(x,y) · The Coordinate Plane
Two numbers pin any spot on the plane — coordinates.
To tell a friend your seat at the movies — how many numbers do you need?
Distinguish the order of coordinates.
The bridge Descartes built between algebra and geometry.
Open the immersive version →A reading path containing this lesson: Work with several numbers at once · 1/3
Why does this question matter?
'Third from the window, second from the front' — describing position in words is long and vague. We needed an agreement for pinning position with numbers.
The fix: draw two reference lines (axes) and record how far across and how far up as an ordered pair (x, y). Legend credits Descartes, watching a fly on his ceiling from bed.
The invention's real power is the marriage of shapes and formulas. Points become numbers, so a line becomes an equation (y=2x+1) — geometry and algebra became one body.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — to tell a friend your seat at the movies, how many numbers do you need?
Find the seat and sit down
round 1/3 · Your friend texts: "I'm at (across 3, up 2)!" — tap that seat.
Read the idea from another angle
A coordinate is an address for position.
Issue the nameless points an address like (3, 2) and 'where?' gets a numeric answer. Latitude/longitude, screen pixels — all descendants of the same invention.
Geometry and algebra meet on Descartes' bridge.
Before coordinates, shape-math and number-math were separate continents. Once the bridge was built, the circle became x²+y²=r² — and every graph and all of calculus happens on this bridge.
Now read the definitions with their conditions
What the seat hunt confirmed, in mathematical language.
The ordered pair
First number horizontal, second vertical — fixed roles, so (2,3) ≠ (3,2).
Origin and quadrants
Where the axes cross. The axes cut the plane into four quadrants, with negatives to the left and below.
Distance between points (preview)
Once position is numeric, distance is computable — Pythagoras reborn as the distance formula.
Make one more distinction
(2, 3) and (3, 2) are the same point.
Different points. The first number is horizontal (x), the second vertical (y) — roles are fixed, hence 'ordered pair'. Just as row 3 seat 2 differs from row 2 seat 3.
Coordinates start at 1, like seat numbers.
The reference is the origin (0,0), and left/below are negative. A seating chart uses only one corner (the first quadrant) — negatives make the whole plane number territory.
Where else can this idea help?
Maps and GPS
Latitude and longitude are Earth-surface coordinates. One ordered pair — 37.5°N, 127°E — pins Seoul.
Screen pixels
Every dot on this screen has an (x, y) pixel coordinate. Game movement and mouse clicks are coordinate arithmetic.
Chess and Go
Chess's e4 and Go's 16-4 — board positions written as letter-number pairs are coordinates' old cousins.
Data visualization
Each dot of a scatter plot is an ordered pair like (study hours, score). Correlation analysis lives on the coordinate plane.
Explain it in your own words
Try another question
Q1Which quadrant holds the point (−2, 3)?Show answer ▾
The second. x negative (left), y positive (up) — the upper-left region. Quadrants run counterclockwise from the upper right: 1→2→3→4.
Q2How far is the point (3, 4) from the x-axis?Show answer ▾
4 — the distance to the x-axis is |y|, how far up you are. Distance to the y-axis is |x| = 3.
Q3Distance from (1, 1) to (4, 5)? (Try the distance formula.)Show answer ▾
√((4−1)² + (5−1)²) = √(9+16) = √25 = 5. The hypotenuse of a 3-by-4 right triangle — Pythagoras.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Work with several numbers at once · 1/3
Previous concept
Negative Numbers
Only after the number line extends below zero can numbers cover the whole plane.
← Negative Numbers labLeads to next
Linear Functions
Plot a rule (y=2x+1) on the plane and the points form a line — the moment formulas become pictures.
Go to the Linear Functions lab →Related lab
Negative Numbers
Writing left of and below zero takes negatives — the raw material of coordinates.
Go to the Negative Numbers lab →Related lab
Pythagorean Theorem
The distance formula on the plane is this theorem's modern outfit.
Go to the Pythagorean Theorem lab →