Subtract $(3+2 i)$ from $(-9-8 i)$.
Answer (2)
Distribute the negative sign: ( − 9 − 8 i ) − ( 3 + 2 i ) = − 9 − 8 i − 3 − 2 i .
Combine the real parts: − 9 − 3 = − 12 .
Combine the imaginary parts: − 8 i − 2 i = − 10 i .
The result is − 12 − 10 i .
Explanation
Understanding the problem We are asked to subtract the complex number 3 + 2 i from the complex number − 9 − 8 i . This means we need to compute ( − 9 − 8 i ) − ( 3 + 2 i ) .
Distributing the negative sign To subtract complex numbers, we subtract the real parts and the imaginary parts separately. First, distribute the negative sign: ( − 9 − 8 i ) − ( 3 + 2 i ) = − 9 − 8 i − 3 − 2 i
Combining real parts Now, combine the real parts: − 9 − 3 = − 12
Combining imaginary parts Next, combine the imaginary parts: − 8 i − 2 i = − 10 i
Final result Finally, write the result as a complex number: − 12 − 10 i So, subtracting ( 3 + 2 i ) from ( − 9 − 8 i ) gives us − 12 − 10 i .
Examples
Complex numbers are used in electrical engineering to represent alternating current (AC) circuits. The voltage and current in an AC circuit can be represented as complex numbers, and the impedance of the circuit, which is the opposition to the flow of current, is also a complex number. By using complex numbers, engineers can analyze and design AC circuits more easily. For example, they can calculate the total impedance of a circuit by adding the individual impedances of the components, which is a complex number addition. Similarly, subtracting complex numbers can help determine the difference in voltage or current between two points in a circuit.
To subtract ( 3 + 2 i ) from ( − 9 − 8 i ) , we rewrite it as ( − 9 − 8 i ) − ( 3 + 2 i ) and distribute the negative sign. By combining the real parts and the imaginary parts, we find the result is − 12 − 10 i . The final answer is oxed{-12 - 10i}$.
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