Before we can even start to answer this question, we need to know what commutative means.
You may have heard about the commutative property before.
Even if we do not know the formal definition of commutative property yet, you have used it before with addition and multiplication.
2 x 3 = 3 x 2
11 + 5 = 5 +11
In math, we strive to set solid definitions in order to provide logical structure.
Let's construct a working definition for commutative property to guide us to see if division is commutative.
First, we need have an operation defined.
For this lesson, we are only going to use the integers to help us out.
Recall, integers are positive and negative whole numbers where zero is also included.
Zero is not positive or negative.
Zero is an interesting number, which we will explore in future lessons. However, it is important to know one other cool fact about zero.
Zero is even.
We are going to use the term binary operation to make a general definition of the commutative property, so let's define what a binary operation is now.
A binary operation is simply a set of rules applied to two numbers where the result produces another number.
Binary = two numbers
Operation = rules applied to numbers to produce an output number
In formal mathematics, we have to use sets to define terms.
Sets are just a collection of items.
In our case, our set will be integers.
I will use this notation for integers:
I = {..., -2, -1, 0, 1, 2, ...}
In math, the goal is to have a general definition that applies for all numbers (elements) in your set.
We do not want to test every pair of numbers to confirm if they have the commutative property or not.
That would take a long time to do...an infinite amount of time.
Instead, we want to make broad statements on sets to generalize the results to all numbers.
One other note, in math the term star is used as a placeholder for operations.
It is used to say that some operation star is applied to a set to produce another element in the set.
The notation for star is just that ... a star *.
Not all operations on sets will produce an element within the set.
Just think about taking the square root of -1.
A whole new set of numbers had to be created, imaginary numbers, since there is no solution for square root of -1 in the real number system.
Back to commutative property...
Now we can define what commutative property means in regard to the set of integers.
Def:
I = {..., -2, -1, 0, 1, 2, ...}
A binary operation * on a set I is commutive for all elements in the set if the following holds true:
a*b = b*a, where a, b are elements in set I.
Notice again the * operation.
In our case, we already know that addition and multiplication are commutative.
Ex:
a = 3
b = 5
* = +
We must show that a*b equals b*a.
a*b = (3)+(5) = 8
b*a = (5)+(3) = 8
Therefore, a*b = b*a.
The same logic applies if we change the operation * to multiplication.
For practice, replace the logic from the example above to show a case where multiplication is commutative.
Now we know formally that the operations addition and multiplication are commutative for numbers in the integers. However, what about the other operations we are used to?
Let's check if subtraction and division have the commutative property in the integers.
But before we start, we need to go back to our definition of commutive property on integers from earlier.
Def:
I = {..., -2, -1, 0, 1, 2, ...}
A binary operation * on a set I is commutive for all elements in the set if the following holds true:
a*b = b*a (where a, b are elements in the set I)
If we find one example where the logic is false from our definition, the operation will not have the commutive property.
Let's look at subtraction now for the same numbers a = 3 and b = 5 we used before.
Ex:
a = 3
b = 5
* = -
We need to show that a*b =b*a.
a*b = (3) - (5) = -2
b*a = (5) - (3) = 2
-2 ≠ 2
Therefore, a*b ≠ b*a for subtraction.
Thus, we have shown a case where the property does not hold true for all values of integers for the operation subtraction.
What we just saw was called a counterexample.
In math, counterexamples are used to show cases that are false.
If we find one counterexample, it disproves the entire definition.
In the subtraction operation example, we found a counterexample to our original definition, which means subtraction does not have the commutive property for integers.
Note, this is true for other sets of numbers (rational, irrational number sets) for subtraction. However, since our definition is based on integers, I am just discussing properties of integers now.
Now, we can finally answer our question:
Is Division Commutative?
Let's do the same process as we did for subtraction. However, change the operation * to be ÷.
Ex:
a = 3
b = 5
* = ÷
We need to show that a*b =b*a.
a*b = (3) ÷ (5) = (3/5)
b*a = (5) ÷ (3) = (5/3)
(3/5) ≠ (5/3)
Therefore, a*b ≠ b*a for the operation division.
Since we found one counterexample where the property does not hold true, we can conclude that division is not commutative.
Mark Braxton, Lead Instructor, UPskill Network
