Problem 30 - Beginning and Intermediate Algebra

Question

Write an example that shows that subtraction is not commutative.

Step-by-Step Solution

Answer

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

Step 1: Understand Commutative Property

A commutative operation is one in which changing the order of the operands does not change the result. For example, in addition, if we have two numbers, say 3 and 5, both 3 + 5 and 5 + 3 equal 8. Therefore, addition is commutative.

Step 2: Define Non-Commutative Operation

To show that subtraction is not commutative, we need to find an example where changing the order of the numbers changes the result.

Step 3: Choose Two Numbers for Subtraction

Let's choose the numbers 7 and 4 for our subtraction example.

Step 4: Subtraction in Original Order

Subtract 4 from 7, yielding:\[ 7 - 4 = 3 \]

Step 5: Subtraction in Reverse Order

Subtract 7 from 4, yielding:\[ 4 - 7 = -3 \]

Step 6: Compare Results

Observe the results: \( 7 - 4 = 3 \) and \( 4 - 7 = -3 \). Since 3 is not equal to -3, subtraction changes the result when the order is switched, demonstrating that it is not commutative.

Key Concepts

Commutative Property

The commutative property is a fundamental principle in mathematics that applies to certain operations, like addition and multiplication. This property states that you can change the order of the numbers involved in the operation without changing the result. Here are some key points about the commutative property:

• It applies to addition: For any two numbers, say \(a\) and \(b\), the equation \(a + b = b + a\) holds true.
• It also applies to multiplication: For any two numbers \(a\) and \(b\), you have \(a \times b = b \times a\).
• Order does not matter in commutative operations.

Notably, the commutative property does not hold for all mathematical operations, which we'll explore further, especially with subtraction.

Subtraction

Subtraction is one of the basic operations in mathematics, but unlike addition, it does not follow the commutative property. This means that changing the order of the numbers in subtraction results in different outcomes. Here is how subtraction is distinct:

• Non-commutative by nature: If you take two numbers, \(7 - 4\), it equals 3, but \(4 - 7\) results in -3.
• Order matters in subtraction: For subtraction, swapping the minuend and subtrahend leads to different results.

Thus, unlike addition, subtraction cannot be rearranged without affecting the outcome, making it non-commutative.

Mathematical Operations

Mathematical operations are procedures you perform on numbers. The basic operations include addition, subtraction, multiplication, and division. Understanding these helps in solving a wide range of problems. Here are some characteristics of these operations:

• Add and Multiply: Both are commutative, allowing flexible ordering of numbers, i.e., \(a + b = b + a\) and \(a \times b = b \times a\).
• Subtract and Divide: Both are non-commutative. For subtraction, different order leads to different results, like \(7 - 4 eq 4 - 7\). For division, \(a \div b eq b \div a\) in most cases.

Knowing whether an operation is commutative or not helps in predicting outcomes and creating effective problem-solving strategies.