Lesson Five: In Which Form Emerges.

In some ways, the integers are a fundamental example of patterns we will see repeated throughout mathematics. They can interact with each other through addition, there is a way to leave them alone denoted by 0, and anything you do can be undone with a negative.

Addition:

Math would be quite boring were there no way for numbers to interact with each other. Addition is our first example of numbers interacting, you can use addition to take two integers and produce a third. This is what is called an operation. An operation, in common usage, is something you do to something, such as when a surgeon operates on a patient, or a farmer operates a tractor. You use addition to do something to two integers, namely change them into a third integer.

Notice that if you add two integers, the result is always an integer, if you think about all the integers as being constructed from 1's and -1's you can see why this should be true. Because using addition on two integers always produces an integer, we say that the integers are closed under addition, meaning that we cannot use addition to escape the integers. Notice that the whole numbers are also closed under addition. Now we know that addition in the integers has three properties, it is associative, (1+2)+3=1+(2+3), commutative, 1+2=2+1, and closed, two integers always add to an integer.

Zero:

Zero plays a special role with respect to addition, it leaves things unchanged. This is quite useful, as sometimes you don't want to alter the number you are considering. Because n+0=n we call 0 the identity. The term identity is used because combining a number with 0 preserves the number's identity, ie it doesn't change the number, sometimes 0 is called the additive identity to denote what operation has 0 as the identity.

Negatives:

We know that 1+-1=0, so let us extend the definition of the '-' symbol to say that n+(-n)=0. This agrees with what we know so far in that 1+-1=0 and 4+-4=0, but remember n is allowed to be any integer. So -1+(-(-1))=0 must also be true, but what is -(-1)? Since addition is commutative, we know that -1+1=0 because 1+(-1)=0, so 1 might be -(-1). Could there be another?

Suppose that -1+n=0. We also know -1+1=0, so -1+n=-1+1, since they both are 0. Since they are the same number, we can add 1 to both of them and they will stay the same, so 1+(-1)+1=1+(-1)+n. Since addition is associative, I may add the first two numbers together first on both sides of the equal sign, so (1+(-1))+1=(1+(-1))+n which is the same as 0+1=0+n since 1+(-1)=0. Since 0 does not change numbers when added to them, we now know that 1=n, or that n must in fact be 1. This means that -(-1) must be 1, or that 1 is the only value which when added to -1 yields zero. One can make a similar argument for any number, replacing 1 with the number in question, so every number has a unique negative.

In general, if n is a whole number, -n will be a negative number, and -(-n) will be the number n again. This is because n+(-n)=0 also means that -n+n=0, so n must be -(-n). We call -n the inverse of n, because adding -n is the opposite of adding n, or to put it another way, adding -n undoes adding n. It is very nice for operations to have inverses, because they allow you to undo things that have been done!

Structure:

I make note of these concepts, operations, identities, and inverses, because they help define the structure of the integers, that is, how the integers relate to each other. I also mention the concepts because they are important themes underlying mathematics and shall soon come into play in a new manner.

Remember that I said the whole numbers are closed under addition. The reason we want to also include the negative numbers to "fill out" the integers is in order to have inverses. If we did not have inverses we could add 2+7 to get 9, but there would be no way to undo this, and get back to the 2 from the 9. With inverses we know that we just need to undo adding 7, which means adding -7, and we get that 9+(-7)=2.

A Note on Abstraction

I am tired of saying that something is true for all numbers. It would be much simpler to have a symbol that is used to represent all the numbers. To that end, I introduce the variable 'n'. Unless I specify otherwise, when I use n as a number it is allowed to be any integer.

So, for example, if I were to say that the next integer after a number n is n+1, what I am saying is that for any integer, the next integer after it is itself plus 1. See, isn't it a lot more convenient to just use a shorthand like calling them all n? If I need to talk about more than one integer, I may also use variables 'm' and 'l'.

Lesson Four: In Which Less Than Nothing Happens

Continuing our example from Lesson Three, suppose the school day ends and students begin leaving. We are now faced with the challenge of making the total number of students at the school smaller. Suppose just one student leaves, when she/he arrived we added 1 to the total number of students at the school. Now we need a number that we can add to undo adding 1 earlier. To address this problem, we introduce the negative numbers.

Negative Numbers:

Negative one, which we write -1, is the number that we add to something when we want to undo adding 1 to it. This means that 1+(-1)=0 since the -1 cancels out the 1 so nothing should happen, and the way we write nothing numerically is 0. Since 2=1+1, if we want to undo 2 we need to use (-1)+(-1) which we call -2, or negative two. Just like we could use 1 repeatedly added to itself to form the whole numbers, we can build the negative whole numbers by repeatedly adding -1 to itself. Should we encounter 3+(-2) we might pause momentarily, for there is no 2 for the -2 to undo, but if we write everything in terms of 1 and -1 things are resolves. We obtain 1+1+1+(-1)+(-1) which allows two of the 1/-1 pairs to be cancelled, leaving only 1. Since adding by 0 changes nothing, -0, which undoes addition by 0, also changes nothing. Thus -0=0, since 0 is our symbol for nothing changing.

"Subtraction":

Assuming this is not your first time learning math, you probably have heard of something called subtraction. Subtraction is a way to make numbers smaller without needing to introduce the negative numbers. Basically you have two choices on how to make things smaller, introduce negative numbers and leave everything as addition, or leave everything as whole numbers and introduce subtraction. Subtraction has two very big problems. The first is that the nice properties of associativity and commutativity that addition has are not true for subtraction. That is, 3-(2-1) is not (3-2)-1, and 3-2 is not 2-3. The second problem is that it is perfectly natural to wonder, once the notion of subtraction is on the table, what IS 2-3? Then, of course, we must introduce the negative numbers after all, making all the inconvenience we went through to keep them out of things wasted effort. This is our first real lesson from thinking about math systematically and logically, subtraction is an inconvenient lie!

Integers:

Just like the numbers you get by adding together 1's has a special name, the whole numbers, the numbers you get by adding combinations of 1's and -1's has its own name, the integers. The integers are made up of the whole numbers, zero, and the negative whole numbers. Please feel free to explore for yourself that you can make all of these by adding 1's and -1's. Slightly more tricky is that these are the only things you can make by adding 1's and -1's, but it is true, and I invite you to ponder upon it.

A Note on Order

When it is necessary or useful to indicate that something is happening before something else, I shall enclose the things to be done first in parentheses. For example, I might have broken down 3+4 into (1+1+1)+(1+1+1+1), indicating that you can recover 3+4 by adding the first three 1's and the last four 1's together before you finally add their sums. Of course, since each time you add a one you simply move to the next number, no matter what order you add them, seven ones must add up to 7. This is why we may write 1+1+1+1+1+1+1 unambiguously, it does not matter how you add them. Since you can do addition in any order, this means that 1+(2+3)=(1+2)+3. It doesn't matter whether the 2 is grouped, or associated, with the 1 or the 3, the sum is still 6. Because addition works the same no matter how the numbers are associated, we say that addition is associative.

Lesson Three: In Which Nothing Happens

Suppose people are arriving at school in the morning, and we are keeping track of how many arrive each ten minutes. For a while life is good, people arrive and we are adding them to the total number of people at school. After a while however, school starts and the arrivals stop. We eventually need to add nothing to the number of people at school!

Zero:

If we want to add nothing to our total, it would be helpful to have a numerical representation of nothing. This is of course 0. Anything plus 0 remains the same. This means that 0+1=1, so, since adding one gives us the next number, we can think about 0 as being the number before 1.

Optional Tangent:

For some rather advanced reasons, which I may get to later, 0 is actually a more philosophically sound place to start our numbers than 1. However, 1 is the starting place that I chose for three reasons: the argument for 0 is advanced, but not complicated, and I want to keep things simple for now; if I remember correctly, math education starts with the positive numbers then introduces 0 and I think that is worthy of emulation and; it makes sense historically, giving nothing its own symbol is something that came after the other numbers.

A Note on Equality

I am slightly tired of writing 3+4 is 7, so I think it is time to introduce the equal sign. The symbol '=' means that the values on either side of it are actually the same. As another of my colleagues pointed out a while back, due to its use on a calculator, students sometimes think of '=' as having the same meaning as the Enter key, in that it signifies that a computation need be performed. However '=' is simpler than that, it just indicates that the left side and right side are quantitatively the same. So 3+4=4+3=5+2=1+1+1+1+1+1+1.

Lesson Two: In Which We Add

So far, whenever we have changed whole numbers we have done so by the smallest amount possible, incrementing them 1. However, it is not entirely without precedent that two largish groups of whole numbers should collide and need to be combined. When this happens it would be convenient if we didn't have to do so each 1 at a time. This is the problem which we create addition to solve.

Addition and the Next Number Generator:

Given what we already know, the easiest to define addition is to use the Next Number Generator. Any number plus one is the next number, so 1+1 is 2 and 2+1 is 3. Notice that since 1+1 and 2 are the same thing and 2+1 and 3 are the same thing, 3 must also be 1+1+1. Using the Next Number Generator in this way, any whole number can be considered as an addition of units. Thinking about whole numbers this way means that we already know how to add them together.

Addition:

Consider 3+4, while you may know that it is 7, we do not yet know exactly what 3+4 means. However, we do know that 3 means 1+1+1, and that 4 means 1+1+1+1, so 3+4 must also be 1+1+1+1+1+1+1. Aha, that we know the meaning of! Start with the unit, then find the next number six times, so 3+4 must be 7. Hmm, but if you look at 4+3, converting it to 1's you also get 1+1+1+1+1+1+1 so it looks like 4+3 is the same as 3+4. In fact, adding together any two numbers will be the same, no matter which order you add them in, because of this we say that addition is commutative.

Lesson One: In Which We Create Some Numbers

Before we do some mathematics, most of you would probably be more comfortable if we had some numbers with which to do it. All the numbers that we will need for now can be created using two simple concepts, a unit, and a next number generator.

The Unit:

If you are forced to do math by some circumstance of life, odds are that the numbers you are using represent some real world objects, be they cars, meters, pizza boxes, or people. The unit, which we denote by '1', represents a single object. In some sense, it is the smallest amount of a thing that you can have. If you have a parking lot full of cars, you can use your bulldozer to push some off a cliff and still have cars left. But, if you have 1 car and you use your plasma cutter to get rid of some of it, you no longer have any cars left, although some of it may remain, if it is less than 1 car it is no longer a car, in Oregon we call such things redneck flowerpots.

When 1 isn't out representing some real world object, it will perform a similar duty for our numbers. That makes 1 the smallest whole number that you can have, cut it up for scrap and you no longer have a whole number, you have something else.

The Next Number Generator:

Since 1 is the least amount of a thing that you can have, the next smallest amount of a thing that you can have is two 1's, which we denote as 2. Consider, if you have a room with 1 person in it and you want to have more people in the room, you must have at least 2 people in the room. Anything less and you don't have more people in the room, you have 1 person and an organ donation, which is something entirely different than a person. So, since 1 is the smallest amount of whole number we can have, the next smallest is 2, and we say that 2 is the next whole number after 1.

In fact, if we have any whole number, we can arrive at the next whole number by increasing our amount by 1. This is all that the Next Number Generator does, it takes a whole number and makes the next one by increasing it by 1, the smallest amount that it can be increased.

The Whole Numbers:

I while back I casually italicized the term, "whole number," assuming that you had a notion what those might be. However, here is a nice quick way of defining them in terms of things we already understand. Starting with our unit, 1, and using the Next Number Generator to make other numbers, we can make all the numbers we shall need for now. Any number that can be made in this way shall be called a whole number.

Math is Beautiful

Since it has come up so much recently, I have decided to attempt to explain math in a way closer to how I think it were taught. I am going to try to emphasize the interconnectedness of the concepts and the almost organic way math grows from simple ideas to more complicated ones. In doing so, it is my hope that math not only makes more sense, but also that you come to find some beauty in its intricate, interlocking logic.

My intended audience is not the mathematical novice, but rather the person who, while having been exposed to math, finds it unintuitive and unappealing. My reason for this is simple, while I wish math were taught more like this, I do not feel confident enough to want to be someone's first experience in math. I think that elementary mathematics educators are dealt a bad hand, both having their love of math stifled by the dull methods that they were taught and being trained to teach in the same vein, but it is a testament to their care for their students that any of us take our first tentative steps in the realm of mathematics.

Because my thesis is that math makes more sense when approached from a holistic, interconnected point of view, I advise against skipping posts. Although you may be both familiar and comfortable with the very fundamentals, I may have a new perspective on what is occurring that ties in with later concepts. I do heartily appreciate feedback on what is insufficiently clear, or any other subject, and will endeavor to make revisions incorporating said feedback.

Let us wish each other good luck on our journey!