 Simple equations on a graph

Algebra is the way Islamic mathematicians figured out to connect our ideas about geometry with our ideas about numbers. You can use equations to represent geometric shapes. You can also use equations to represent the movement of numbers (or real objects like trains) through time or space.

The simplest equations have only one variable. They look like y = 3 or x = 4 + 15. These equations define a line in space: they tell you which line someone is talking about.

Equations can also have two variables. These look like y = 2x + 1 or x = 4y – 12. These equations also define a straight line, but the line can be tilted in various ways, and can move up or down, to the left or to the right.

What if the line you want to talk about isn’t straight, but curvy? Quadratic equations help you to define a curved line. They look like y = x2 or x = 8y3 + 16.

These equations each define one particular line, but you can also use algebra to describe a whole set of lines that are all in the same category. We call simple categories algebraic groups, and more complicated kinds of groups are called rings and fields.