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 final answer is 9 + i .
Explanation
Understanding the Problem We are asked to add two complex numbers, ( 4 − 3 i ) and ( 5 + 4 i ) , and express the result in the form a + bi , where a and b are real numbers.
Adding Complex Numbers To add complex numbers, we add their real parts and their imaginary parts separately.
Adding Real Parts The real part of the sum is the sum of the real parts: 4 + 5 = 9 .
Adding Imaginary Parts The imaginary part of the sum is the sum of the imaginary parts: − 3 + 4 = 1 .
Final Result Combining the real and imaginary parts, we get the complex number 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 a circuit can be represented as complex numbers, and the impedance of the circuit elements (resistors, capacitors, and inductors) can also be represented as complex numbers. By adding and multiplying these complex numbers, engineers can calculate the behavior of the circuit. For example, the total impedance of a series circuit is the sum of the individual impedances, just like adding complex numbers.
To add the complex numbers ( 4 − 3 i ) and ( 5 + 4 i ) , we first add the real parts (4 + 5 = 9) and then the imaginary parts (-3 + 4 = 1). Thus, the resulting complex number is 9 + i .
;
