- Addition of complex numbers is associative: \((x + y) + z = x + (y + z)\) (True).
- Multiplication of complex numbers is associative: \((x \times y) \times z = x \times (y \times z)\) (True).
- Addition of complex numbers is commutative: \(x + y = y + x\) (True).
- Subtraction is not associative or commutative, making statements 2 and 4 false.
- Final correct answers: \(\boxed{1, 3, 5}\).
Explanation
- Analyzing the Properties of Complex Numbers
We are given three complex numbers: \(x = a + bi\), \(y = c + di\), and \(z = f + gi\). We need to determine which of the five given statements are true based on the fundamental algebraic properties of complex numbers. Complex numbers follow the same structural rules as real numbers under addition and multiplication, forming a field. Specifically, both operations are associative and commutative. Subtraction is not associative or commutative in general.
- Evaluating the Statements
- Statement 1: \((x + y) + z = x + (y + z)\)
This is the associative property of addition for complex numbers. Addition in complex numbers is associative, meaning the grouping of terms does not affect the sum. This statement is true.
- Statement 2: \((x - y) - z = x - (y - z)\)
Let us expand both sides. On the left, we have \((x - y) - z = x - y - z\). On the right, we have \(x - (y - z) = x - y + z\). Since \(x - y - z\) is not equal to \(x - y + z\) in general, this statement is false.
- Statement 3: \((x \times y) \times z = x \times (y \times z)\)
This is the associative property of multiplication for complex numbers. Similar to addition, the multiplication of complex numbers is associative, meaning the result remains the same regardless of how the factors are grouped. This statement is true.
- Statement 4: \(x - y = y - x\)
This claims the commutative property for subtraction. Subtraction is not commutative; for example, if \(x=2\) and \(y=1\), \(x - y = 1\) but \(y - x = -1\). This statement is false.
- Statement 5: \(x + y = y + x\)
This is the commutative property of addition for complex numbers. Since the addition of complex numbers is defined component-wise (adding the real parts and the imaginary parts separately), the order of addition does not matter. This statement is true.
- Final Conclusion
Based on our evaluation, the correct statements are the associative property of addition (1), the associative property of multiplication (3), and the commutative property of addition (5). Therefore, the correct answers are 1, 3, and 5.
Examples
In computer graphics and game development, complex numbers (or their generalization, quaternions) are frequently used to handle rotations in 3D space. Understanding whether an operation like multiplication or addition is associative is critical when chaining multiple transformations, such as rotating an object and then moving it, or combining two consecutive rotations. Knowing that multiplication is associative allows developers to multiply matrices or complex representations in any order of groups, ensuring that the final orientation of an object is consistent.