Why coordinates?
Suppose you want to tell a friend exactly where a dot is on a sheet of paper. "Near the top" is vague. "5 cm from the left edge and 3 cm from the bottom edge" is exact. Two numbers, measured from two fixed lines, pin down a position precisely.
The French mathematician René Descartes developed this idea in the 17th century. In his honour it is called the Cartesian system.
The Cartesian plane
Draw two number lines that cross at right angles:
the horizontal one is the x-axis,
the vertical one is the y-axis,
they meet at the origin O, where both values are 0.
On the x-axis, numbers to the right of O are positive and those to the left are negative. On the y-axis, numbers above O are positive and those below are negative.
Coordinates of a point
Every point P in the plane is described by an ordered pair (x, y):
x, the abscissa or x-coordinate, is the point's perpendicular distance from the y-axis, with a sign.
y, the ordinate or y-coordinate, is the point's perpendicular distance from the x-axis, with a sign.
The order matters: (3, 5) and (5, 3) are different points.
Points on the axes
Any point on the x-axis has y = 0, so it looks like (a, 0).
Any point on the y-axis has x = 0, so it looks like (0, b).
The origin is (0, 0).
The four quadrants
The axes divide the plane into four regions called quadrants, numbered anticlockwise starting from the top right.
Quadrant | Position | Signs of (x, y) | Example |
|---|---|---|---|
I | top right | (+, +) | (4, 2) |
II | top left | (−, +) | (−3, 5) |
III | bottom left | (−, −) | (−2, −6) |
IV | bottom right | (+, −) | (5, −1) |
Points that lie on an axis do not belong to any quadrant.
Plotting a point
To plot (−3, 4):
Start at the origin.
Move 3 units to the left along the x-axis (x is negative).
From there, move 4 units up, parallel to the y-axis (y is positive).
Mark the point. It lies in quadrant II.
Worked examples
Example 1. In which quadrant or on which axis does each point lie?
(−2, −7): both coordinates negative, so quadrant III.
(0, −4): x = 0, so it is on the y-axis.
(6, 0): y = 0, so it is on the x-axis.
(3, −3): x positive and y negative, so quadrant IV.
Example 2. A point is 5 units from the x-axis and 2 units from the y-axis, and lies in quadrant II. What are its coordinates?
In quadrant II, x is negative and y is positive. Its distance from the y-axis is |x| = 2, and its distance from the x-axis is |y| = 5. So the point is (−2, 5).
Example 3. Plot A(1, 1), B(4, 1), C(4, 3) and D(1, 3), and join them in order. What shape do you get?
AB and DC are horizontal and 3 units long. BC and AD are vertical and 2 units long. All the angles are right angles, so ABCD is a rectangle with area 3 × 2 = 6 square units.
Common mistakes
Mixing up the order: x always comes first.
Thinking the abscissa is the distance from the x-axis. It is the distance from the y-axis.
Numbering the quadrants clockwise. They go anticlockwise.