Suppose we better explain what we mean when we say "algebra" it doesn't quite mean that stuff you hated at school, usually we say "an algebra", GA is an algebra, a geometric algebra.

The numbers you learned at school are an algebra; all it means is that you have all these elements (numbers in this case) and that they all obey some set of rules. And if you replace the numbers with letters they still obey the same set of rules.

Elements and rules together is an algebra. Sounds pointless? Well, no because you can have different elements, they don't have to be numbers; and you can have different sets of rules, they don't have to be the same rules that apply to numbers.

Normally with numbers you don't really think too much about these rules, it's all automatic. So you "know" that 6*4 = 4*6 for instance, that's a rule that says the order in which you multiply two numbers doesn't matter.

But what if we made up an algebra that didn't have this rule? That is, ab is not necessarily equal to ba.

And what if a and b are not numbers but something else, vectors for instance.

On that basis, there are a whole load of different algebras out there and people stay up late at night studying these for their properties and possible applications.

Now there is a whole bunch of theory that goes along with the above explanation to make everything all nice and formal mathematically speaking and you can go look it up if you are interested in that.

For now, all I want you to do is write up a short paragraph or two as a reply in here that identifies the rules governing normal arithmetic and show examples of how they work (I will tell you that two of these rules are called Associativity and Commutativity).