Quadrant
A quadrant is one of the four regions of the coordinate plane divided by the x-axis and y-axis, numbered I through IV counterclockwise.
Definition
The coordinate plane is divided into four regions called quadrants, numbered I through IV starting in the upper right and going counterclockwise, defined by the signs of the coordinates: Quadrant I ($x > 0$, $y > 0$), Quadrant II ($x < 0$, $y > 0$), Quadrant III ($x < 0$, $y < 0$), Quadrant IV ($x > 0$, $y < 0$); points on the axes belong to no quadrant. In complex analysis, quadrants of the complex plane correspond to regions where both the real and imaginary parts of $z$ have specified signs, relevant for branch cuts and analytic continuation.
Example
The point $(3, 4)$ is in Quadrant I, and $(-2, 5)$ is in Quadrant II; the point $(-4, -7)$ has both coordinates negative, so it is in Quadrant III, while $(5, -3)$ is in Quadrant IV, and a point on an axis like $(0, 5)$ is in no quadrant. The principal square root function is analytic on $\mathbb{C}$ minus the negative real axis (the branch cut), a cut that separates Q2 from Q3, with the function defined on upper and lower half-planes combining Q1+Q2 and Q3+Q4.
Key Insight
To remember which quadrant is which, start at the upper right (both positive = Quadrant I) and go counterclockwise like counting from I to IV. In trigonometry, the quadrant of an angle determines the signs of sine, cosine, and tangent, with the mnemonic "All Students Take Calculus" (ASTC) helping: All positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4. Sign patterns of quadrants encode parity conditions fundamental to Fourier analysis, and integrals over specific quadrants arise naturally in multivariable calculus, such as the Gaussian integral over the first quadrant equaling $\pi/4$.