtraction of integers is not commutative, i.e., for any two integers \(a\) and \(b\), \(a-b \neq b-a\).

Q.4: What is the successor and predecessor of \(200\)?
Ans: Every integer has a successor and a predecessor, i.e., for every integer \(a, a+1\) is the successor and \(a-1\) is its predecessor.
Therefore, the successor of \(200\) is \(201\), and the predecessor of \(200\) is \(199\).

Q.5: What is the associative property of addition?
Ans: The addition of integers is associative, i.e., for any three integers \(a, b, c\), we have \((a+b)+c=a+(b+c)\).

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