The process of obtaining the sum of two or more numbers is known as the Addition of Integers. The value may grow or decrease depending on whether the integers are positive, negative, or a combination. The number system is the backbone of Mathematics. Integers can be said as the subset of the number system. There are four arithmetic operations that we can apply to integers, and Addition is one of them. 

The addition is the common operation used in normal life to count the money or to count the number of items we buy from any supermarket, or so on. So, addition is to bring two or more numbers together. We can add integers using the number line as well. Now let us know more about the addition of integers in this article.

Definition of Integers

Integers can be defined as the set of natural numbers and their additive inverse, including zero. The set of integers is \({ \ldots .. – 3,\, – 2,\, – 1,\,\,0,\,\,1,\,\,2,\,\,3 \ldots .}\). Integers are numbers that can not be a fraction.

Integers are an extension of whole numbers and natural numbers. Natural numbers along with zero and negative natural numbers make integers. Whole numbers, along with negative natural numbers, make integers.

That is,

Whole Numbers \( + \) Negative Natural numbers \( = \) Integers
Natural Numbers \( + \) Zero \( + \)Negative Natural Numbers \( = \) Integers

Definition of Addition

Addition, generally indicated by the \( + \) sign, is a method of combining two or more numbers. In other words, finding the sum of two or more numbers or objects is known as an addition.

For example, to find the sum of \(5\) and \(7\) we will write it as \({\rm{5 + 7 = 12}}\).

Addition of Integers

Unlike natural numbers or whole numbers, integers include negative numbers as well.

So, the addition of integers has to be understood in detail when positive and negative integers are involved.

Addition of Two Positive Integers

The way we add natural numbers and whole numbers when two positive integers are added, we get a positive integer.

That is, \({\rm{Positive}} + {\rm{Positive}} = {\rm{Positive}}\)

For example, \({\rm{2 + 3 = 5}}\)

\({\rm{6 + 4 = 10}}\)

Addition of a Positive and Negative Integer

Before learning the addition of a positive and negative integer, we should first understand the concept of absolute value. The absolute value of a number is the numerical value of the number regardless of its sign. That absolute value is neither negative nor positive, and it is just the numerical value of the number.

For example,

The absolute value of \({\rm{5 = 5}}\)

The absolute value of \({\rm{5 = 5}}\)

The absolute value of \({\rm{0 = 0}}\)

When a positive and negative integer has to be added, we should first take the absolute value of the two integers and take the difference between them. Then, the answer will take the sign of the integer, which has the bigger absolute value.

For example,

1. \({\rm{ – 2 + 3}}\)

Here, the absolute value of \({\rm{3 = 3}}\) and the absolute value of \({\rm{-2 = 3}}\)

So after finding the difference of absolute values, the answer will take the sign of the integer with the greater absolute value. That is of \(3\).

2. \({\rm{ – 5 + 1}}\)

Here, the absolute value of \({\rm{ – 5 = 5}}\) and the absolute value of \({\rm{ 1 = 1}}\)

So, the difference of absolute values \({\rm{ = 5 – 1 = 4}}\)

The answer will take the sign of the integer with the greater absolute value, that is of \(-5\).

Addition of Two Negative Integers

The addition of two negative integers is similar to how we add two positive integers, and the only change is in the sign of the answer. When two negative integers are added, the result will be a negative integer. So, when adding two negative integers, add the absolute value of both the integers and include a negative sign to the answer.

For example,

\({\rm{( – 3) + ( – 2) = ( – 5)}}\)

We can observe that when we add two integers, the sum is again an integer. That is, integers are closed under addition.

Representation of Addition of Integers Using Number Line

The representation of the addition of integers using number line are given below:

Definition and the Uses of Number Line

A number line can is a straight line with numbers placed at equal intervals or segments along its length. A number line can be extended infinitely in any direction and is usually represented horizontally.

We can represent the integers on the number line as

The positive integers are represented to the right side of zero on the number line, and negative integers are represented to the left side of zero.

The farther the integers move to the right from zero on the number line, the value of the integers increases and the farther the integers move to the left on the number line from zero, the value of the integer decreases.

When we compare any two integers on a number line, the integer on the right side of the integer line will be greater.

There are two steps to be followed to add integers on a number line.

1. Move right from zero to add the positive integers.
2. Move left from zero to add the negative integers.

Number lines help students to understand the concept of positive and negative integers, and it helps to learn how to add or subtract the integers.

Importance of the Number Line

Let us see how we can add the integers using the number line. Example: Let us add \(5\) and \(7\) using the number line.

First, mark point \(5\) on the number line. Then move \(7\) points to the right since we are adding positive integers.

From the above number line, we find that \({\rm{5 + 7 = 12}}\).

Now, let us consider negative integers. \(-6\) and add \(3\) to it and find the value.

We know that \(-6\) lies left to zero in a number line and \(3\) lies to the right side of zero in a number line. So, mark the point \(-6\) on the number line and move \(3\) points to the right as we are adding a positive number with \(-6\)

So, from the above number line, we can write \({\rm{ – 6 + 3 = – 3}}\).

Now, let us add \(-6\) and \(-3\) using a number line. Since both are negative integers, mark the point \(-6\) to the left of zero. Because \(-6\) lies left to zero in a number line.

Now, move \(3\) points to the left of -6 on the number line (\(-3\) is a negative integer).

So, from the above number line, we can write \({\rm{ – 6 – 3 = – 9}}\)

So, we have learned how to add the given integers using a number line.

Properties of the Integers over Addition

The \(4\) main properties of integers are

1. Closure Property
2. Commutative Property
3. Associative Property
4. Existence of Identity
5. Existence of Inverse

These are the four properties of addition. Let us discuss these using some examples.

Closure Property

Integers are closed under addition, which means if we add two integers, we will get an integer as a result.

We can easily represent this as

\({\rm{Integer}} + {\rm{Integer}} = {\rm{Integer }}\)

For example, \({\rm{3 + 1 = 4}}\)

Here, 3 and 1 are integers, and on adding them you get an integer that is \(4\).

Commutative Property

The commutative property states that when the order or numbers are changed or interchanged, the answer will remain the same.

If \(p\) and \(r\) are two integers, then if \(p + r = r + p\) , the addition is commutative in integers.

For example, \({\rm{1 + 2 = 2 + 1 = 3}}\)

So, addition is commutative in integers.

Associative Property

The associative property states that when three or more numbers are operated together, the order of numbers did not affect the result.

If \(p,\,q\) and \(r\) are three integers, then if \(\left( {p + q} \right) + r = p + \left( {q + r} \right)\) , the Addition is associative in integers.

For example, \({\rm{1 + (2 + 3) = 6 = (1 + 2) + 3}}\)

Existence of Identity

Identity property exists for the Addition of integers.

Additive identity states that the sum is the integer itself when an integer is added to the additive identity. The additive identity is \(0.\)

For any integer \(a,\,\,a + 0 = 0 + a = a.\)

Existence of Inverse

Inverse property exists for the addition of integers.

Additive inverse states that when an integer is added to the additive inverse, the sum yields zero.

For any integer \(a,\,\,a + ( – a) = 0\)

So, the additive inverse of \(a\) is \(-a\) and the additive inverse of \(-a\) is \(a\).

Solved Examples on 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 \(3\) to the right side of \(0\) and start moving \(4\) points to the right.

So, \({\rm{3 + 4 = 7}}\)

Q.2. 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, \({\rm{6 + 5 = 11}}\)

Q.3. Add \(23\) and \(-45.\)
Ans: We need to find \({\rm{23 + ( – 45)}}\)
When a positive and negative integer has to be added, we should first take the absolute value of the two integers and take the difference between them.
Here, the absolute value of \(23 = 23\) and the absolute value of \({\rm{ – 45 = 45}}\)
So after finding the difference of absolute values, the answer will take the sign of the integer with the greater absolute value. That is \(45.\)
Now, the difference between \(-45\) and \(23\) is \(-22.\)
Hence, the answer is \(-22.\)

Q.4. Add \(-46\) and \(-65\).
Ans: When two negative integers are added, the result will be a negative integer. So, when adding two negative integers, add the absolute value of both the integers and add a negative sign to the answer.
So, the absolute value of \(-46\) is \(46\) and \(-65\) is \(65\).
Now, adding the absolute values of both the numbers we get, \({\rm{46 + 65 = 111}}\)
Hence, \({\rm{( – 46) + ( – 65) = – 111}}\).

Q.5. Do \({\rm{12 + 13}}\) and \({\rm{13 + 12}}\) give the same result? Justify.
Ans: According to the commutative property, when the order of integers is changed or interchanged, the answer will remain the same.
So, \({\rm{12 + 13 = 25,}}\,{\rm{13 + 12 = 25}}\)
Hence, \({\rm{12 + 13 = 13 + 12}}\)

Summary

In this article, we have learned about the addition of integers and some properties of integers. We discussed how we add integers using number lines and without number lines.

FAQs on Addition of Integers

Q.1. How do the number lines help in the addition of two integers?
Ans: The positive integers are represented to the right side of zero on the number line, and negative integers are represented to the left side of zero.
Example: Let us add \(p\) and \(q\) using the number line, \(p,\,q\) are positive integers.
First, mark point \(p\) by the number line. Then move \(q\) points to the right since we are adding positive numbers. In the case of a negative integer, we have to move to the left side.

Q.2. What is Addition?
Ans: Addition is a process of finding the sum of two or more numbers or objects or something else. The addition is a mathematical operation that uses \(+\) as the symbol to add the given things.

Q.3. The integers are closed under Addition. Justify the statement.
Ans: The closure property of integers under addition states that the sum of two integers is always an integer.
\({\rm{Integer}} + {\rm{Integer}} = {\rm{Integer}}\)
For example, \({\rm{5 + 3 = 8}}\)
Here, \(5\) and \(3\) are integers, and adding them also gets an integer that is \(8\).

Q.4. If p is an integer, then find the additive inverse of it.
Ans: Additive inverse states that when an integer is added to its additive inverse, the result will be zero.
 For integer \(p + ( – p) = 0\)
So, the additive inverse of \(p\) is \(-p\).

Q.5. What are the different properties of the addition of integers?
Ans: The different properties of the addition of integers are:

1. Closure property
2. Commutative property
3. Associative property
4. Existence of Identity
5. Existence of Inverse

Learn Addition and Subtraction of Integers

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