- Analyze the given equation \(x \cdot y = y \cdot x\) for complex numbers.
- Define and compare the equation with the standard definitions of mathematical properties (identity, associative, commutative, distributive).
- Identify that swapping the order of factors while maintaining the product is the definition of the commutative property.
- Conclude that the equation represents the commutative property.
- Final Answer: \(\boxed{c}\)
Explanation
- Analysis of the Problem
We are given two complex numbers, \(x = a + bi\) and \(y = c + di\), and the equation \(x \cdot y = y \cdot x\). We need to determine which property of multiplication this equation represents among the given options: identity property, associative property, commutative property, and distributive property.
- Defining the Properties
Let's define each of the properties listed:
- Identity Property: For multiplication, this property states that there is an element, 1, such that any number multiplied by 1 remains unchanged: \(x \cdot 1 = x\).
- Associative Property: This property concerns the grouping of elements in an operation. For three complex numbers \(x, y, z\), it states: \((x \cdot y) \cdot z = x \cdot (y \cdot z)\).
- Commutative Property: This property states that the order in which two numbers are multiplied does not change the product: \(x \cdot y = y \cdot x\).
- Distributive Property: This property concerns the interaction between multiplication and addition: \(x \cdot (y + z) = x \cdot y + x \cdot z\).
- Identification of the Property
The equation provided in the problem is \(x \cdot y = y \cdot x\). By comparing this to the definitions above, we can see that the order of the operands \(x\) and \(y\) is swapped without changing the result of the multiplication. This is the definition of the commutative property. Thus, the equation \(x \cdot y = y \cdot x\) represents the commutative property of multiplication for complex numbers.
- Conclusion
Based on the analysis, the equation \(x \cdot y = y \cdot x\) for complex numbers represents the commutative property. The correct option is c. Therefore, the property is the commutative property. \(\boxed{c}\)
Examples
The commutative property is fundamental in everyday mathematics, such as when calculating the total cost of multiple items. For example, if you buy 3 apples for \(\$2\) each, the total is \(3 \times 2 = 6\). If you buy 2 bags that each contain 3 apples, you also have \(2 \times 3 = 6\) apples. Because of the commutative property, you know that the order in which you multiply the number of items by their unit cost doesn't change the total amount you need to pay. Understanding this property helps simplify calculations in shopping, budgeting, and resource management.