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A comon misconception about mathematics is that it is all about numbers. If that was the case, mathematicians would have gone out of business years ago when humanity programmed the first calculator. Rather, mathematics is the science behind patterns. So, although we will be talking about numbers here, keep in mind that we are less interested in the numbers themselves, and more interested in how the numbers ‘behave’.

Last time we introduced the idea of the Natural Numbers, N. If you will recall, N is the set of numbers we use to count with. In a more mathematical way we would write “N = {0, 1, 2, …}”. To speak this out loud we would say, “N is the set containing 0, 1, 2 and so on.” But what does that “…” or “and so on” mean? When mathematicians use the notation “…” it means that they are skipping some terms or entries. When used correctly, they have given you enough information to fill in the missin information. In some cases they will write “{0, 1, 2, … , n}”, in others, they will write like we did above: “{0, 1, 2, …}”. The difference is a subtle one, but understanding it will go a long way to building a solid foundation for more advanced math. Simply, the first expression is finite and the second expression is infinite. In the first one, no matter how large ‘n’ is, we could eventually write down the whole set (it may take hundreds of years, but we could do it). The second one, however, goes on forever and so we would never be able to list off every element.

So now, how does this set of numbers behave? Well, we know from working with objects of various quantities that there are certain things that we can do with those objects. In the best case senario we want to see our high abstract numbers behave a lot like bunches of objects in real life. With that said, there are basically only 6 opperations that we can perform with these numbers, either with out brains or with computers. They are, quite simply: addition (+), subtraction(-), multiplication(x or *), division(/), roots and exponents. On top of that, we can also compare any two numbers (<, >, =). We should be able to compare any two numbers and say that either one is larger than the other or they are equal. Now, notice that for our comparison and for all 4 opperations, we are only seeing how two elements combine at a time.

Think about that for a second. Do you agree that we can only use two elements at a time?

The astute readers among you will probably have stood up and come up with a counter to this. For instance, we can do “1 + 2 + 3″ or “5 < 6 < 20″, and in each of those cases we are using 3 numbers. If you came up with that yourself, give yourself a pat on the back. Asking questions like that about assertations is key to mathematic development. If you didn’t come up with that question, don’t worry… the people who are ask that are wrong. When we add three numbers together we are, in fact, adding two numbers and then adding a number to their sum. Likewise, when we are comparing numbers we are simply stating the relation between the first and second, and then the second and thrid. When mathematicians want to sound smart they call these “binary relations” or “binary opperations”.

With this new found information about “binary opperations” we are left with a problem. What do we do when we have to put together more than 2 elements with more than 1 relation. Theoretically, anyway you want to do it is just as valid as any other so long as we keep certain rules. (For instance, we want 1 + 2 to be the same as 2 + 1). However, in order to avoid confusions we refer to something call the order of opperations, or, if you remember from school, “PREMDAS”. What does a non-sense word like that mean? Simply put it stands for “Parenthesis, Roots/Exponents, Multiplication/Division, Addition/Subtraction.” Notice that we have all 6 of our binary opperations listed here along with some grouping item called parenthesis. Basically we allow for Parenthesis in our order of opperation so that we can do some opperations out of order.

Also, notice that PREMDAS takes our 6 opperations and puts them into pairs. This is something we will look further into on the next entry as we will see that our 6 opperations can get boiled down into 2 if we try hard enough. However, in order to do that we will first need to learn about numbers beyond the Naturals. For anyone who wants to read ahead, they will be called the Integers and the Rationals.

When I said that I would start at the beginning I wasn’t kidding.

Due to the simple fact that we are all members of a modern society (that is, I am assuming you are because you have a computer with access to the internet) most of us have been exposed to math our entire lives. In many cases, such as playing checkers or any number of children’s games, we don’t recognise it as math. However, one thing that everyone recognizes as math is numbers. I’m sure you have heard of them, or at least heard some of them. However, I would wager a guess that you have never really thought about them unless you are a graduate student or Ph.D. Try to answer this question: “What is 5?”

… Seriously… answer it before you keep reading. Don’t read ahead to see where I am going with this. Just do it.

Now, odds are you held up your fingers or somehow showed me 5 objects. (Alternatively, you may think yourself clever and said, “The number that comes after 4.” I hope you didn’t though because besides being right, it is also completely useless for trying to define the term number in the first place.) Although this understanding of 5 has served you really well in your day to day life, all you have done is given me one example of what a ‘5′ is. In reality, 5 does not exist (it will when we get to talk about Ordinals, but that will be much later). We can have 5 of a quantity of something, but we can’t have just a 5. 5, like all numbers, is an abstract concept we have for counting things.

Is your mind blown yet? Don’t worry, things about about to become more familiar. The numbers that we first learn about in life, be it through counting our fingers or counting our money, are called ‘Counting Numbers’. This, as you will see, is very appropriate because they are the numbers we use to count items. Often times mathematicians will use the term ‘Natural Numbers’ or just the ‘Naturals’. This is a little less telling because the Natural Numbers don’t actually appear in nature. Rather, the Naturals are part of the nature of our minds. As humans we have a tendency to want to put things into order so that we can understand them.

To mathematicians, the term Naturals has stuck. Due directly to the fact that they are so important, many mathematicians will use the short hand, N, to mean the natural numbers. Often times, they will double slash the N or write it in block lettering to further show that they are talking about the Naturals. Often times mathematicians will also refer to N as “the set of all Naturals”. What, specifically, a set is will be covered in the future, but for now just think of a set as bunch of items (which may or may not have something in common). Unfortunately, not everyone agrees on what “all of the Naturals” actually means. Specifically there is some disagreement about the inclusion of 0 in N. Some do, some don’t, but in the end it makes very little difference to most of the results we get from using N. In this blog I will try to consistently put 0 in N (and I’ll remind you of what N is often).

Be sure to check back tomorrow when we will start to explore the naturals and discus how they interact and form some of the non-Natural numbers.

“Hello world!” seems an appropriate title as any considering that this will be the first post in a hopefully long lasting blog. My main goal here is to create a blog which will encourage mathematics research and education. It should not matter if you are a struggling high school student, an undergrad looking for an honor’s thesis, a high school teacher looking to expand your knowledge to answer the dreaded “Why do we need to know this?” question, or a college professor looking for ways to get your students to understand more about your subject, all should find something interesting and useful about this blog.

In the long term I hope to cover pretty much all of the mathematics one would learn while attaining a Batchelors Degree in Mathematics. I will, however, start at the very beginnings of math education… and I mean the way beginnings. Luckily for the advanced folks out there, I do not foresee math up to high school taking much longer than a few weeks as the main ideas are few and far between. Once we wrap up basic algebra and trigonometry, we will move into calculus. From there, we will branch out based on questions and comments.

The short term will look slightly different. I will start with the main core of the blog for 3 or 4 days. Immediately following that I will provide answers to problems given during the week. The next two days will revolve around applications and/or open questions of the math presented. Now, at the beginning the math will be fairly simple to both read and write on this blog. However, as time goes on, and our notations get increasingly complicated, I may start posting small TeX files in PDF form (unless, of course, I find a good way to get TeX to output HTML code). In either case, I will occasionally talk about how to use TeX for mathematics. My favorite TeX editor, and the one I will be using, is called WinEdt. You can download a trial version of it from www.winedt.com, but if you are even slightly interested in writing for math research, I strongly advise buying a full version. Mac and Linux users, unfortunately will not be able to use WinEdt. My experience there is somewhat limited so unfortunately I cannot make any good recommendations for TeX editors for those OS types.

So, for all of you who read this, I thank you and I look forward to accompanying you on your adventure through the beautiful, but often bizarre, world of mathematics.