The formula for the Commutative Property of Multiplication
\( a\times b=b\times a \)
Where \( a \) and \( b \) are any two numbers, algebraic terms or polynomials.
The commutative property for numbers can be understood using arrays.
Here is an example multiplication with the help of arrays:
Clearly in both these cases the product is the same. Whether objects are arranged in 4 rows of 6 objects, or 6 rows of 4 objects, the product remains the same as 24. This is a demonstration of the commutative property of multiplication using array multiplication.
Now let us see the application of the commutative property of multiplication in algebraic expressions.
Consider two algebraic terms, say \( 2a \) and \( 3b \).
Let us multiply them:
\( 2a\times 3b=6ab \)
Now, let us multiply them by changing the order:
\( 3b\times 2a=6ab \)
Now, clearly by using the commutative property of multiplication formula we know that:
\( a\times b=b\times a \)
Or,
\( ab=ba \)
So we can say that:
\( 2a\times 3b=3b\times 2a \) as, \( 6ab=6ab \)
Hence, the commutative property of multiplication formula can also be used for algebraic expressions.
Let us quickly have a look at the commutative property of the multiplication formula for algebraic expressions.
The formula for the commutative property of multiplication is:
\( a\times b=b\times a \)
But here a and b represent algebraic terms.
The commutative property of multiplication formula works for polynomials as well instead of just being applicable to algebraic terms.
Let us consider two binomials, \( (2a+b) \) and \( (3c-d) \)
Now let us multiply them:
So, \( (2a+b) \times (3c-d) \)
\( =2a\times (3c-d)+b\times (3c-d) \)
\( =2a\times 3c-2a\times d+b\times 3c-b\times d \) Using the distributive property and simplifying
\( =6ac-2ad+3bc-bd \)
Now let us change the order and multiply:
So, \( (3c-d) \times (2a+b) \)
\( =3c\times (2a+b)-d\times (2a+b) \)
\( =3c\times 2a+3c\times b-d\times 2a+(-d\times b) \) Using the distributive property and simplifying
\( =6ca+3cb-2da-db \)
\( =6ca-2da+3cb-db \) By rearranging the terms
So by using the commutative property of multiplication formula we know that:
\( ab=ba \)
Similarly,
\( ca=ac,~cb=bc,~da=ad \) and \( bd=db \)
Therefore,
\( 6ac-2ad+3bc-bd= 6ca-2da+3cb-db \)
Now, we can say that the commutative property formula for multiplication can be used for polynomial multiplication as well.
Solved Examples
Example 1: Find the array multiplication of figure 1 and then without solving find for figure 2:
Figure 1:
Figure 2:
Solution:
This array can be written in numbers as follows:
9 rows X 8 columns = 72
Now, let us look at figure 2:
To find the array multiplication for figure 2 without calculating, we can use the commutative property formula for multiplication :
Figure 2:
This can numerically be written as:
8 rows X 9 columns
As we found out for figure 1
9 X 8 = 72
So by using the he commutative property formula for multiplication we know that:
\( a\times b=b\times a \)
or,
\( 8\times 9=9\times 8 \)
\( =72 \)
So without calculating we can say that the array multiplication shown in figure two is 72.
Example 2: There is a charity event at school for which chairs need to be arranged in rows and columns. There will be 48 guests in total who will be occupying these chairs. The seating arrangement plan done by Ms. Stacy involves 6 rows and 8 columns. However, the plan done by Mrs. Cooper involves 8 rows and 6 columns. Whose seating arrangement plan is better?
Solution:
If we consider the seating plan made by Ms. Stacy then we will have \(6\times 8 = 48\) chairs which satisfies the seating requirement of 48 guests.
Now, if we consider the seating plan made by Mrs. Cooper then we will have \(8 \times 6 =48\) chairs which also satisfies the seating requirement of 48 guests.
Clearly, both Ms. Stacy’s and Mrs. Cooper’s plans work. We can use either of the one.
Now, instead of solving we can also answer this by using the commutative property formula for multiplication:
As we know that:
\( a\times b=b\times a \)
We can say that:
\( 6\times 8=8\times 6=48 \)
This also suggests that both the seating arrangement plans work.
Example 3: Multiply the following terms and check if they are following the commutative property for multiplication.
1. \( 3ab\times 4c \)
2. \( 8d\times x \)
3. \( 16\times 20xyz \)
4. \( -23mn\times 65 \)
Solution:
1. \( 3ab\times 4c=12abc \) ,
Now, let us see if the commutative property is being followed.
Let us calculate \( 4c\times 3ab=12cab \)
Clearly by using the commutative property for multiplication formula
\( a\times b=b\times a \)
Let \( a=3ab \) and \( b=4c \) ,
So clearly, \( 3ab\times 4c=4c\times 3ab \)
or,
\( 12cab=12abc \)
Hence, the commutative property is being followed.
2. \( 8d\times x=8dx \)
Now, let us see if the commutative property is being followed.
Let us calculate \( x\times 8d=8xd \)
Clearly by using the commutative property for multiplication formula
\( a\times b=b\times a \)
Let \( a=x \) and \( b=8d \) ,
So clearly, \( 8d\times x=x\times 8d \)
or,
\( 8dx=8xd \)
Hence, the commutative property is being followed.
3. \( 16\times 20xyz=320xyz \)
Now, let us see if the commutative property is being followed.
Let us calculate \( 20xyz\times 16=320xyz \)
Clearly by using the commutative property for multiplication formula
\( a\times b=b\times a \)
Let \( a=16 \) and \( b=20xyz \),
So clearly, \( 16\times 20xyz=20xyz\times 16 \)
or,
\( 320xyz=320xyz \)
Hence, the commutative property is being followed.
4. \( -23mn\times 65=-1495mn \)
Now, let us see if the commutative property is being followed.
Let us calculate \( 65\times -23mn=-1495mn \)
Clearly by using the commutative property for multiplication formula
\( a\times b=b\times a \)
Let \( a=-23mn \) and \( b=65 \),
So clearly, \( -23mn\times 65=65\times -23mn \)
or,
\( -1495mn=-1495mn \)
Hence, the commutative property is being followed.
Example 4: Multiply the following polynomials and check if they are following the commutative property for multiplication.
1. \( 3b(2a+3c) \)
2. \( (a+b)(a-b) \)
Solution:
1. \( 3b(2a+3c) \)
\( =3b\times 2a+3b\times 3c \) Using the distributive property.
\( =6ab+9bc \)
Now, let us see if the commutative property is being followed.
Let us calculate:
\( (2a+3c)\times 3b \)
\( =2a\times 3b+3c\times 3b \) Using the distributive property.
\( =6ab+9cb \)
Clearly by using the commutative property for multiplication formula
\( ab=ba \) and \( bc=cb \)
So, \( 6ba+9bc=6ab+9cb \)
Hence, the commutative property is being followed.
2. \( (a+b)\times (a-b) \)
\( =a\times a-a\times b+b\times a-b\times b \) Using the distributive property.
\( =a^2-ab+ba-b^2 \) By simplifying.
\( =a^2-b^2 \)
Now, let us see if the commutative property is being followed.
Let us calculate:
\( (a-b)\times (a+b) \)
\( =a\times a+a\times b-b\times a-b\times b \) Using the distributive property.
\( =a^2+ab-ba-b^2 \) By simplifying.
\( =a^2-b^2 \)
Clearly by using the commutative property for multiplication formula
\( a\times b=b\times a \) where , \( a=(a+b) \) and \( b=(a-b) \) and their product is \( a^2-b^2 \)
So, \( (a+b)\times (a-b)=(a-b)\times (a+b) \)
Hence, the commutative property is being followed.
The commutative property for addition can be written as:
\( a+b=b+a \)
It also holds true and can be seen in the example below:
If we consider two numbers 30 and 20 and add them,
\( 30+20=50 \)
Now, if we change the order and add them,
\( 20+30=50 \)
We can see that the commutative property for addition is followed as: \( 30+20=20+30 \)
The commutative property of multiplication formula is surely applicable to integers. This can be written as:
Let us consider two integers -2 and -3
Now, by using the commutative property of multiplication formula:
\( -2\times -3=6 \) and \( -3\times -2=6 \)
Hence, integers follow the commutative property of multiplication formula.
Commutative property for multiplication can be seen in multiplication tables. Here are multiplication tables till 10. We can observe that the numbers following commutative property are highlighted with the same color.
The commutative property for subtraction does not hold true
Let us understand this further.
If the commutative property for subtraction was true, then, this would satisfy the following:
\( a-b=b-a \)
Let us check this with an example if this property holds true:
Consider two numbers 30 and 20 and subtract them.
\( 30-20=10 \)
Now, we change the order and subtract them.
\( 20-30=-10 \)
We can see that the commutative property is not followed while subtracting numbers. This is because, \( 30-20\neq 20-30 \).
The commutative property for division does not hold true.
Let us understand this further,
If the commutative property for division was true, then this would satisfy the following equation:
\( a\times b=b\times a \)
Let us check with an example if this holds true:
Take two numbers 30 and 15
Divide them as shown:
\( 30\div 15=2 \)
Now, change the order and divide these numbers.
\( 15\div 30=0.5 \)
We can see that the commutative property is not followed as when we divide numbers. As, \( 30\div 15\neq 15\div 30\).