Perform the indicated operations, and write the result in the form $a+b i$. (Simplify your answers completely.)
$(4-3 i)+(5+4 i)$
Answer (2)
Add the real parts: 4 + 5 = 9 .
Add the imaginary parts: − 3 + 4 = 1 .
Combine the results to form the complex number 9 + i .
The simplified expression is 9 + i .
Explanation
Understanding the Problem We are asked to perform the addition of two complex numbers and express the result in the form a + bi . The given expression is ( 4 − 3 i ) + ( 5 + 4 i ) .
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 4, and the real part of the second complex number is 5. The imaginary part of the first complex number is -3, and the imaginary part of the second complex number is 4.
Calculating the Sums Adding the real parts, we have 4 + 5 = 9 . Adding the imaginary parts, we have − 3 + 4 = 1 .
Final Result Therefore, the sum of the two complex numbers is 9 + 1 i , which is usually written as 9 + 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.
To add the complex numbers ( 4 − 3 i ) and ( 5 + 4 i ) , we first add their real parts (4 + 5 = 9) and then their imaginary parts (-3 + 4 = 1). The final result is combined as 9 + 1 i , typically written as 9 + i . Therefore, the result is 9 + i .
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