Simplify. Write your response in $a+b i$ form.
$(16-8 i)+(8-3 i) =$
Answer (1)
Add the real parts: 16 + 8 = 24 .
Add the imaginary parts: − 8 i + ( − 3 i ) = − 11 i .
Combine the real and imaginary parts to form the complex number.
The simplified expression is 24 − 11 i .
Explanation
Understanding the problem We are asked to simplify the expression ( 16 − 8 i ) + ( 8 − 3 i ) and write the result in the form a + bi , where a and b are real numbers. This involves adding two complex numbers.
Adding real and imaginary parts To add complex numbers, we add their real parts and their imaginary parts separately. The real part of the first complex number is 16, and the real part of the second complex number is 8. The imaginary part of the first complex number is -8, and the imaginary part of the second complex number is -3.
Performing the addition Adding the real parts, we have 16 + 8 = 24 . Adding the imaginary parts, we have − 8 + ( − 3 ) = − 11 .
Final result Therefore, the simplified expression is 24 − 11 i .
Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. The voltage and current in an AC circuit can be represented as complex numbers, and the impedance of circuit elements (resistors, capacitors, and inductors) can also be represented as complex numbers. By using complex numbers, engineers can easily calculate the behavior of AC circuits.
