Course 6.1Lesson 1 of 3

Fixing a position with two numbers

One number fixes a position on a line. Two perpendicular number lines, the x-axis and the y-axis, fix a position on a flat surface. They meet at the origin, and together they are the Cartesian plane.

about 25 min1 simulation2 practice questions1 Python program4 questions to check yourself

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Try the code and the simulations beside the lesson.

One number, or two?

How do you tell a friend where a dot is on a straight line, such as the edge of a ruler? One number is enough: "the dot is at $3\ \text{cm}$ ". We measure from a fixed starting point, the zero.

Now think of a dot on a sheet of paper. One number is not enough. "The dot is $3\ \text{cm}$ from the left edge" is true for many dots, all along a vertical line. We need a second number, for example "and $2\ \text{cm}$ from the bottom edge". Now only one dot fits.

You already use this idea:

A seat in a cinema is "row $4$ , seat $7$ ".
A square on a chessboard is "e4".
A place on a map is found by a grid reference.

The two axes

The French mathematician René Descartes had the same idea, and it is named after him. We draw two number lines that cross at right angles.

The horizontal line is the x-axis. Numbers to the right of zero are positive, and numbers to the left are negative.
The vertical line is the y-axis. Numbers above zero are positive, and numbers below are negative.
The point where they cross is the origin, written $O$ . It is zero on both axes.

Both axes use the same unit of length. Together they make the Cartesian plane, also called the coordinate plane or the xy-plane.

The Cartesian plane. A horizontal x-axis and a vertical y-axis cross at the origin O. They divide the plane into four quadrants. Quadrant I, at the top right, has both coordinates positive. Quadrant II, at the top left, has x negative and y positive. Quadrant III, at the bottom left, has both negative. Quadrant IV, at the bottom right, has x positive and y negative.

The axes cut the plane into four quadrants, numbered I, II, III and IV, going anticlockwise from the top right. We study them in the next lesson.

Put the point on an axis. One dashed line shrinks to nothing, because the point is $0$ units from that axis.

Why this matters

Once every point has an address of two numbers, we can describe a shape by a few numbers, and we can use algebra to study it. A circle, a line and a parabola all become equations. This is how a computer draws a picture on the screen, with a pair of numbers for each dot.

Practise with new numbers

Try it in Python

This program plots four points on the plane.

Remember these

One number fixes a point on a line. Two numbers fix a point on a plane.
The x-axis is horizontal. The y-axis is vertical. They cross at the origin, $O$ .
Both axes use the same unit, and together they make the Cartesian plane.