Associative Law: Definition, Addition, Multiplication and Formula

Associative law or associative property is a mathematical property applied to the addition and subtraction of three numbers. Associative law can be applied to problems regarding addition and multiplication and not to subtraction and division because there will be a change in result. The position of integers in addition and multiplication do not change the sign of the integers. Therefore, associative law can be applied. In case of addition and multiplication, the change in order or position of numbers cannot affect the result. Associative property can also be used in vector algebra to solve vector-related problems.

Read More: Applications of Determinants and Matrices

Key Terms: Matrix, Associative law, Rational numbers, Equations, Numbers, Division, Subtraction, Addition, Multiplication

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Associative Law formula

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The formula for Associative law can be derived easily from the definition and examples. Associative law is a mathematical law that can be followed for addition and multiplication of three numbers, without any pattern of grouping or particular pattern. Three numbers can be grouped and regrouped with two of them in brackets and one outside, and solved starting from the bracketed numbers. The grouping or combination of numbers in any way, through associative law, does not yield a different result than the original one. According to Associative Law, the three numbers can be grouped in the following ways:

a+(b+c)=(a+b)+c

a*(b*c)=(a*b)*c

Also Read: Sequence and Series

The associative laws for each of addition and multiplication will be explained below:

• Associative Law of Addition

Addition operation in mathematics always follows associative property. Which means that the results do not depend on the combination of numbers and any grouping or regrouping will give out the same result. Therefore, addition can be represented in the following way through associative property if the numbers taken are x,y and z.

x+(y+z)=(x+y)+z=x+y+z

• Associative Law of Multiplication

Multiplication, like addition, also follows associative property and the operation does not depend on how the numbers are clubbed. Taking any association or grouping, it will bring the same result. Taking the numbers x,y and z, the associative law can be represented as follows.

x*(y*z)=(x*y)*z=x*y*z

Also Read: Complex Numbers and Quadratic Equations

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Proof of Associative Law

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After learning the formula for associative law, the proof of associative law can be established. The proofs for the law of addition and the law of multiplication are explained below.

• Proof of Associative Law of Addition

Associative law for addition can be proven through examples. Let’s understand how this property works.

Example 1: Prove that 2+(3+5)=(2+3)+5

Taking LHS first, 2+(3+5)=2+8=10

Now, we can take RHS,

(2+3)+5=5+5=10

After comparing both sides of the equation,

LHS=RHS

Therefore, it can be proved that,

2+(3+5)=(2+3)+5

Example 2: Prove that 4+(-5+8)=(4+(-5))+8

Taking the LHS first, we can see that,

4+(-5+8)=4+(3)=7

Now, taking RHS, it can be seen that

(4+(-5))+8=(4-5)+8=-1+8=7

After comparing both sides of the equation,

LHS=RHS

Therefore, it can be proved that,

4+(-5+8)=(4+(-5))+8

Also Read: Real valued functions

• Proof of Associative Law of Multiplication

The proof of the associative law of multiplication can also be presented in a similar way.

Example 1: Prove that 2*(3*5)=(2*3)*5

Taking LHS first, 2*(3*5)=2*15=30

Now, we can take RHS,

(2*3)*5=6*5=30

After comparing both sides of the equation,

LHS=RHS

Therefore, it can be proved that,

2*(3*5)=(2*3)*5

Example 2: Prove that 4*(-5*8)=(4*(-5))*8

Taking the LHS first, we can see that,

4*(-5*8)=4*(-40)=-160

Now, taking RHS, it can be seen that

(4*(-5))*8=(-20)*8=-160

After comparing both sides of the equation,

LHS=RHS

Therefore, it can be proved that,

4*(-5*8)=(4*(-5))*8

Also Read: Types of Matrices

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Why does Associative Law not work for Subtraction and Division?

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Associative law is applicable only to addition and multiplication and not to subtraction and division. Why it does not work in subtraction and division is discussed below.

1. Subtraction

Let’s take into account three numbers 2,3 and 5.

Now, let us assume that associative law applies to subtraction. Thus, by the law,

2-(3-5) should be equal to (2-3)-5.

On solving LHS we get 2-(3-5)=2-(-2)=2+2=4

On solving RHS we get (2-3)-5=-1-5=-6

4 does not equate to -6, therefore,

LHS is not equal to RHS.

Therefore, it can be stated that,

2-(3-5) is not equal to (2-3)-5.

Therefore, the assumption that associative law can be applied to subtraction is not valid.

2. Division

Suppose, we take into account three numbers 8,4 and 2.

Now we can assume that associative property applies to division. So, 8/(4/2)=(8/4)/2. And we can see whether the regrouping changes the result or not.

From LHS, we get 8/(4/2)=8/2=4.

From RHS, we get (8/4)/2=2/2=1.

From the two equations it can be justified that

4 is not equal to 1.

Therefore, RHS is not equal to LHS or 8/(4/2) is not equal to (8/4)/2.

Therefore, the assumption that associative law applies to division can be deemed null since this example clearly proves that it does not.

Also Read: Determinant of a Matrix

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Associative Law of Rational Numbers

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Rational numbers in the number system possess the associative property-which means that when three rational numbers are added or multiplied, in any order or combination, they will give the same result. But in case of subtraction or division, the result will differ. For addition, three numbers: ?,¼ and ½ can be taken as examples. When we add (?+¼)+½ or ?+(¼+½) we always get a result of 13/12. In case of multiplication, if we take the three numbers and if multiply them as (?*¼)*½ or ?*(¼*½) we always get a result of 1/24. Therefore, it can be proven that associative property of rational numbers is a noticeable aspect. For subtraction and division, associative law does not hold true because the change in the grouping of numbers will always yield a different result because of the arrangement of numbers.

Also Read: Linear Equation in Two Variable

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Examples of Associative Law

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There are several examples of associative law which can be taken into account to understand the property of numbers better regarding addition and multiplication. There are some examples which are noted below.

Example 1: Let’s take three numbers 10,25 and 30. 

Now according to associative property, we have to prove that 10+(25+30)=(10+25)+30.

Taking the LHS,

10+(25+30)=10+55=65.

Now, taking RHS,

(10+25)+30=35+30=65.

So, as a result, LHS=RHS.

Therefore, it proved that 10+(25+30)=(10+25)+30.

Example 2: Let’s take three numbers 23,37 and 55.

Now according to associative property, we have to prove that 23+(37+55)=(23+37)+55.

Taking the LHS,

23+(37+55)=23+92=115.

Now, taking RHS,

(23+37)+55=60+55=115.

So, as a result, LHS=RHS.

Therefore, it proved that 23+(37+55)=(23+37)+55.

Also Read: Operations on Matrices

Example 3: Let’s take three numbers 9,12 and 14.

Now according to associative property, we have to prove that 9*(12*14)=(9*12)*14

Taking the LHS,

9*(12*14)=9*168=1512

Now, taking RHS,

(9*12)*14=108*14=1512

So, as a result, LHS=RHS.

Therefore, it proved that 9*(12*14)=(9*12)*14.

Example 4: Let’s take three numbers -4,-8 and -12.

Now according to associative property, we have to prove that -4*(-8*-12)=(-4*-8)*-12

Taking the LHS,

-4*(-8*-12)=-4*96=-384.

Taking the RHS,

(-4*-8)*-12=32*-12=-384.

So, as a result, LHS=RHS.

Therefore, it proved that -4*(-8*-12)=(-4*-8)*-12.

Also Read: Onto Function

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Things to Remember

• Associative law indicates that three numbers added or multiplied together can be done in no specific order and they will always yield the same result.
• According to the associative law of addition, the following equation is true.: a+(b+c)=(a+b)+c
• According to the associative law of multiplication, the following equation is true: a*(b*c)=(a*b)*c
• Associative property in mathematics is very useful since through this property while adding and multiplying big numbers, they can be grouped or assembled to create smaller parts irrespective of the order to solve bigger equations.
• Addition and multiplications of rational numbers can be solved easily through associative law.

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