Legend says René Descartes, lying in bed, watched a fly crawl across the ceiling and realised he could pin down its position with just two numbers. That idea fused algebra and geometry forever.
Cartesian coordinates locate a point on a plane with an ordered pair (x, y): x measures horizontal distance from the origin, y measures vertical distance.
Graphs, maps, screens (every pixel has coordinates), physics, game engines, data plots — the entire visual world of mathematics runs on this system.
The four quadrants
- Quadrant I — x > 0, y > 0 (top-right).
- Quadrant II — x < 0, y > 0 (top-left).
- Quadrant III — x < 0, y < 0 (bottom-left).
- Quadrant IV — x > 0, y < 0 (bottom-right).
- Origin — (0, 0), where the axes cross.
Plot (3, −2): which way from the origin do you go?
How far is (3, 4) from the origin?
Pythagoras: √(3² + 4²) = √25 = 5. Coordinates turn distance into arithmetic.
(x, y) order is fixed: x then y. Swapping them lands you somewhere else entirely. 'Along the corridor, then up the stairs' — horizontal first.
Descartes published this in 1637. Before it, algebra and geometry were separate worlds; after it, an equation became a curve and a curve became an equation — the birth of analytic geometry, and the road to calculus.
- A point = an ordered pair (x, y): horizontal, then vertical.
- Four quadrants by the signs of x and y; origin is (0, 0).
- Turns geometry into algebra — distance, lines, curves all become equations.