The Cartesian Plane

On a number line, one number tells you exactly where you are. Say four and everybody knows the spot. But on a flat page one number is useless, because there are hundreds of places that are four steps across. You need a second number to say how far up or down as well. So we lay two number lines at right angles to each other. The horizontal one is called the x-axis and the vertical one is the y-axis. The point where they cross is called the origin, and it has coordinates zero, zero. That whole arrangement is the Cartesian plane, named after the French thinker René Descartes, and it is how every graph you will ever draw is built.

A pair of coordinates is written in round brackets with a comma between, like three comma two. To plot it, start at the origin. The first number is the x-coordinate and tells you how far to go across, so move three to the right. The second number is the y-coordinate and tells you how far to go up, so from there move two up. Where you stop is the point three, two. The memory hook nearly everybody uses is that you walk along the hall before you climb the stairs: across first, then up. Count the ticks on the axes rather than guessing by eye, and mark your point with a small neat dot or a cross.

This is the single biggest thing to get right. Coordinates are an ordered pair, which means the order is part of the information. Look at three, five and five, three. They use the same two digits, but three, five sits three across and five up, while five, three sits five across and only three up. They are clearly two different places on the grid. Swapping the numbers moves the point, so you can never treat a pair as just a set of digits. The only time swapping makes no difference is when the two numbers happen to be equal, like four, four. A good habit is to say the pair out loud as you plot it: across three, up five.

Both axes are number lines, so they carry on past zero into the negatives. A negative x-coordinate means move left instead of right, and a negative y-coordinate means move down instead of up. The two axes cut the plane into four regions called quadrants, numbered starting at the top right and going round anticlockwise. In the first quadrant both coordinates are positive. In the second, x is negative and y is positive. In the third, both are negative. In the fourth, x is positive and y is negative. Notice that negative two, two and two, negative two land in different quadrants, which is the order rule showing up again. A point sitting on an axis belongs to no quadrant.

Reading a point off a grid is plotting run backwards. Slide from the point straight up or down until you hit the x-axis, and the number you land on is the x-coordinate. Then slide straight left or right to the y-axis for the y-coordinate. Write them in brackets, x first. Point A here sits two right and three up, so it is two, three. Point B is two left and two down, giving negative two, negative two. Point C is trickier because it sits right on the y-axis: it went nowhere across, so its x-coordinate is zero, and it is nought, two. Always check the signs as you read. Left means negative x, and below means negative y.