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Algebraic Expressions

Algebraic Expressions

At a Glance - I Like Abstract Stuff; Why Should I Care?

Wait, we thought you just said you liked...oh my. So fickle.

That's okay, there is also plenty of abstract to go around if that is the sort of thing you are into.

When we talked about the language of math, we were talking in particular about the language of algebra. Each different area of mathematics has its own dialect with its own symbols, and there are many different areas of mathematics. See the Math Atlas for descriptions of these areas.

Sometimes mathematicians who work in different branches don't understand each other's symbols, and sometimes different areas of math have different meanings for the same word. It is not that strange, really, when you think about it. In England, "pants" means "underwear," and why shouldn't "normal" mean different things to different mathematicians? All we can hope is that most of them are wearing normal pants.

All areas of math share an underlying idea of "proof." As do all trial prosecutors, but that is neither here nor there. In math, one starts with a collection of statements called a hypothesis (Ex: "We hypothesize that you will read the rest of this paragraph"), and from these determines that some other statement, called the conclusion (Ex: "You will enjoy this paragraph so much that you will move quickly and eagerly on to the next"), must be true. The area of mathematical logic studies reasoning itself. Since it is necessary to make precise statements in order to get good proofs (what is a "side" of a circle?), logic is also concerned with language. Logic has "terms" and "formulas" as well, although they aren't the same as terms and formulas in algebra. While we are at it, politicians have terms and formulas as well. They spend a certain number of "terms" in office, and an example of a formula they may use is 8(10,000v) = w, in which the v stands for votes, the 8 indicates ballot-stuffing by 10,000 of the voters, and the w represents an election win.

Most of the geometric formulas we have given you were understandable...that is, we were able to explain where the formulas came from. The exceptions were the formulas for circumference and area of a circle. These are quite hard to explain, because they involve π, which people have been trying to understand for millennia. If you have ever got a spare millennium, drop us a line and we will tell you all about it.

Seriously though, you will be able to understand the formula for the circumference of a circle when we get to geometry. By the time we get to calculus, you will be able to figure out the formula for the area of a circle. Sure, mathematicians have already done that, but knowing that someone has already climbed a mountain won't stop you from reaching the top, is it? If you are not the adventurous type, please don't answer that.

So far we have only looked at geometric formulas for two - dimensional shapes. As the dimensions become higher, objects become harder to visualize. Three dimensions aren't too bad—a rectangular box is an example of a three - dimensional object—but what about 4 dimensions? 15? Could you have an object with infinitely many dimensions? Even if you did, where in the world would you store it?

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