What is the additive inverse of the complex number $-8+3 i$ ?
The problem asks for the additive inverse of the complex number \(z = -8 + 3i\). By definition, the additive inverse of a complex number \(z\) is another complex number \(w\) such that their sum equals zero: \(z + w = 0\). If we represent \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers, the additive inverse \(w\) is given by \(-z\). Specifically, \(-z = -(a + bi) = -a - bi\). In this case, \(a = -8\) and \(b = 3\).
We apply the definition to our specific complex number \(z = -8 + 3i\). The additive inverse \(-z\) is calculated by multiplying the entire expression by \(-1\):
\[-z = -(-8 + 3i)\]
Distributing the negative sign to both the real part and the imaginary part, we get:
\[-z = -(-8) - (3i) = 8 - 3i\]
Thus, the additive inverse is \(8 - 3i\).
Comparing our calculated result \(8 - 3i\) with the provided options:
We see that our result matches option d. Therefore, the additive inverse of \(-8 + 3i\) is \(\boxed{8 - 3i}\), which corresponds to option d.
The concept of an additive inverse is fundamental in algebra, acting like the "opposite" of a number. Think of it like balancing a scale: if you place a weight of \(-8\) units in one direction and a force of \(3i\) units in another on a complex plane, the additive inverse represents the exact force needed in the opposite direction to return the system to an equilibrium of zero. This is used in electrical engineering when analyzing AC circuits using phasors, where canceling out specific signal components is necessary to filter noise or achieve a desired wave output.