Problem 107 - Elementary and Intermediate Algebra

Question

Explain what it means when we say that subtraction is not commutative.

Step-by-Step Solution

Answer

Subtraction is not commutative because changing the order of numbers changes the result.

Step 1: Understanding Commutativity

Commutativity is a property of some operations where the order of the numbers involved doesn't change the result. For example, in addition, both \( a + b \) and \( b + a \) yield the same result. This property is known as commutativity.

Step 2: Applying Commutativity to Subtraction

To determine if subtraction is commutative, we test whether changing the order of the numbers affects the result. Consider two numbers, \( a \) and \( b \). For subtraction to be commutative, \( a - b \) should equal \( b - a \).

Step 3: Example Calculation

Let's choose simple numbers to illustrate this. Take \( a = 5 \) and \( b = 3 \). Calculate \( a - b = 5 - 3 = 2 \) and then \( b - a = 3 - 5 = -2 \). Since \( 2 eq -2 \), this indicates a change in the result when the numbers are reversed.

Step 4: Conclusion on Commutativity of Subtraction

Because reversing the numbers in subtraction results in a different answer, unlike addition, subtraction is not commutative. In general, \( a - b eq b - a \) which means subtraction doesn't have the commutative property.

Key Concepts

Subtraction

Subtraction is one of the basic arithmetic operations where one quantity is taken away from another. Imagine you have 5 apples, and you give away 3 of them to a friend. Now, you're left with 2 apples. This can be mathematically represented as:

• 5 - 3 = 2

This simple operation helps us determine how much is left when we remove something from an initial amount.

But what happens when we reverse the order in subtraction? If you try the reverse, giving away 5 apples when you only have 3, it doesn't quite make sense without borrowing! Mathematically:

• 3 - 5 = -2

Here, you get a negative number, which shows a debt or deficiency. This difference when the order changes is a key point in understanding why subtraction behaves differently than addition.

Order of Operations

In mathematics, when you have different operations to perform in a problem, the sequence in which you do them matters. This is especially true when subtraction is involved along with other operations like addition, multiplication, or division.

The sequence to follow is usually remembered by the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Consider the expression: 8 - 5 + 2. To solve this, you start from left to right:

• First, do 8 - 5 = 3
• Then add 2 to the result: 3 + 2 = 5

Switching orders here, such as doing the addition first, would lead to inaccurate results. So, keeping the order correct is crucial for getting the right answer.

Non-Commutative Operations

Non-commutative operations, such as subtraction, show that the order of items affects the final outcome. Unlike commutative operations like addition and multiplication, changing the sequence can drastically change the result you get.

• In subtraction, reversing terms like from: a - b to b - a gives a different number.
• This is similar to division, where dividing a by b is not equal to dividing b by a.

Let's compare subtraction and addition quickly:

• For addition: 3 + 5 = 5 + 3 = 8
• For subtraction: 3 - 5 is not equal to 5 - 3

Recognizing which operations are non-commutative helps in solving problems accurately and appreciating the unique nature of each operation in arithmetic.