- Identify the real part \(a = -2\) and the imaginary part \(b = 8\) of the given complex number.
- Apply the additive inverse rule: \(-(a + bi) = -a - bi\).
- Substitute the values: \(-(-2) - 8i = 2 - 8i\).
- The additive inverse of \(-2+8i\) is \(\boxed{2-8i}\).
Explanation
- Analysis of the Additive Inverse Definition
To find the additive inverse of a complex number, we use the definition that for any complex number \(z = a + bi\), the additive inverse, denoted by \(-z\), is \(-a - bi\). This is because the sum of a complex number and its additive inverse must equal zero: \((a + bi) + (-a - bi) = (a - a) + (b - b)i = 0 + 0i = 0\). We are given the complex number \(-2 + 8i\), where the real part \(a = -2\) and the imaginary part \(b = 8\).
- Calculating the Additive Inverse
Using the formula for the additive inverse, we negate both the real part and the imaginary part of the given complex number. The additive inverse of \(-2 + 8i\) is calculated by taking the negative of the real part \(-(-2)\) and the negative of the imaginary part \(-(8i)\). This yields \(2 - 8i\). To verify, we can add the original number and the result: \((-2 + 8i) + (2 - 8i) = (-2 + 2) + (8 - 8)i = 0 + 0i = 0\). This confirms that \(2 - 8i\) is indeed the additive inverse of \(-2 + 8i\).
- Conclusion and Final Answer selection
Comparing the result \(2 - 8i\) with the provided options:
A\(-2 + 8i\)
B\(2 - 8i\)
C\(-2 - 8i\)
D\(2 + 8i\)
The correct option is b. Thus, the additive inverse of \(-2 + 8i\) is \(\boxed{2-8i}\).
Examples
Additive inverses are fundamental in algebra and complex analysis for solving equations. For example, if you are modeling a circuit with alternating currents represented by complex numbers, knowing the additive inverse allows you to calculate the balance point where the total current or voltage becomes zero. This helps in understanding how waves of energy can cancel each other out in various physics and engineering applications.