Complex numbers and quadrants: In the complex plane, determine the quadrant in which the number 14 − j5 (real part 14, imaginary part −5) lies.

Introduction / Context:
Quadrant identification for complex numbers is a foundational skill in AC circuit analysis and phasor algebra. A complex number a + jb corresponds to a point with horizontal coordinate a (real axis) and vertical coordinate b (imaginary axis). Knowing the sign of each part immediately tells you the quadrant.

Given Data / Assumptions:

• Complex number: 14 − j5.
• Real part Re = 14 (positive).
• Imaginary part Im = −5 (negative).
• Standard quadrant convention: I (+,+), II (−,+), III (−,−), IV (+,−).

Concept / Approach:

Plot the complex number as a point (Re, Im) on the Argand plane. The sign of the real part determines left/right, and the sign of the imaginary part determines up/down. Combine these to pick the correct quadrant quickly without any trigonometry.

Step-by-Step Solution:

Verification / Alternative check:

Compute the angle if desired: θ = arctan(Im/Re) = arctan(−5/14) ≈ −19.65°, which is indeed in quadrant IV (angles between −90° and 0°).

Why Other Options Are Wrong:

First quadrant requires Im > 0. Second quadrant requires Re < 0 and Im > 0. Third quadrant requires both Re < 0 and Im < 0—none match our signs.

Common Pitfalls:

Swapping the role of real and imaginary axes; forgetting that a negative imaginary part places the point below the real axis.

Final Answer:

fourth quadrant