Subtract $(3+2 i)$ from $(-9-8 i)$.
We are tasked with subtracting the complex number \(z_2 = 3 + 2i\) from \(z_1 = -9 - 8i\). A complex number is typically written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Complex number subtraction is performed by subtracting the real parts from each other and the imaginary parts from each other. That is, if we have two complex numbers \(z_1 = a + bi\) and \(z_2 = c + di\), their difference is defined as \(z_1 - z_2 = (a - c) + (b - d)i\).
We identify the real and imaginary parts of the given complex numbers:
Now we apply the subtraction rule:
\[z_1 - z_2 = (-9 - 8i) - (3 + 2i)\]
Distributing the negative sign:
\[z_1 - z_2 = -9 - 8i - 3 - 2i\]
Grouping the real and imaginary parts:
\[z_1 - z_2 = (-9 - 3) + (-8i - 2i)\]
Calculating the sums:
\[-9 - 3 = -12\]
\[-8i - 2i = -10i\]
Thus, the result is \(-12 - 10i\). Comparing this to the given options:
The calculated result matches option d.
Complex number subtraction is essential in fields like electrical engineering, specifically in AC circuit analysis. When you need to calculate the difference in impedance or voltage between two points in a circuit, you often work with complex numbers. For instance, if you have a voltage drop represented by \(-9 - 8i\) Volts and you need to subtract another component's contribution of \(3 + 2i\) Volts, you use the subtraction method shown here to find the total net voltage difference, which helps in understanding how electricity flows through different elements in the circuit.