A map for numbers
The Cartesian plane is a grid made by two perpendicular number lines:
- The x-axis (horizontal)
- The y-axis (vertical)
They cross at the origin, (0, 0). It’s named after René Descartes, the 17th-century French mathematician who linked algebra and geometry.
Ordered pairs
Every point has an ordered pair (x, y):
- x: how far left (−) or right (+) from the origin
- y: how far down (−) or up (+)
The four quadrants
The axes divide the plane into four quadrants, numbered counterclockwise starting at the top right:
| Quadrant | x | y | Example |
|---|---|---|---|
| I | + | + | (3, 2) |
| II | − | + | (−3, 2) |
| III | − | − | (−3, −2) |
| IV | + | − | (3, −2) |
Points on an axis aren’t in any quadrant: (0, 5) is on the y-axis; (−4, 0) is on the x-axis.
Plotting points with negatives
To plot (−4, 3):
- Start at the origin.
- Move 4 left (x is negative).
- Move 3 up (y is positive).
Distances along a line
When two points share an x- or y-coordinate, find the distance by subtracting:
- (2, 5) and (2, −3): same x, distance = 5 − (−3) = 8 units
- (−6, 1) and (4, 1): same y, distance = 4 − (−6) = 10 units
Shapes on the plane
Example: Plot A(−2, 1), B(3, 1), C(3, 4), D(−2, 4). Join them.
It’s a rectangle 5 units wide and 3 units tall: area 15 square units.
Why it matters
The Cartesian plane lets us draw equations as graphs, describe transformations with rules, and calculate distances. GPS, video games and computer graphics all use coordinate systems.
Worked examples
Example 1: Which quadrant is (−5, −1) in?
III.
Example 2: What are the coordinates of a point 6 left and 2 down from the origin?
(−6, −2).
Example 3: Distance between (−1, −4) and (−1, 3)?
3 − (−4) = 7 units.
Common mistakes
- Going up/down first instead of across first.
- Numbering quadrants clockwise.
- Ignoring negative signs when finding distances.
Try it yourself
- In which quadrant is (7, −2)?
- Name a point on the negative x-axis.
- Distance between (3, 4) and (−5, 4)?
- Three corners of a square are (0, 0), (4, 0), (4, 4). The fourth?
Answers: 1) IV 2) e.g. (−3, 0) 3) 8 units 4) (0, 4)
Quick recap
- The Cartesian plane has an x-axis, a y-axis and an origin.
- Four quadrants are numbered counterclockwise from the top right.
- Each point is an ordered pair (x, y).