In which quadrant is the number [tex]$6-8 i$[/tex] located on the complex plane?
A. I
B. III
C. II
D. IV
Answer (2)
Identify the real and imaginary parts of the complex number: a = 6 and b = − 8 .
Determine the signs of the real and imaginary parts: 0"> a > 0 and b < 0 .
Recall the quadrant definitions based on the signs of the real and imaginary parts.
Conclude that the complex number 6 − 8 i is in Quadrant IV, since 0"> a > 0 and b < 0 . I V
Explanation
Understanding the Complex Plane We are given the complex number 6 − 8 i and asked to determine which quadrant it lies in on the complex plane. The complex plane has a real axis (x-axis) and an imaginary axis (y-axis). The quadrants are numbered I, II, III, and IV in a counterclockwise direction, starting from the upper right.
Identifying Real and Imaginary Parts The complex number is in the form a + bi , where a represents the real part and b represents the imaginary part. In our case, a = 6 and b = − 8 .
Determining the Signs Now, let's determine the signs of a and b . We have 0"> a = 6 > 0 and b = − 8 < 0 .
Identifying the Quadrant The quadrants are defined as follows:
Quadrant I: 0"> a > 0 , 0"> b > 0
Quadrant II: a < 0 , 0"> b > 0
Quadrant III: a < 0 , b < 0
Quadrant IV: 0"> a > 0 , b < 0 Since 0"> a > 0 and b < 0 , the complex number 6 − 8 i is located in Quadrant IV.
Final Answer Therefore, the complex number 6 − 8 i is located in Quadrant IV.
Examples
Complex numbers and their representation on the complex plane are used in electrical engineering to analyze AC circuits. The impedance of a circuit, which includes resistance and reactance, can be represented as a complex number. The quadrant in which the impedance lies helps engineers understand the nature of the circuit's reactance (inductive or capacitive) and its effect on the circuit's behavior. This understanding is crucial for designing and optimizing electrical circuits.
The complex number 6 − 8 i is located in Quadrant IV of the complex plane since its real part is positive and its imaginary part is negative.
;
