- The problem asks to identify the property that allows the transformation \((5-2i) - (1+8i) = 5-2i-1-8i\).
- We identify the negative sign as a factor of \(-1\) multiplying the complex number \((1+8i)\).
- We apply the distributive property, \(-1(1+8i) = -1(1) + -1(8i) = -1 - 8i\).
- This calculation concludes that the distributive property is the correct choice, and the final result of the subtraction is \(\boxed{4-10i}\).
Explanation
- Analyzing the Expression
To subtract two complex numbers, we follow the general algebraic rule for subtraction: \((a + bi) - (c + di) = a + bi - c - di\). The expression given is \((5-2i) - (1+8i)\). To simplify this, we treat the negative sign as a multiplication by \(-1\), which is applied to the entire complex number inside the parentheses, \((1+8i)\).
- Identifying the Property
The expression can be written as \((5-2i) + (-1) \cdot (1+8i)\). According to the distributive property, multiplication is distributive over addition, which means that for any real or complex numbers \(a, b,\) and \(c\), we have \(a(b+c) = ab + ac\). Applying this to our expression with \(a = -1\), \(b = 1\), and \(c = 8i\), we get:
\[(-1) \cdot (1 + 8i) = (-1 \cdot 1) + (-1 \cdot 8i) = -1 - 8i\]
This is exactly the property that allows us to remove the parentheses while distributing the negative sign to each term inside.
- Final Calculation and Conclusion
Now that we have successfully distributed the negative sign, the full expression becomes:
\[5 - 2i - 1 - 8i\]
Next, we group the real parts and the imaginary parts:
\[(5 - 1) + (-2i - 8i) = 4 - 10i\]
The transformation is justified specifically by the distributive property, which corresponds to option (d).
Examples
In engineering, complex numbers are frequently used to model alternating current circuits. When analyzing the total impedance of a circuit, you might need to subtract the impedance of one component from another, represented as \(Z_1 - Z_2\). Understanding how to correctly distribute the negative sign ensures that you accurately combine the resistive (real) and reactive (imaginary) parts, allowing you to correctly design circuits and predict their behavior under different frequencies.