Algebra/What is math, exactly?

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Algebra
Algebra
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Of all the subjects taught in schools throughout the world, mathematics is possibly the one which has collected the image of being most feared and disliked.

So, to start, what is math? What makes it so different from other areas of study, like languages or history? And more importantly, why on Earth do you need to know anything about it?

Definition[edit]

Mathematics is such a wide and broad field of study. To define it would be very hard.

Study of patterns[edit]

Mathematics is a study of patterns—finding patterns and explaining why such patterns exist. Patterns are everywhere: shapes (What is the area of this? What is the volume of that?), counting (How many ways are there to do this? How many are there of that?), and more.

One particularly interesting class of patterns is the patterns of numbers. Whole numbers, for example, look simple but they're not: 1, 2, 3... everyone knows what they are, everyone is familiar with their addition and multiplication. But there are subtle and profound patterns lurking. For example, we can look at what numbers are formed when we repeatedly add 2. 2, 4, 6, 8... We can look at what numbers are formed when we repeatedly add 3: 3, 6, 9, 12... It's easy to see that every number falls into at least one of these sequences, but how many does it fall into, and which ones? For example, 12 is in the sequence 2, 4, 6, 8, 10, 12...; 3, 6, 9, 12...; 4, 8, 12...; 6, 12...; and 12... What numbers fall into only one sequence? These numbers get a special name, prime numbers, because they are only divisible by one and themselves. Finding prime numbers is an extremely tricky question and forms the mechanism for protecting your privacy on the internet. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. How many of them are there? 3 and 5 are 2 apart, so are 5 and 7, and 11 and 13, and 17 and 19. How many pairs are there like this?

There are all sorts of questions to ask about these simple sequences, and some of them lead to very profound statements about the structure of numbers.

But why are there such patterns? Why is it that there has to be an infinite number of primes? It's possible to imagine that this stream of numbers just stops, maybe after billions and billions of primes, maybe there are so many that people will never be able to compute them all—but why is that impossible? Why is it that if you imagine that, then, if you are very imaginative, you will see that it is absurd?

The asking of questions about pure patterns is mathematics, as well as answering why it must be that way.

Art of making conjectures[edit]

Mathematics is an art of making conjectures and theorems—guesses about patterns. It is about explaining patterns like primes (How many are there?). There are so many conjectures and theorems, especially in geometry.

Conjectures and theorems are important in math. They say what can be done and what cannot. These things are discoveries, made by people who looked at patterns and found rules about them. For example, the numbers formed when we multiply by 2 are called the even numbers. There are special rules about them. For example, if you add any two even numbers together, you get another even number. That is what is called a theorem. It is a discovery about a pattern.

Study of logic[edit]

Mathematics is a study of logic—proving the conjectures. It is about showing why the conjectures and theorems are true.

This is important in geometry. In fact, many mathematicians want to prove many things, even up to today. There are many conjectures about patterns that still aren't proven.

Conclusion[edit]

Now, why would you want to know about it? Simple. Because it is fun and because it is interesting.

Read this if you're not convinced ...[edit]

In a recent conversation I had with an intelligent but incredulous young man about the philosophical concept of causality he claimed that there was such a thing as cause and effect. He stated that there were too many causes for events to always have the same effect. I tried to continue this conversation on general terms, but realized that I was arguing for a probabilistic view of the world. I asked the young man to take a coin from his pocket and flip it until he got three heads in a row. I then asked him if he would give me odds that the next flip would be tails. He said yes, and when he flipped, the coin came up tails and I had to pay him.

When you get to the last chapter in this book hopefully you will be able to agree with me that the odds were even (1/1, 50/50) that the coin would come up heads or tails. I was hoping that the coin would come up heads so that we could continue our conversation. It seems that in math there is always a "right" answer, and depending on how much we practice getting to this answer is a matter of luck and hard work. In algebra, we will be replacing numbers with letters in some equations, and the "right" answer will depend on how well we manipulate those numbers and letters. For instance we might use algebra to determine how much money we need to take a date to the movies. The young man I was talking to could easily solve this problem, but he would also know that his date would go smoother if he took a little extra. Mathematics gives you the power to predict giving known information. For instance, when going to a movie the young man could plan on taking his date to the matinee show when tickets are half price. A question that he could ask himself might be: What time should I pick my date up? Ideally I would be able to point out to him that by going to the matinee early he is more likely to be able to buy tickets and to use his extra money to visit the snack stand to keep his date entertained. If the matinee is sold out when he and his date arrive, then the young man is faced with buying more expensive tickets and keeping his date entertained while they wait. Practicing with mathematics allows us to see potential patterns ahead of time, and plan accordingly.

Because I lost my wager with the young man I decided to continue our discussion at a later time. When you use this book I hope that you will have some moments when you think "Aha! By thinking mathematically these circumstances make sense." When you have a moment like this I hope you will encapsulate it in a word problem and add it to the chapter you are reading. When thinking mathematically you will often feel that you have the right answer, but with enough study you will find that mathematics has theories about why "the best laid plans of mice and men often go astray." The young man was correct to assume that if you use mathematics your plans will not always work out, but if you study it long enough you will find that it is a tool that helps you have things work out the way you want more often than not, and to learn how to do better with your next try.