Which addition expression has the sum $8-3 i$?
A. $(9+4 i)+(-1-7 i)$
B. $(7+2 i)+(1-i)$
C. $(7+4 i)+(-1-7 i)$
D. $(9+2 i)+(1-i)$
Answer (1)
Add the real and imaginary parts of each complex number.
Expression 1: ( 9 + 4 i ) + ( − 1 − 7 i ) = 8 − 3 i .
Expression 2: ( 7 + 2 i ) + ( 1 − i ) = 8 + i .
Expression 3: ( 7 + 4 i ) + ( − 1 − 7 i ) = 6 − 3 i .
Expression 4: ( 9 + 2 i ) + ( 1 − i ) = 10 + i .
The expression with the sum 8 − 3 i is ( 9 + 4 i ) + ( − 1 − 7 i ) .
Explanation
Problem Analysis We are asked to find which of the given addition expressions results in the complex number 8 − 3 i . We will evaluate each expression and compare the result to 8 − 3 i .
Evaluating Expression 1 Let's evaluate the first expression: ( 9 + 4 i ) + ( − 1 − 7 i ) . We add the real parts and the imaginary parts separately: ( 9 + 4 i ) + ( − 1 − 7 i ) = ( 9 − 1 ) + ( 4 − 7 ) i = 8 − 3 i
Evaluating Expression 2 Now let's evaluate the second expression: ( 7 + 2 i ) + ( 1 − i ) .
( 7 + 2 i ) + ( 1 − i ) = ( 7 + 1 ) + ( 2 − 1 ) i = 8 + i
Evaluating Expression 3 Next, we evaluate the third expression: ( 7 + 4 i ) + ( − 1 − 7 i ) .
( 7 + 4 i ) + ( − 1 − 7 i ) = ( 7 − 1 ) + ( 4 − 7 ) i = 6 − 3 i
Evaluating Expression 4 Finally, we evaluate the fourth expression: ( 9 + 2 i ) + ( 1 − i ) .
( 9 + 2 i ) + ( 1 − i ) = ( 9 + 1 ) + ( 2 − 1 ) i = 10 + i
Finding the Correct Expression Comparing the results, we see that the first expression, ( 9 + 4 i ) + ( − 1 − 7 i ) , equals 8 − 3 i .
Final Answer Therefore, the addition expression that has the sum 8 − 3 i is ( 9 + 4 i ) + ( − 1 − 7 i ) .
Examples
Complex numbers are used in electrical engineering to represent alternating currents. The addition of complex numbers is used to find the total impedance of circuits connected in series. For example, if you have two circuits with impedances 9 + 4 i and − 1 − 7 i , the total impedance is ( 9 + 4 i ) + ( − 1 − 7 i ) = 8 − 3 i . This calculation helps engineers design and analyze electrical circuits effectively.
