Quadrant
Geometry & MeasurementA 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 sections called quadrants. They are numbered I, II, III, and IV, starting in the upper right and going counterclockwise.
Example
Quadrant I is upper right $(+, +)$. Quadrant II is upper left $(-, +)$. Quadrant III is lower left $(-, -)$. Quadrant IV is lower right $(+, -)$. The point $(3, 4)$ is in Quadrant I; the point $(-2, 5)$ is in Quadrant II.
Key Insight
To remember which quadrant is which: start at the upper right (both positive = Quadrant I) and go counterclockwise like a count from I to IV.
Definition
The four quadrants are the four regions of the Cartesian plane 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.
Example
The point $(-4, -7)$ has both coordinates negative, so it is in Quadrant III. The point $(5, -3)$ has positive $x$ and negative $y$, so it is in Quadrant IV. Points on an axis (like $(0, 5)$) are not in any quadrant.
Key Insight
In trigonometry, the quadrant of an angle determines the signs of sine, cosine, and tangent. The mnemonic "All Students Take Calculus" (ASTC) helps: All positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4.
Definition
The four quadrants partition $\mathbb{R}^2$ into four open sets based on sign patterns of coordinates. 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 principal square root function is analytic on $\mathbb{C}$ minus the negative real axis (the branch cut). The branch cut separates Q2 from Q3, and the function is defined on the upper and lower half-planes, which combine Q1+Q2 and Q3+Q4.
Key Insight
In multivariable calculus, integrals over specific quadrants arise naturally, for example the Gaussian integral over the first quadrant equals $\pi/4$. The sign patterns of quadrants encode parity conditions fundamental to Fourier analysis.