Cartesian Coordinates and the Plane

One of the great unifying ideas in mathematics. Before Descartes (1637), geometry and algebra lived in separate worlds. He had the insight to label every point in the plane with a pair of numbers, turning geometric questions into algebraic ones and vice versa. Suddenly the locus "all points equidistant from the origin" became the equation $x^2 + y^2 = r^2$, and computation could replace clever construction.

The Cartesian setup

Draw two perpendicular number lines that cross at the origin $O = (0, 0)$. The horizontal one is the $x$-axis, the vertical one is the $y$-axis. Together they form the Cartesian coordinate system, named for René Descartes.

Every point $P$ in the plane gets two numbers: its $x$-coordinate (signed horizontal distance from the $y$-axis) and its $y$-coordinate (signed vertical distance from the $x$-axis). We write $P = (x, y)$. Order matters: $(2, 3)$ and $(3, 2)$ are different points.

• Cartography: Latitude and longitude form a curved-surface coordinate system; flat-map projections (Mercator, Robinson, Peters) are all attempts to lay this onto a Cartesian plane, each making different trade-offs.
• Spreadsheets: An Excel cell address like D7 is a 2D coordinate (column letter, row number); every formula that references another cell is doing the same plane-geometry indexing as plotting $(x, y)$.
• Computer Vision: Pixel coordinates $(i, j)$ in an image are a 2D coordinate system, usually with the origin at the top-left and $j$ growing downward, a sign convention that has tripped up generations of programmers.

Drag $P$ and $Q$ around in the widget above and watch their coordinates update. Place $P$ at $(3, 0)$: it lies on the $x$-axis, three units to the right.

The four quadrants

The axes carve the plane into four quadrants, numbered counter-clockwise starting from the upper right:

• Quadrant I: $x > 0, y > 0$, upper right.
• Quadrant II: $x < 0, y > 0$, upper left.
• Quadrant III: $x < 0, y < 0$, lower left.
• Quadrant IV: $x > 0, y < 0$, lower right.

Points lying exactly on either axis do not belong to a quadrant. A point on the $x$-axis has form $(a, 0)$; a point on the $y$-axis has form $(0, b)$.

The bridge between geometry and algebra

The Cartesian system sets up a one-to-one correspondence between points in the plane and ordered pairs of real numbers. Two points sharing the same $x$-coordinate lie on the same vertical line, two points sharing the same $y$-coordinate lie on the same horizontal line, and the straight-line distance becomes the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ that we develop in the next section.

Try it

• Drag $P = ($, $)$. Predict first: $(-3, 5)$, which quadrant should that be? Drag the point and verify (answer: Quadrant II, with negative $x$ and positive $y$).
• Drag $P = ($, $)$. Before dragging: with $x = 0$, which axis does the point sit on? Drag and verify it lands on the $y$-axis.
• Place $P = ($, $)$ and $Q = ($, $)$. Predict first: how are $(4, -3)$ and $(-4, 3)$ related geometrically? Place both points and verify they are reflections of each other through the origin.

A trap to watch for

The convention that the first coordinate is horizontal ($x$) and the second is vertical ($y$) is universal in mathematics, but beginners regularly transpose them. Triple-check: $(2, 5)$ means "go 2 to the right, then 5 up." A related trap: some screen-coordinate systems in computer graphics put $y$ pointing downward, the opposite of the mathematical convention.

What you now know

You can place any point in the plane using its ordered pair, identify which quadrant (or axis) it lies in, and switch between geometric descriptions and algebraic coordinates. The next section derives the distance formula.

Quick check

Mark section complete →

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 8, §1.
• Descartes, R. (1637). La Géométrie (Smith and Latham, Dover, 1954). The original publication of analytic geometry.
• Boyer, C. B. (2004). History of Analytic Geometry. Dover. Detailed history of how coordinates came into mathematics.
• Stewart, J. (2015). Calculus, 8th ed. Cengage. Appendix B reviews the coordinate plane and its conventions.