At a Glance - I Like Abstract Stuff; Why Should I Care?
The Theory of Notation
We've been translating English to math using what's called infix notation. You didn't even know that, did you? It's true. Using infix notation means that we put operators in between the numbers they're acting on. For example, to add 3 and 5 we write a plus sign in between them:
3 + 5
Infix isn't the only way to write mathematical expressions, though. There are also ways called prefix notation and postfix notation. "Pre" means before and "post" means after, so we'll bet you can guess what these entail. In case you're not in a guessing mood, we'll tell you.
In prefix notation we write the operator before the numbers. To add 3 and 5, we write + 3 5. In postfix notation we write the operator after the numbers. To add 3 and 5, we write 3 5 +. You may never have occasion to use these, but it's a nice tidbit to keep in the back of your mind in case you ever do. Better to learn it pre you needing it rather than post.
These different types of notation are studied in some computer science classes. If you plan on spending much of your life in cyberland, it'll benefit you to know them. There are some calculators that use postfix notation, also called reverse Polish notation, because that notation makes it easier to design the internal workings of the calculator.
A lot of the mathematics we study in school has been around since Euclid, when the years were still counting down to zero. No wonder he was so good with negatives. Although you might not believe it to see all the math textbooks that exist, mathematicians today are still doing research and learning new things about math. If Euclid were alive today, our advancements would blow his mind.
A large part of mathematical research involves figuring out how to translate the questions you want answered into problems that can be tackled with mathematics. Translating the question "what shapes can a soap bubble take?'' into mathematics is no easy task.
When you've got a problem where the answer isn't known by anybody, it's called an open problem. We're not talking about something that is baffling merely you and your eighteen classmates. If your teacher knows the answer or it's in a book somewhere, that puppy is closed.