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Properties of Addition with Solved Examples.

addition properties

Overview

Addition is one of the most fundamentally used operations in mathematics. Where you go, whether to purchase a thing, or calculate the amount of money you have in your bank, addition becomes a crucial operation. 

There are a few properties for addition. Mathematics applies four basic properties for adding two or more numbers. There are other properties besides these four, which we will study in this article. This article aims to make you aware of all the properties you need for addition. 

The properties of addition specify how we can combine the given integers. In mathematics, “addition” is one of the most fundamental arithmetic operations. The process of summing items together is known as an addition. The sign “+” is used to add the integers together. The numbers we’ll add are called “addends,” and the result we’ll get is called “sum” or “total”. Addition requires at least two addends. These can be any integer (positive or negative), fractions, and decimals. 

Need for Addition Properties 

The properties of addition specify how we can combine the given integers. In many algebraic problems, addition characteristics are used to reduce complex statements to a much simpler form. These qualities are extremely beneficial to pupils because they follow all sorts of integers. We’ll go over the most important addition properties in this article with definitions and examples. 

What are the Four Properties of Addition?

There exist several laws and conditions of addition; properties of addition are defined. These properties also indicate the addition’s closure property. Other arithmetic operations like subtraction, multiplication, and division have well-defined properties in Mathematics, similar to addition. However, the properties of each operation may differ from each other. In Math, four fundamental properties of addition are defined. The name of the four basic properties are:

  1. Commutative property of addition 
  2. Associative Property of Addition 
  3. Distributive property of addition 
  4. Additive Identity Property of addition 

Read below the detailed explanation of each property of addition.

Commutative Property of Addition

The commutative property comes from the word commute, which means to move around. This property of addition states that on adding two or more numbers, the position of the numbers doesn’t matter. You can change the order of the numbers according to your choice. The result will remain unaltered despite the arrangement of the numbers. This property also holds for multiplication. Representation of this property is given below:

  • K + L = L + K
  • K + L + M = K + M + L = L + M + K = L + K + M and so on.

Example of the commutative property of addition: Add the numbers given as K = 32 and L = 12 using commutative property. 

Solution: Commutative property tells that K + L = L + K

               Therefore, 32 + 12 = 12 + 32

                                         44 = 44

We can notice that the left-hand side of the equation is equal to the right-hand side. The result doesn’t depend on the arrangement of numbers. Hence we can say that addition is commutative. Remember it with ‘commute’, which means to switch places. 

Associative Property of Addition

The associative property is yet another basic and easy-to-understand property of addition. If we split the words of this property, we get the word ‘associate’. It means to group or come together. Thus this property is related to the grouping of numbers during addition. 

This property is only applicable when adding three or more numbers. The addition with two terms doesn’t follow this law. According to the associative property of addition, the numbers can be associated with one another. You can first group the first and second numbers or associate the first and third numbers first. The grouping doesn’t matter. The result will always be the same. Long story short, this law implies that when three or more numbers are added together, the total/sum remains the same, even if the order of the addends is modified. This property can be represented as follows:

  • K + (L + M) = (K + L) + M

Let us take an example to understand this property.

Example: Let the numbers be K = 4, L = 21 and M = 12

We know, according to the associative law of addition. 

K + (L + M) = (K + L) + M

Therefore LHS = K + (L + M) = 4 + (21 + 12) = 4 + 33 = 37

                 RHS = (K + L) + M = (4 + 21) + 12 = 25 + 12 = 37

Thus we can see that LHS = RHS = 37 

As a result, the associative property is established. This characteristic applies to multiplication as well. Remember that the parenthesis or round brackets () are used to group the numbers in this property. It creates operations using a set of numbers. 

Distributive Property of Addition

The distributive property of addition is far different from the commutative and associative property. It doesn’t involve the movement of the terms in the expression. This property, as the name suggests, distributes the terms for addition. The distributive property is essential because it combines both the addition and multiplication operations. For this property to happen, we must have a number multiplied by the sum of 2 numbers. This is represented as:

  • K × (L + M) = K × L + K × M

As you can see, K is multiplied by both the terms respectively. Here K is also termed as a monomial factor. The term (L + M) is the binomial term. 

Example: Find the answer to the expression using distributive property where K = 4, L = 17, and M = 3. 

Solution: According to distributive property K × (L + M) = K × L + K × M

First solving the left hand side

K × (L + M) = 4 x (17 + 3) = 4 x 20 = 80.

Now solving the right hand side 

K × L + K × M = 4 x 17 + 4 x 3 = 68 + 12 = 80

Hence we have proved that according to distributive property, the multiplication to the sum of numbers is equal to the sum of individual products.  

Additive Identity Property of Addition

This characteristic states a unique positive integer for each number, which yields the number itself when added to the integer. Zero is a one-of-a-kind real number added to a number to create the number. As a result, zero is the identity element of addition. This is represented as: 

K + 0 = K or 0 + K = K

For example: K = 13 therefore 13 + 0 = 13 and 0 + 13 = 13.

The identity property of addition is easily remembered by thinking about it and asking and answering questions. We must consider which integer should be increased to the supplied number to preserve the original number’s value. If you believe that, your response should be zero.

Hitherto we have completed the four basic addition properties. Go through them once again if you have any shadow of a doubt.

Some More Properties of Addition

  • Property of Opposites

This property states that if there exists a real number ‘K’, then there also exists a unique number ‘-K’, which when added to the original number yields zero as a result. This can be seen as 

  • K + (-K) = 0 or (-K) + K = 0

This is known as the property of opposites. The term ‘-K’ is known as the opposite. Since the addition of the opposite causes the result to be zero, thus ‘-K’ is also known as the additive inverse of the number. This property is also termed as the inverse property of addition. Remember that every real number on the number line has one additive inverse. 

Example: Find the additive inverse of K = 8. 

From the property of opposites, we know that the additive inverse of any number is the number that gives zero when added to the original integer. We have K= 8. 

Therefore 

K + (X) = 0

8 + X = 0

X = -8 

Thus the additive inverse of K is -8.

What if the number given to us is already negative. For example, find the additive inverse of K = -23.

We will still apply the law of additive inverse. We know that

K + X = 0

-23 + X = 0

X = 23.

Therefore, the additive inverse of -23 is plus 23.

Sum of Opposite of Numbers

This is a derivative of the above property. Consider two numbers K and L, with their additive inverses equal to -K and -L. This property states that the inverse sum of the two numbers is equal to the sum of the individual opposites. The properties representation is given below: 

-(K + L) = (-K) + (-L)

Let us prove this with an example: 

Example 1: Suppose K = 9 and L = 11.

Solution: The additive inverse of K = -9 and L = -11

According to the sum of opposite numbers, property

-(K + L) = (-K) + (-L)

LHS → -(K + L) = -(9 + 11) = -20

RHS→ -K + -L = -9 – 11 = -20

The LHS = RHS; hence the property is true. 

Example 2: Prove that – (-2+-7) = (2)+(7)

Solution: According to the property of sum of additive inverse 

LHS

– (-2 + -7) = – (-9) = 9

RHS

2 + 7 = 9

LHS = RHS hence (- (-2 + -7)) = (2) + (7)

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