What is the additive inverse of the complex number -8 +3i?
A. 8+3i
B. -8-3i
C. -8+3i
D. 8-3i
Answer (2)
The problem asks for the additive inverse of the complex number − 8 + 3 i .
The additive inverse of a complex number a + bi is − a − bi .
Applying this to − 8 + 3 i , the additive inverse is − ( − 8 ) − 3 i .
Simplifying, the additive inverse is 8 − 3 i .
Explanation
Understanding the Problem We are given the complex number z = − 8 + 3 i and we want to find its additive inverse. The additive inverse of a complex number a + bi is simply − a − bi , which when added to the original number, results in zero.
Finding the Additive Inverse To find the additive inverse of z = − 8 + 3 i , we need to find a complex number z ′ such that z + z ′ = 0 . If z = a + bi , then z ′ = − a − bi . In our case, a = − 8 and b = 3 .
Calculating the Inverse So, the additive inverse of − 8 + 3 i is − ( − 8 ) − 3 i = 8 − 3 i .
Final Answer Therefore, the additive inverse of the complex number − 8 + 3 i is 8 − 3 i .
Examples
Complex numbers are used in electrical engineering to represent alternating current (AC) circuits. The additive inverse is useful when calculating the net impedance of a circuit. For example, if one component has an impedance of − 8 + 3 i ohms, another component with an impedance of 8 − 3 i ohms would cancel out the impedance of the first component, resulting in a net impedance of 0. This concept is crucial for designing circuits that operate efficiently and predictably.
The additive inverse of the complex number -8 + 3i is 8 - 3i. This is calculated by negating both the real and imaginary parts of the complex number. Therefore, the correct answer is option D: 8 - 3i.
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