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What comes to mind when you hear the word associative? Associate means to ‘connect’ or ‘to group’. In the same way, the associative property allows us to group terms that are joined by addition or multiplication in various ways. Parentheses are used to group the terms, and they establish the order of operations. Work inside the parentheses is always done first.
Let’s understand what is associative property and how it is used.
What is Associative Property?
The associative property states that the sum or the product of any three or more numbers is not affected by the way in which the numbers are grouped by parentheses. In other words, if the same numbers are grouped in a different way for addition and multiplication, their result remains the same.
Mathematically it can be stated as
If $A$, $B$, and $C$ are any three numbers then,
- For addition: $\left(A + B \right) + C = A + \left(B + C \right)$
- For multiplication: $\left(A \times B \right) \times C = A \times \left(B \times C \right)$
Note
- The associative law is applicable to addition and multiplication
- The associative law is not applicable to subtraction and division
A. Associative Property of Addition
The associative property of addition states that $\left(A + B \right) + C = A + \left(B + C \right)$.
Let’s understand the associative property of addition by this example.
Consider three numbers $A = 5$, $B = 7$ and $C = 3$.
Left Hand Side of the statement becomes $\left(5 + 7 \right) + 3 = 12 + 3 = 15$.
And, the Right-Hand Side of the statement is $5 + \left(7 + 3 \right) = 5 + 10 = 15$.
The result of both is the same ($=15$).
Consider one more example by taking $A = 18$, $B = 23$, and $C = 32$.
Left Hand Side of the statement becomes $\left(18 + 23 \right) + 32 = 41 + 32 = 73$.
And, the Right-Hand Side of the statement is $18 + \left(23 + 32 \right) = 18 + 55 = 73$.
In this case, also the result of both is the same ($=73$).
So, based on the associative property of addition we can say that the sum of three or more numbers always remains the same whatever way we group the numbers to add them.
B. Associative Property of Multiplication
The associative property of multiplication states that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$.
Let’s understand the associative property of multiplication by this example.
Consider three numbers $A = 3$, $B = 2$ and $C = 5$.
Left Hand Side of the statement becomes $\left(3 \times 2 \right) \times 5 = 6 \times 5 = 30$.
And, the Right-Hand Side of the statement is $3 \times \left(2 \times 5 \right) = 3 \times 10 = 30$.
The result of both is the same ($=30$).
Consider one more example by taking $A = 12$, $B = 15$, and $C = 40$.
Left Hand Side of the statement becomes $\left(12 \times 15 \right) \times 40 = 180 \times 40 = 7200$.
And, the Right-Hand Side of the statement is $12 \times \left(15 \times 40 \right) = 12 \times 600 = 7200$.
In this case, also the result of both is the same ($=7200$).
So, based on the associative property of multiply we can say that the product of three or more numbers always remains the same whatever way we group the numbers to multiply them.
Is Associative Property Applicable to Numbers of All Categories?
The associative property in both the two forms – associative property of addition and associative property of multiplication works well with any real number.
Note: A set of real numbers $R$ is a superset of sets of natural numbers, a set of whole numbers, a set of integers, a set of rational numbers, and a set of irrational numbers.
Associative Property of Natural Numbers and Whole Numbers
Consider any three whole numbers $45$, $67$, and $13$.
Associative Property of Addition
We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$
$\left(45 + 67 \right) + 13 = 45 + \left(67 + 13 \right)$
$=>112 + 13 = 45 + 80 => 125 = 125$.
Also, $\left(0 + 12 \right) + 19 = 0 + \left(12 + 19 \right)$
$=>12 + 19 = 0 + 31 => 31 = 31$.
Associative Property of Multiplication
We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$
$\left(45 \times 67 \right) \times 13 = 45 \times \left(67 \times 13 \right)$
$=>3015 \times 13 = 45 \times 871 => 39195 = 39195$.
Also, $\left(0 \times 12 \right) \times 19 = 0 \times \left(12 \times 19 \right)$
$=>0 \times 19 = 0 \times 228 => 0 = 0$.
Associative Property of Integers (Signed Numbers)
Consider any three integers $-11$, $B=+16$ and $C=-14$.
Associative Property of Addition
We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$
$\left(-11 + 16 \right) + \left(-14 \right) = -11 + \left(16 + \left(-14 \right) \right)$
$=> 5 + \left(-14 \right) = -11 + \left(16 – 14 \right)$
$=> 5 – 14 = -11 + 2$ $=>-9 = -9$
Associative Property of Multiplication
We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$
$\left(-11 \times 16 \right) \times \left(-14 \right) = -11 \times \left(16 \times \left(-14 \right) \right)$
$=> -176 \times \left(-14 \right) = -11 \times \left(-224 \right)$
$=> 2464 =2464$
Associative Property of Decimal Numbers
Consider any three decimal numbers $A = 2.5$, $B = 5.6$ and $C = 0.8$.
Associative Property of Addition
We want to verify that We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$
$\left(2.5 + 5.6 \right) + 0.8 = 2.5 + \left(5.6 + 0.8 \right)$
$=> 8.1 + 0.8 = 2.5 + 6.4 => 8.9 = 8.9$
Associative Property of Multiplication
We want to verify that We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$
$\left(2.5 \times 5.6 \right) \times 0.8 = 2.5 \times \left(5.6 \times 0.8 \right)$
$=>14 \times 0.8 = 2.5 \times 4.48 => 11.2 = 11.2$
Associative Property of Fractions
Consider any three fractions $A = \frac {1}{2}$, $B = \frac {2}{3}$ and $C = \frac {3}{4}$.
Associative Property of Addition
We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$
$\left(\frac {1}{2} + \frac {2}{3} \right) + \frac {3}{4} = \frac {1}{2} + \left(\frac {2}{3} + \frac {3}{4} \right)$
$=> \frac{23}{12} = \frac{23}{12}$
Associative Property of Multiplication
We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$
$\left(\frac {1}{2} \times \frac {2}{3} \right) \times \frac {3}{4} = \frac {1}{2} \times \left(\frac {2}{3} \times \frac {3}{4} \right)$
$=> \frac {1}{3} \times \frac {3}{4} = \frac {1}{2} \times \frac {1}{2}$
$=> \frac {1}{4}= \frac {1}{4}$
Associative Property of Irrational Numbers
Consider any three irrational numbers $A = 2\sqrt{3}$, $B = 3\sqrt{2}$ and $C = 5\sqrt{2}$.
Associative Property of Addition
We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$
$\left(2\sqrt{3} + 3\sqrt{2} \right) + 5\sqrt{2} = 2\sqrt{3} + \left(3\sqrt{2} + 5\sqrt{2} \right)$
$=>2\sqrt{3} + 3\sqrt{2} + 5\sqrt{2} = 2\sqrt{3} + 8\sqrt{2}$
$=>2\sqrt{3} + 8\sqrt{2} = 2\sqrt{3} + 8\sqrt{2}$
Associative Property of Multiplication
We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$
$\left(2\sqrt{3} \times 3\sqrt{2} \right) \times 5\sqrt{2} = 2\sqrt{3} \times \left(3\sqrt{2} \times 5\sqrt{2} \right)$
$=>6\sqrt{6} \times 5\sqrt{2} = 2\sqrt{3} \times 30$ $=>30\sqrt{12} = 60\sqrt{3}$
$=>30\sqrt{4 \times 3} = 60\sqrt{3}$ $=>30\times 2 \sqrt{3} = 60\sqrt{3}$ $=>60\sqrt{3} = 60\sqrt{3}$
Is the Associative Property Applicable to Subtraction and Division?
Let’s verify whether the associative property is applicable to subtraction and division also, i.e.,
- $\left(A – B \right) – C = A – \left(B – C \right)$
- $\left(A \div B \right) \div C = A \div \left(B \div C \right)$
Again consider any three numbers $A = 5$, $B = -2$ and $C = 0.5$
Subtraction
LHS = $\left(5 – \left(-2 \right) \right) – 0.5 = 5 + 2 – 0.5 = 6.5$
RHS = $5 – \left(-2 – 0.5 \right) = 5 – \left(-2.5 \right) = 5 + 2.5 = 7.5$
Since, LHS $\ne$ RHS, therefore, the associative property does not hold for subtraction.
Division
LHS = $\left(5 \div \left(-2 \right) \right) \div 0.5 = -\frac {5}{2} \div 0.5 = -\frac {5}{2} \times \frac{10}{5} = -5$
RHS = $5 \div \left(-2\div 0.5 \right) = -5 \div \frac {2}{0.5} = -5 \times \frac {0.5}{2} = -5 \times \frac {5}{20} = -\frac {5}{4}$
Since, LHS $\ne$ RHS, therefore, the associative property does not hold for division.
Note:
- The associative property is applicable to the operation addition
- The associative property is applicable to the operation multiplication
- The associative property is not applicable to the operation subtraction
- The associative property is not applicable to the operation division
Practice Problems
Verify Associative Property of Addition and Multiplication for the following set of numbers
- $A = 5$, $B = -7$, $C = 2$
- $A = \frac {2}{5}$, $B = -7$, $C = \frac{1}{3}$
- $A = 0.95$, $B = 1.5$, $C = -8.9$
FAQs
What is the associative property? Give an example.
The associative property states that the sum or the product of any three or more numbers is not affected by the way in which the numbers are grouped by parentheses. In other words, if the same numbers are grouped in a different way for addition and multiplication, their result remains the same.
For example, in the case of addition $12 + \left(14 + 18 \right) = \left(12 + 14 \right) + 18$. (The sum in both the cases is $44$ and in the case of multiplication $7 \times \left(5 \times 4 \right) = \left(7 \times 5\right) \times 4 = 140$.
What is the associative property formula?
There are two variations of associative property in math:
Associative property of addition: It states that for any three numbers $A$, $B$, and $C$, $A + \left(B + C \right) = \left(A + B \right) + C$.
Associative property of multiplication: It states that for any three numbers $A$, $B$, and $C$, $A \times \left(B \times C \right) = \left(A \times B \right) \times C$.
What is the difference between associative and commutative property?
The associative property deals with the grouping of numbers whereas the commutative property deals with the ordering of numbers while performing addition or subtraction.
The associative property states that in the case of addition and multiplication the grouping of the numbers does not matter i.e., for any three numbers $A$, $B$, and $C$,
Addition: $A + \left(B + C \right) = \left(A + B \right) + C$
Multiplication: $A \times \left(B \times C \right) = \left(A \times B \right) \times C$
The associative property states that in the case of addition and multiplication the ordering of the numbers does not matter i.e., for any two numbers $A$, and $B$,
Addition: $A + B = B + A$.
Multiplication: $A \times B = B \times A$.
To which operations Associative property is applicable?
The associative property is applicable to the operations of addition and multiplication. It does not hold for subtraction and division.
Is the associative property applicable to division and subtraction?
No, the associative property does not apply to subtraction and division.
Is multiplication always associative?
The operation multiplication always follows associate property for all the categories of numbers such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
Conclusion
The associative property states that the result remains the same irrespective of the order in which the arithmetic operation is performed. The associative property is applicable to addition and multiplication only and not for subtraction and division.
