Properties of addition is considered to be one of the basic arithmetic operations in Mathematics. The process of adding the numbers or things together is called addition. In addition, to add the numbers together, the sign “\( + \)” is used. After adding two or more numbers, the outcome is called “sum”. Properties of addition are used to solve different mathematical problems. Addition properties are very helpful as these properties are easy in terms of application and makes the task easier. In this article, there will be an in-depth discussion on properties of addition.

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Properties of Addition: Definition

Properties of addition are defined for the different conditions and rules of addition. The properties of addition also indicate the closure property of addition. In fact, in addition, it adds two or more numbers together. The properties of addition define the way of adding two or more numbers. It defines the various rules and condition for addition.

Addition: Rules to be Followed

1. Suppose two positive or two negative integers/numbers are added. We add their values regardless of their signs and give the sum their common sign.
   Examples:
   a. \({\rm{3 + 5 = 8}}\)
   b. \({\rm{( – 3) + ( – 5) =  – 8}}\)
   c. \({\rm{( – 9) + ( – 6) = ( – 15)}}\)
2. To add a positive and a negative integer, we find the difference between their numerical values regardless of their signs, and give the sign of the integer with the greater value to it.
   Examples:
   a. \({\rm{( – 8) + 5 = ( – 3)}}\)
   b. \({\rm{8 + ( – 5) =  + 3}}\)
   c. \({\rm{( – 12) + 10 = ( – 2)}}\)

Addition: Properties

There are different Properties of Addition, which are,

• 1. Closure Property
• 2. Commutative Property
• 3. Associative Property
• 4. Distributive Property
• 5. Additive Identity
• 6. Additive Inverse

Closure Property of Addition

In the Closure Property of Addition, the sum of two integers is always an integer/ number. It can be represented as, \(a + b = c\)

Examples:
1. \({\rm{5 + 3 = 8}}\)
2. \({\rm{ – 6 + 2 = ( – 4)}}\)
3. \({\rm{8 + ( – 2) = 6}}\)
So, the sum of two integers is always an integer.

In the Commutative Property of Addition, if the two numbers or integers are added, the sum remains the same even if we change the order of numbers/integers.

It can be represented as, \(a + b = b + a\)

Example:
Let us consider \(a = 8\) and \(b = 7\)
\( \Rightarrow 8 + 7 = 7 + 8\)
\( \Rightarrow 15 = 15\)
From the above example, when we add the two numbers, \(8\) and \(7\) and if we interchange the numbers, the obtained results remain the same as \(15\).
We can conclude that addition is commutative for integers/numbers.

Addition: Associative Properties

In Associative Property of Addition, if we add three numbers, the association of numbers in a different pattern does not change the sum. This means that while adding three or more numbers, the obtained result/ sum will be the same, even when the grouping of addends is changed.

In general, for any number/integers \(a,\,b\) and \(c\), we can say,
\(a + (b + c) = (a + b) + c\)

Example-1: Let us consider  \(a = 3,\,b = 5,\,c = 8\)
\(a + (b + c) = (a + b) + c\)
\( \Rightarrow 3 + (5 + 8) = (3 + 5) + 8\)
\( \Rightarrow 3 + 13 = 8 + 8\)
\( \Rightarrow 16 = 16\)
\( \Rightarrow {\rm{L}}{\rm{.H}}{\rm{.S}} = {\rm{R}}{\rm{.H}}{\rm{.S }}\)

Example-2: Let us consider  \(a = 10,\,b = 15,\,c = 20\)
\(a + (b + c) = (a + b) + c\)
\( \Rightarrow 10 + (15 + 20) = (10 + 15) + 20\)
\( \Rightarrow 10 + 35 = 25 + 20\)
\( \Rightarrow 45 = 45\)
\( \Rightarrow {\rm{L}}{\rm{.H}}{\rm{.S}} = {\rm{R}}{\rm{.H}}{\rm{.S }}\)

In the above examples, the left-hand side is equal to the right-hand side. Hence, the associative property is proved. This shows that addition is associative for integers.

Distributive Property of Multiplication over Addition

From the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together. This property is called the distributive property of multiplication over addition.

In general, for any integers \(a, b\) and \(c\), we can say,
\(a \times (b + c) = (a \times b) + (a \times c)\)

Example-1: Let us take  \(a = 4,\,b = 6\) and \(c=5\)
\(a \times (b + c) = (a \times b) + (a \times c)\)
\( \Rightarrow 4 \times (6 + 5) = (4 \times 6) + (4 \times 5)\)
\( \Rightarrow 4 \times 11 = 24 + 20\)
\( \Rightarrow 44 = 44\)
\({\rm{L}}{\rm{.H}}{\rm{.S}} = {\rm{R}}{\rm{.H}}{\rm{.S}}\)

Example-2: Let us take \(a = 7,\,b = 8\) and \(c = 9\)
\(a \times (b + c) = (a \times b) + (a \times c)\)
\( \Rightarrow 7 \times (8 + 9) = (7 \times 8) + (7 \times 9)\)
\( \Rightarrow 7 \times 17 = 56 + 63\)
\( \Rightarrow 119 = 119\)
\({\rm{L}}{\rm{.H}}.{\rm{S}} = {\rm{R}}.{\rm{H}}.{\rm{S}}\)

Additive Identity or Identity Property or Property of Zero

When we add zero to an integer, we get the same integer. Zero is an additive identity for integers.
If \(a\) is any integer, \(a + 0 = 0 + a\)

Example:
\({\rm{9 + 0 = 0 + 9}}\)
\( \Rightarrow 0\)
Therefore, \(0\) called the identity element for the addition of integers.

Additive Inverse

If the sum of any two integers is zero \((0)\), then the two integers are called additive inverse or negatives of each other.
If \(a, b\) are integers such that \(a + b = 0\) then \(a\) and \(b\) are the additive inverse of each other.

Example: \({\rm{5 + ( – 5) = 0}}\)
Therefore, we conclude that for any integer \(a,\, – a\) is the additive inverse.
\( \Rightarrow  – ( – a) = a\)
Note: \({\rm{0 + 0 = 0}}\);  therefore, \(0\) is its own negative, i.e., \({\rm{0 = 0}}\).

Solved Example Problems on Properties of Addition

Q.1. Write the additive inverse of \(8\).
Ans: We know, if the sum of any two numbers is zero \({\rm{(0)}}\), then the two numbers are called additive inverse or negatives of each other.
So, the additive inverse of \(8\) is \(-8\).

Q.2. Prove that \({\rm{ – (4 + 6) = ( – 4) + ( – 6)}}\).
Ans: From the given,
\({\rm{ – (4 + 6) = ( – 4) + ( – 6)}}\)
\( \Rightarrow  – 10 =  – 4 – 6\)
\( \Rightarrow  – 10 =  – 10\)
\({\rm{L}}{\rm{.H}}.{\rm{S}} = {\rm{R}}.{\rm{H}}.{\rm{S}}\)

Q.3. In a quiz, team \(A\) scored \({\rm{ – 40,}}\,{\rm{10,}}\,{\rm{0}}\) and team \(B\) scored \({\rm{10,}}\,{\rm{0,}}\,{\rm{ – 40}}\) in three successive rounds. Which team scored more?
Ans: Score of team \(A{\rm{ = ( – 40) + 10 + 0 =  – 30}}\)
The score of team \(B{\rm{ = 10 + 0 + ( – 40) =  – 30}}\)
Hence, both the team scored the same marks.

Q.4. Verify that \((a + b) + c = a + (b + c)\) by taking: \(a =  – 2,\,b = 4,\,c = 5\).
Ans: From the given,
\((a + b) + c = a + (b + c)\)
\( \Rightarrow ( – 2 + 4) + 5 = ( – 2) + (4 + 5)\)
\( \Rightarrow 2 + 5 = ( – 2) + 9\)
\( \Rightarrow 7 = 7\)
\({\rm{L}}{\rm{.H}}.{\rm{S}} = {\rm{R}}.{\rm{H}}.{\rm{S}}\)
Hence, it is verified.

Q.5. Verify that \(a + b = b + a\) by taking: \(a = 9,\,b = 5\)
Ans: From the given,
\(a + b = b + a\)
\( \Rightarrow 9 + 5 = 5 + 9\)
\( \Rightarrow 14 = 14\)
\({\rm{L}}{\rm{.H}}.{\rm{S}} = {\rm{R}}.{\rm{H}}.{\rm{S}}\)
Hence, it is verified.

Q.6. At \({\rm{8}}\,{\rm{am}}\), the temperature was \({\rm{ – }}{{\rm{6}}^{\rm{o}}}{\rm{F}}\). At noon, the temperature rose \({9^{\rm{o}}}{\rm{F}}\).  What was the temperature at noon?
Ans: The temperature in the morning was \({{\rm{9}}^{\rm{o}}}{\rm{F}}\) and it raised in the afternoon to \({{\rm{9}}^{\rm{o}}}{\rm{F}}\).
Therefore, the temperature in the noon was \({\rm{ – 6 + 9 =  – }}{{\rm{3}}^{\rm{o}}}{\rm{F}}\).

Q.7. Evaluate \({\rm{( + 423) + (253)}}\)
Ans: We know that, in the Closure Property of Addition, the sum of two numbers/integers is always an integer/ number.
Therefore, \({\rm{( + 423) + (253) = 676}}\)
Therefore, the obtained sum is \(676\).

Q.8. Add the following numbers: \({\rm{( – 9) + 15}}\)
Ans: We know thatto add a positive and a negative integer, we find the difference between their numerical values regardless of their signs and give the sign of the Integer with the greater value to it.
So, \({\rm{( – 9) + 15 =  + 6}}\)
Therefore, the obtained sum is \(+6\).

Summary

Addition is the fundamental arithmetic operation in mathematics. The properties of addition define the way of adding two or more numbers. It defines the various rules and condition for addition. This article includes the different properties of addition like Closure Property, Commutative Property, Associative Property, Distributive Property, Additive Identity, Additive Inverse in detail. Properties of Addition explained in this article helps a lot in solving the different mathematical problems.

Frequently Asked Questions (FAQs) on Properties of Addition

Frequently asked questions related to properties of addition is listed as follows:

Q.1. What do you mean by the additive inverse of integer?
Ans:  In the additive inverse of integers, if the sum of any two numbers is zero \((0)\) then the two numbers are called additive inverse or negatives of each other.
If \(a, b\) are integers/numbers such that \(a + b = 0\) then \(a\) and \(b\) are the additive inverse of each other.
Examples: \({\rm{6 + ( – 6) = 0}}\) and \({\rm{12 + ( – 12) = 0}}\)

Q.2. What number is the additive identity?
Ans: We know thatwhen we add zero to any number, we get the same number as a result.
Example: \({\rm{8 + (0) = 8}}\)
Therefore, Zero is an additive identity for whole numbers.

Q.3. What are the \(4\) basic properties of Addition?
Ans: The \(4\) basic properties of Addition are,
a. Closure Property of Addition
b. Commutative Property of Addition
c. Associative Property of Addition
d. Distributive Property of Addition

Q.4. What does the commutative property of addition tell us?
Ans: In the Commutative Property of Addition tells us that if the two numbers or integers are added, the sum remains the same even if we change the order of numbers/integers.
It can be represented as,
\(a + b = b + a\)
Example: Let us consider \(a=10\) and \(b=6\)
\( \Rightarrow 10 + 6 = 6 + 10\)
\( \Rightarrow 16 = 16\)

Q.5. What is the order property of Addition?
Ans: In the Order Property of Addition, if the two numbers or integers are added, the sum remains the same even if we change the order of numbers/integers.
It can be represented as,
\(a + b = b + a\)
Example:
Let us consider \(a=6\) and \(b=7\)
\( \Rightarrow 6 + 7 = 7 + 6 \Rightarrow 13\)

Q.6. What is the Associative Property of Addition?
Ans: In Associative Property of Addition, if we add three numbers, the association of numbers in a different pattern does not change the sum. This means that when the addition of three or more numbers, obtained result/sum will be the same, even when the grouping of addends is changed.
In general, for any number/integers \(a, b\) and \(c\), we can say,
\(a + (b + c) = (a + b) + c\)
Example: Let us consider \(a = 4,\,b = 6,\,c = 8\)
\(a + (b + c) = (a + b) + c\)
\( \Rightarrow 4 + (6 + 8) = (4 + 6) + 8\)
\( \Rightarrow 4 + 14 = 10 + 8\)
\( \Rightarrow 18 = 18\)
\( \Rightarrow {\rm{L}}.{\rm{H}}.{\rm{S}} = {\rm{R}}.{\rm{H}}.{\rm{S}}\)
In the above example, the left-hand side is equal to the right-hand side. Hence, the associative property is proved. This shows that addition is associative for numbers/integers.

Q.7. What are the different properties of Addition?
Ans: There are different Properties of Addition are:
a. Closure Property
b. Commutative Property
c. Associative Property
d. Distributive Property
e. Additive Identity
f. Additive Inverse

Q.8. Which property of addition does \(0\) illustrate?
Ans: We know that Additive Identity (Property) explains about \(0\).
When we add zero to any whole number, we get the same whole number. Zero is an additive identity for whole numbers.
If \(a\) is any number/integer, \(a + 0 = 0 + a\)
Example:
\({\rm{8 + 0 = 0 + 8}}\)
\( \Rightarrow 8 = 8\)
Therefore, \(0\) called the identity element for the addition of integers/numbers.

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