An eighth-grade student claims she can prove that subtraction
of integers is commutative. She points out that if a
and b are integers, then a-b = a+ -b. Since addition is
commutative, so is subtraction. What is your response?
My response is that the claim made by the eighth-grade student is incorrect. While it is true that adding the opposite of a number, denoted by -b, is the same as subtracting that number, the commutative property of addition does not directly imply the commutative property of subtraction.
To prove commutativity, we need to show that a-b is equal to b-a for all values of a and b. Let's consider an example to illustrate this:
Let a = 5 and b = 3. According to the student's claim, 5-3 should be equal to 5+(-3), which is 2. However, if we calculate 3-5, we get -2. Therefore, subtraction is not commutative.
To further clarify, the commutative property states that changing the order of the numbers being added or multiplied does not change the result. For example, 2+3 is the same as 3+2, and 4*5 is the same as 5*4. However, with subtraction, changing the order of the numbers does change the result.
Therefore, to argue that subtraction is commutative based on the commutative property of addition would be a false assumption. Subtraction is not commutative in general.