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Algebraic Expressions

Algebraic Expressions

I Like Abstract Stuff; Why Should I Care?

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

That's okay, there's also plenty of abstract to go around if that's the sort of thing you're 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 tons of different areas of mathematics.

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's 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 a "proof." As do all big-time lawyers, but that's neither here nor there. In math, we start with a collection of statements called a hypothesis (ex: "We hypothesize that you'll read the rest of this paragraph"), and from there we determine that some other statement, called the conclusion, must be true (ex: "You'll enjoy this paragraph so much that you'll move quickly and eagerly onto the next").

The area of mathematical logic studies reasoning itself. Since it's 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're at it, politicians have terms and formulas, too. 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've shown you were understandable...that is, we were able to explain where the formulas came from. The exceptions were the formulas for the circumference and area of a circle. These are reeeeal hard to explain because they involve π, which people have been trying to understand for millennia. If you ever have a spare millennium, drop us a line and we'll tell you all about it.

Seriously though, you'll 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'll be able to pick apart 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, will it? If you're not the adventurous type, please don't answer that.

So far we've 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 3D object—but what about 4 dimensions? Fifteen? 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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