Addition and Subtraction of Integers: We use the addition and subtraction of integers in our day-to-day life without even realizing it. For example, suppose the temperature in a city was \(5\,^\circ {\rm{C}},\) and it falls by \(10\,^\circ {\rm{C}}.\) In that case, the current temperature of that city will be negative, i.e., \(- 5\,^\circ {\rm{C}}.\) Arithmetic is the fundamental area of Mathematics that studies numbers, especially the properties of operation such as addition, subtraction, multiplication and division. Arithmetic also includes additions and Subtraction of Integers in highly complex computations such as percentages, logarithms, exponentiation, square roots, etc. 

Both whole numbers and negative numbers are considered integers. Similar to whole numbers, we can add or subtract integers also. In this article, we will explore addition and subtraction using integers. But before that, we will know and understand the need for negative integers.

Addition and Subtraction of Integers

Integers are a collection of natural numbers, the negatives of natural numbers, and zero. An integer is a complete entity with no fractional part. Thus, we can add or subtract integers. Addition and subtraction of integers mean carrying out addition and subtraction operations on two or more integers by putting the addition and subtraction operator in between.

Need for Negative Integers

We have learned to perform mathematical operations of addition, subtraction, multiplication, and division on whole numbers. When we subtract a smaller whole number from a greater whole number, the result is always a whole number. For example, \(20-13=7,\) the resultant is a whole number. But when we subtract a greater whole number from a smaller whole number, the result is not a whole number. Therefore, a need was felt to have a set of numbers that includes negative integers, that is, the numbers with a minus sign. It resulted in the creation of a new set of numbers, which are called integers.

Addition of Integers

Addition generally means to increase the value. But, at times in integers, the addition operation might lead to a decrease in the value of the given number. For example, if we add a negative integer, the value of the given number will decrease, and if we add a positive integer, the value will increase.

Let us understand this with the help of examples.

\(17+7=24\) (After addition, the common sign of the integers is given to the sum)

\(-35+20=-15\) (After subtraction, the sign of the integer having the greater absolute value is given to the difference)

\(-200+-150 = -350\) (After addition, the common sign of the integers is given to the sum)

\(300+-210 = 90\) (After subtraction, the sign of the integer having the greater absolute value is given to the difference)

Addition of Integers on Number Line

The number line is a horizontal straight line on which numbers are marked at regular intervals. Thus, the number line extends indefinitely on both sides of zero.

Have a look at the example given below.

To add \(3\), jump three steps towards the right.

Look at another example.

In the above examples, we used the concept of the addition of integers. While showing the addition of integers on a number line, we have to move towards the right side or the positive side when we add a positive integer to a given number. On the other hand, when we add a negative number, we move towards the left side of the number line.

Rules for Addition of Integers

The following are the \(2\) main rules of the addition of integers.

Rule \(1:\) To add two positive integers or two negative integers, add their absolute values and assign the common sign of the integers to the sum.

Rule \(2:\) To add a positive and a negative integer, subtract their absolute values and assigned to the difference the sign of integer having a greater absolute value.

Properties of Addition of Integers

All properties of the addition of whole numbers hold true for integers also.

1. Closure property: The sum of any two integers is always an integer.

Example: \(15+3=18\), and \(18\) is an integer. Similarly, \(17+-20 = -3\), and \(-3\) is integer.

2. Commutative property: The sum of any two integers remains the same even if the order in which they are added is changed.

Example: \(-19+15=15+(-19)=-4\)

3. Associative property: The sum of three or more integers does not depend on how the integers are grouped.

Example: \(-13+(-15+16)=[-13+(-15)+16]=-12\)

4. When \(0\) is added to an integer, the result is the integer itself. The integer \(0\) is called the additive identity.

Example: \(10+0=16, 0+-2 =-2\)

5. For every integer, there exists a number which, when added to this integer, makes it \(0\). Thus, the two opposite integers are called the additive inverse of each other.

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

6. Every integer has a successor and predecessor. We get the successor of an integer by adding \(1\) to it, whereas we get the predecessor by subtracting \(1\) from it.

Example: \(4\) is the successor of \(3\). Similarly, \(-5\) is the successor of \(-6\). Finally, \(8\) is the predecessor of \(9\).

Subtraction of Integers

We know that subtraction is the reverse of addition. Subtraction is the operation of finding the difference between two numbers. When we apply subtraction to a collection, the number of things in the collection reduces or becomes less.

In the subtraction problem, \(-100-30=70\), the number \(-100\) is the minuend, the number \(30\) is the subtrahend, and the number \(70\) is the difference. Hence, the minuend, subtrahend and difference are parts of a subtraction problem.

Subtraction of Integers on Number line

Two crucial points to keep in mind about the placement of numbers on a number line are

1. A number on the left is always less than a number on the right.
2. And a number on the right is always greater than a number on the left.

Let us understand the subtraction of integers with the help of examples.

Example: Solve \(25-7\).

Example: Solve \(4-5\).

In the above examples, we use the concept of subtraction of integers. While showing the subtraction of integers on a number line, we have to move towards the left or negative side when we are subtracting a positive integer from a given integer. On the other hand, we move towards the right side or positive side when we subtract a negative integer from a given integer.

Rules for Subtraction of Integers

 The following is the rule of subtraction of integers.

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,

\(a-b=a+(-b)\)

Thus, for subtracting two integers, we take the additive inverse of the integer to be subtracted and add it to the other integer.

Properties of Subtraction of Integers

Let us understand the properties of subtraction of integers.

1. Closure property: The difference between any two integers is always an integer.

Example: \(13-17=-4\), and \(-4\) is an integer. Similarly, \(-5-8=-13\) and \(-13\) is an integer.

2. Commutative property: The difference between two integers is changed if their order is reversed.

Example: \(6-3=3\) but \(3-6=-3\). Thus, \(6-3≠3-6\).

3. Associative property: In subtraction, the result changes if the way in which the three or more integers are grouped is changed.

Example: \((80-30)-60=-10\) but \([80-(30-60)]=110\). Thus, \((80-30)-60≠ [80-(30-60)]\).

4. For any integer, let us say \(8, 8-0=8\) but \(0-8≠8\). Thus, for subtraction, no identity number exists.

Solved Examples – Addition and Subtraction of Integers

Q.1. Add \(3\) and \(4\) using the number line.
Ans: To add \(3\) and \(4\) using the number line, we need to mark a point \(3\) on the right side of \(0\) and start moving \(4\) points to the right.

So, \(3+4=7\)

Q.2. Subtract \(7\) from \(25\) on a number line.
Ans: Draw a number line and mark both the numbers on the number line and jump \(7\) steps towards the left side. After jumping \(7\) steps, the number you landed on is the required answer.

Q.3. What temperature change will a customer experience in a grocery store when they walk from the vegetable section at \(20\,^\circ {\rm{C}}\) to the other section, which is set to \(- 20\,^\circ {\rm{C}}\)?
Ans: Temperature at the vegetable section \(= 20\,^\circ {\rm{C}}\)

The temperature at the other section \( = \, – {20^{\rm{o}}}{\rm{C}}\)

Hence, the difference in temperature \([ = \, – 20\,^\circ {\rm{C}} – \left( { – 20\,^\circ {\rm{C}}} \right)]\)

\(= 20\,^\circ {\rm{C}} + 20\,^\circ {\rm{C}} = 40\,^\circ {\rm{C}}\)

Q.4. Add \(6\) and \(5\) using the number line.
Ans: To add \(6\) and \(5\) using the number line, we need to mark a point \(6\) to the right side of \(0\) and start moving \(5\) points to the right.

So, \(6+5=11\)

Q.5. Use a number line to find \(78-45\).
Ans: We are asked to subtract two numbers on a number line. The subtraction is done as follows:

Thus, \(78-45=33\)

Summary

In this article, we learned about integers, the need for integers, and adding or subtracting the integers. We also learned the addition and subtraction of integers on a number line. Finally, we also learned the properties of addition and subtraction of integers.

FAQs About Addition and Subtraction of Integers

Let’s look at some of the commonly asked questions about the Addition and Subtraction of Integers:

Q.1. What are the rules for subtracting integers?
Ans: The following is the rule of subtraction of integers.
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,
\(a-b=a+(-b)\)
Thus, for subtracting two integers, we take the additive inverse of the integer to be subtracted and add it to another integer.

Q.2. Explain the addition and subtraction of integers with example?
Ans: Addition and subtraction of integers mean carrying out addition and subtraction operations on two or more integers by putting the addition and subtraction operator in between.
For example, \(11+8=19\) ( After addition, the common sign of the integers is given to the sum)
\(446+(-218)= 228\) ( After subtraction, the sign of the integer having the greater absolute value is given to the difference).

Q.3. What is the difference between the addition and subtraction of integers?
Ans: Addition is one of the basic mathematical operations that represents combining or adding integers. And, subtraction is another basic mathematical operation that represents finding the difference between integers.

Q.4. Name the properties of adding and subtracting positive and negative integers?
Ans: All the properties of addition and subtraction of positive and negative integers are listed below:
a. Closure Property
b. Commutative Property
c. Associative Property
d. Existence of Identity
e. Existence of Inverse

Q.5. What are the rules for adding integers?
Ans: The following are the \(2\) main rules of the edition of integers.
Rule \(1:\) To add two positive integers or two negative integers, add their absolute values and assign the common sign of the integers to the sum.
Rule \(2:\) To add a positive and a negative integer, subtract their absolute values and assigned to the difference the sign of the integer having a greater absolute value.

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