In which quadrant is the number $-14-5 i$ located on the complex plane?
A. I
B. II
C. III
D. IV
Answer (2)
Identify the real and imaginary parts of the complex number: Real part is − 14 , imaginary part is − 5 .
Determine the signs of the real and imaginary parts: Both are negative.
Recall the quadrant definitions: Quadrant III has both real and imaginary parts negative.
Conclude that the complex number is in Quadrant III: III .
Explanation
Problem Analysis We are given the complex number − 14 − 5 i and asked to determine which quadrant it lies in on the complex plane.
Understanding Quadrants The complex plane is divided into four quadrants. The real part of the complex number corresponds to the x-coordinate, and the imaginary part corresponds to the y-coordinate. The quadrants are defined as follows:
Quadrant I: Real part > 0, Imaginary part > 0
Quadrant II: Real part < 0, Imaginary part > 0
Quadrant III: Real part < 0, Imaginary part < 0
Quadrant IV: Real part > 0, Imaginary part < 0
Identifying Real and Imaginary Parts For the complex number − 14 − 5 i , the real part is − 14 and the imaginary part is − 5 .
Determining the Quadrant Since the real part ( − 14 ) is negative and the imaginary part ( − 5 ) is negative, the complex number lies in Quadrant III.
Final Answer Therefore, the complex number − 14 − 5 i is located in Quadrant III.
Examples
Complex numbers and their representation on the complex plane are used in electrical engineering to analyze AC circuits. For example, impedance, which is the opposition to the flow of current in an AC circuit, can be represented as a complex number. The real part of the impedance represents the resistance, and the imaginary part represents the reactance. By plotting the impedance on the complex plane, engineers can visualize the behavior of the circuit and design it to meet specific requirements. This is crucial for designing efficient and stable power systems.
The complex number − 14 − 5 i is in Quadrant III because both its real part (-14) and imaginary part (-5) are negative.
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