Simplify. Write your response in $a+b i$ form.
$(-17+19 i)-(11+2 i)=$
Answer (2)
Distribute the negative sign: ( − 17 + 19 i ) − ( 11 + 2 i ) = − 17 + 19 i − 11 − 2 i .
Combine the real parts: − 17 − 11 = − 28 .
Combine the imaginary parts: 19 i − 2 i = 17 i .
Write the result in a + bi form: − 28 + 17 i .
Explanation
Understanding the problem We are asked to simplify the expression ( − 17 + 19 i ) − ( 11 + 2 i ) and write the result in the form a + bi , where a and b are real numbers.
Distributing the negative sign First, distribute the negative sign to the second complex number: ( − 17 + 19 i ) − ( 11 + 2 i ) = − 17 + 19 i − 11 − 2 i
Combining real parts Next, combine the real parts of the complex numbers: − 17 − 11 = − 28
Combining imaginary parts Then, combine the imaginary parts of the complex numbers: 19 i − 2 i = 17 i
Final answer Finally, write the result in the form a + bi : − 28 + 17 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 complex impedances of the individual components.
The expression ( − 17 + 19 i ) − ( 11 + 2 i ) simplifies to − 28 + 17 i by distributing the negative sign, combining real and imaginary parts. The final result is presented in the form a + bi .
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