# Algebra Cadabra - an occasional look at the magic of maths

## August 2009

David continues his tutorial on Algebra. Regrettable the graphic cannot be re-prpduced in this format so please use your own graph paper to solve his excercises.

The Cartesian coordinate system

In the early 1600s, the French mathematician and philosopher Rene Descartes invented a system of coordinates by which it became possible to define the position of any object or point in space. This became known as the Cartesian system of coordinates. The invention of Cartesian coordinates revolutionized mathematics by providing a systematic link between geometry and algebra. Using this system, every geometric object - lines, circles, parabolas, etc - can be represented by an algebraic equation. We'll see some of those in a future article.

The Cartesian system commonly uses three coordinates, x, y, and z, and can define the position of an object in terms of its distance in three directions from the starting point, called the origin. Imagine a room with a spider hanging from the ceiling. If we want to know the exact position of the spider, we can measure its distance from the side of the room (let's call it x). We can then measure its height from the floor (let's call it y). Finally, we can measure its distance from the end of the room (let's call it z). Given these three measurements, and knowing where we started from, we can define the precise position of the spider. Locations made in this way are written in brackets, in the form (x, y, z). For example, if the spider was 215 cm from the side, 156 cm up from the floor, and 320 cm back , its position would be written as (215, 156, 320).

The Cartesian system can also be used to define positions on a flat surface (called a plane), such as a sheet of paper. In this case, we are dealing with 2 dimensional space rather than 3 dimensional space, and we only use x and y. Imagine our spider sitting on a sheet of paper. We can measure its distance from the left hand side of the paper, and we can call it x. We can measure its distance from the bottom of the paper, and we can call it y. Its position is given as (x, y), with the origin at the bottom left hand corner of the paper. If the spider was 4 cm from the left, and 15 cm up, its position would be given as (4, 15).

Using graph paper

It is quite possible to define coordinates on plain paper, but graph paper is far easier. I have included a piece for you to use in this article. Graph paper comes in all kinds of sizes and scales, but a versatile format has small squares defined by faint lines, with darker lines picking out every tenth line. Graph paper like this can be used for either whole numbers, where each small square is 1 unit, or for decimals, where each small square represents 0.1 units. In fact, you can decide the scale and arrangement of the paper however you want.

I have drawn a line across the graph paper from one side to the other, and another one from top to bottom, so the paper is divided into 4 equal areas. The horizontal line is called the x axis; the vertical line is called the y axis, and they are labelled. The point where the two axes cross is called the origin, and is labelled O. The axes are numbered at intervals, with positive values for x to the right of the origin, and negative values to the left. Positive values for y are above the origin, and negative values below.

Mark positions on the graph paper as accurately as possible, using a small cross or dot. Label the point with its (x, y) coordinates. You will see I have done this for one point already. The point I have marked is (15, 13). This means it is 15 squares in a positive x direction from the origin, and 13 squares in a positive y direction. The origin is always defined as (0, 0).

Now plot the following points. Remember the x coordinate is given first, and represents a horizontal direction, parallel to the x axis. The y coordinate is given second, and represents a vertical direction, parallel to the y axis. Here are the coordinates:

* (5, 13), (5, 9), (5, 1), (13, 9)

If you join the appropriate points with straight lines, including the point I've already given you, you will see we have formed the letter F.

Exercises

Plot the following points to form other letters:

* (-15, 1), (-3, 1), (-3, 7), (-13, 7), (-13, 13), (-3, 13)

* (-13, -3), (-8, -15), (-3, -3)

* (5, -15), (5, -9), (5, -3), (15, -3), (15, -9)

David Smith