Lesson Fourteen: In Which We Factor

So far we have seen how you can go from 1/2 to 3/6 by multiplying both the numerator and denominator by 3. Thus 1/2 = (1/2)*1 = (1/2)*(3/3) = 3/6. But suppose we were given 3/6, how would we know that it can be written as 1/2? This would be something good to know, as 1/2 is arguably "simpler" than 3/6 (the numbers involved are smaller at least).

Factoring:

If you have an integer, m, which can be written as the product of two other integers, say a and b, so m = a*b, then we call a and b factors of m. We also say that m is a multiple of a and that m is a multiple of b. So, if we consider 3, we can think of it as having factors 1 and 3, and 6 has factors 2 and 3. Since 3 is a factor of both the numerator and the denominator, we can remove it, and obtain 1/2. To express it with numbers, 3/6 = (1*3)/(2*3) = (1/2)*(3/3) = 1/2.

We really only need to consider factoring positive integers, because a negative integer is just a positive integer multiplied by -1. So, if we want to factor a negative number, if we first factor out the -1 we can proceed as though we were factoring a positive number. For example -6 = -1*6 = -1*(2*3).

Factoring Tricks:

While figuring out the factors of a number is not always an easy task, there are some tricks that can be helpful. If a number is even, it means that 2 is a factor of the number. Even numbers can be recognized by having a 0,2,4,6, or 8 in the one's place. If the sum of the digits in a number is a multiple of 3, then the original number is a multiple of three. For example consider 33, the sum of the digits is 3+3 = 6, and 6 is a multiple of 3, as we saw above, so 33 should have 3 as a factor. In fact, 33 = 11*3. Finally, if a number has a 0 or a 5 in the one's place, then it has 5 as a factor.

Lesson Thirteen: In Which We Learn Our Place

Up to this point, everything we have talked about has been true of mathematics in general. If you construct numbers in the same way as we have, all things that we have found to be true will still be true. However, for a moment I must take a quick side trip into an idiosyncrasy of our particular number system.

The important fact in this discussion is that we have ten symbols that we use to represent our numbers. They are 0,1,2,3,4,5,6,7,8, and 9. That means that we can count from zero to nine just fine and dandy, but if we want to represent the number ten we must innovate. What we did was invent the ten's place.

The Place System:

The numbers 0-9 represent single things, so 4 apples means that you have four individual apples. What happens when you put another number to the left of the single things is that it represents sets of ten. So if you have a dozen apples, you have 12 apples, which means one set of ten apples, represented by the 1 in the "ten's place," and two individual apples, represented by the 2 in the "one's place." If you happen to be 25 years old, it means you have been alive for two sets of ten years and 5 individual years.

Of course, if you have nine sets of ten and nine individual things, 99, and then you get one more, you have ten sets of ten. Since we do not have a symbol for ten of something, we need to expand our number to the left and create the "hundred's place." Thus 100 represents one set of a hundred, no sets of ten, and no sets of individuals. When we need to represent ten sets of hundreds, we add a new place for thousands, and so on and so forth, allowing us to represent any integer, no matter how large, using our ten symbols.

Base:

Because we have ten symbols, our number system is called base 10. All this means is that we write all our numbers using ten basic symbols. If you are familiar with how computers work, you may have heard of base 2. As you might imagine, base 2 means that every number is represented with two symbols. Zero is still 0 and one is still 1, but if you want the next number, you have run out of symbols already, so you need a new place. Thus the sequence 10 in base 2 represents the number 2, because it is one set of two and no individuals.

Historically there have been variations on what number is the base. Nowadays base 10 is used for mathematics, and most societal calculations, while base 2 and base 16 are used by some computer scientist. How would you express the number nine in base 6?

Lesson Twelve: In Which We Are Rational

Remember that originally we had the natural numbers, which we obtained by starting with 1 and repeatedly looking at the next whole number, obtained by repeatedly adding 1. Then we saw that it would be more useful to consider the integers, so if we added something that later needed to be undone we could do so by adding a negative number. Now we come to the notion of the Rational Numbers which is all of the fractions that can be written with an integer in the numerator and a non-zero integer in the denominator. Notice that every integer is a rational number because, for any integer n, n = (n/1).

The Rationals and Addition:

Conveniently, the rationals are closed under addition, that is, the sum of two rational numbers is always another rational number. Consider (m/n)+(a/b) = (mb+na)/(nb), and since the product of two integers is always an integer, and the sum of two integers is always an integer, (mb+na) and (nb) must both be integers, so the fraction is a rational number. Furthermore, since addition and multiplication of integers are both commutative and associative, addition of rational numbers is both commutative and associative. There is an additive identity, the rational number 0 = 0/1 = 0/n, and every rational number has an additive inverse since (m/n)+(-m/n) = (m+(-m))/n = 0/n = 0.

The Rationals and Multiplication:

Even more conveniently, the rationals are closed under multiplication. This is even easier to show since (m/n)*(a/b) = (ma)/(nb) and the product of two integers is always an integer. As before, multiplication of rationals is both associative and commutative, by now do you have a firm notion of what associativity and commutativity mean respectively? There is also a multiplicative identity, since 1 = 1/1 is a rational number and (1/1)*(m/n) = (1*m)/(1*n) = m/n. Finally, every rational number EXCEPT all forms of 0 has a multiplicative inverse. Hopefully you can convince yourself that a rational number represents the number 0 exactly when its numerator is 0. Now suppose that m/n is a rational number that is not 0, so n is not zero, since m/n is a rational number so zero cannot be in the denominator, and m is not zero, since that would make m/n = 0. In this case n/m is also a rational number, and (m/n)*(n/m) = (mn)/(nm) = (mn)/(mn) = 1. This is a mathematical way of saying that we can "undo" multiplication by any rational number except for 0.

Lesson Eleven: In Which We Do Not Add Apples to Oranges

Suppose you have one bushel of apples, then someone gives you another two apples, how many apples do you have. Although 1+2 = 3, you certainly wouldn't answer that you have three apples. This is because "bushels of apples" and "apples" are different things. However, if you know that every bushel of apples contains twenty apples, then you know that having a bushel of apples and having twenty apples are the same thing, so if you have a bushel of apples, and you get two more apples, you have twenty apples and two more, or 20+2 = 22 apples.

Similarly, if you are asked to add m(1/3)+n(1/2), you are being asked to add different things. You have m "thirds" of something and are adding n "halves," so you cannot immediately add them together. Fortunately, the same factor can have many different ways of being written.

The Many Forms of 1:

Remember that n(1/n) = 1, this means that n/n = 1. This should make a certain amount of sense, because if I split something into 5 pieces, and I have 5 of those pieces, I should have the whole thing. However, since multiplying by 1 does not change a number m*1 = m*(n/n) = (mn)/n = m.

Although we only stated that an integer times one remains the same, you should feel free to verify that (m/n)*1 = m/n. Thus (m/n)*(a/a) = (ma)/(na) = m/n. Sometimes this is called cancellation, as you can think of the a in the numerator and the a in the denominator cancelling each other out. More practically, this gives us a way to add together fractions which, at first, seem to be counting different things.

Adding Fractions:

To add fractions together you MUST have the same denominator, that way the fractions are counting the same sort of thing. So, how do we arrange for 1/3 and 1/2 to have the same denominator? The key is the commutativity of multiplication, we already know that 2*3 = 3*2. In order to make sure not to change the value of 1/3 we can multiply it by 1, but we can write 1 as 2/2, so 1/3 = (1/3)*(2/2) = (1*2)/(3*2) = 2/6. We can do the same thing to 1/2, multiplying it by 3/3 instead, so 1/2 = (1/2)*(3/3) = (1*3)/(2*3) = 3/6. Notice how we made use of the fact that 2*3 = 3*2?

So, if we return to the original problem of adding m(1/3)+n(1/2), we now know that m(1/3)+n(1/2) = m(2/6)+n(3/6) = (m*2)(1/6) + (n*3)(1/6). Since both of our fractions are counting the same thing, in this case sixths, we can successfully add them together to get (2m+3n)(1/6) = (2m+3n)/6. As long as you remember to only add fractions whose denominators match, so you are adding "similar things," and that you can make denominators match by multiplying by a clever form of 1, you will have a long and successful career adding together fractions.

Lesson Ten: There Is No 1/0

Remember when I said that n(1/n) = 1? Well, that is only mostly true, the single exception is that we cannot let n = 0 and expect that to be true, because there is no number (1/0). You might be asking yourself, "what right does he have to tell me what number cannot exist?" Bear with me for a moment and you too will see why (1/0) simply cannot make sense.

The Reciprocal of Zero:

Keep in mind that the purpose of (1/n) is to undo multiplying by n. So if a*n = b, then b*(1/n) = a, because to get from a to be you multiplied by n, so multiplying by (1/n) undoes this and takes b back to a. For example 3*2 = 6, so 6*(1/2) = 3.

However, zero times anything is zero. So 3*0 = 0 and we expect 0 * (1/0) = 3. But 5*0 = 0 so 0*(1/0) = 5 also needs to be true. Since multiplying by zero takes everything to zero, there is no way to undo it, we lose track of where things come from and cannot "send them back."

To use a metaphor, suppose we live in a small town with an airport that has flights that come in from New York. If someone arrives at our airport, we know that they just came from New York, because there is only one place that they can come from. If we consider a larger airport which has incoming flights from multiple places, then we cannot say where an incoming passenger is coming from without knowing more, because there are multiple possibilities. Since multiplying by 0 sends everything to 0, (1/0) does not have enough information to undo multiplication by 0. Because (1/n) is defined to undo multiplication by n, (1/0) cannot exist. Since m/0 would need to be m*(1/0), m/0 cannot exist either. In short, the denominator of a fraction CANNOT be zero

There are two very important concepts in this post. The fact that (1/0), and consequentially (m/0), cannot exist is something that even advanced math students forget or gloss over. You would be surprised at the ways the value 0 can sneak up on you. However, of even deeper importance is the notion that some things cannot be undone, because too much information is lost.

Lesson Nine: In Which We Separate

We have seen that multiplication provides us a powerful tool for adding a number to itself many times. However, suppose we mix things together that we ought have left alone, how can we get back to our original quantity? Asked another way, how do we undo multiplication?

Suppose 40 students arrive in each of 5 buses to their school in the morning. At this point the school has 40(5) = 200 students in it. However, one of them burns the building down, and now all the students need to be sent home, how many should go on each bus so that there are the same number of students on each bus? Because we counted them in the morning, we already know the answer is 40 students, but we would like a way to reach that answer without knowing it before hand.

Reciprocals:

Remember when we needed to undo addition we introduced new numbers, the negatives, which undid addition. This worked because n+(-n) = 0, and adding zero to a number does not change that number. We do a similar thing to undo multiplication.

Since multiplying by 1 does not change a number, to undo multiplying by n, we need another number which, when multiplied by n, becomes one. Then these two numbers will cancel each other out. We call this new number the reciprocal of n, and write it as (1/n). So far all we know is that n(1/n)=1.

However, we can quickly discern that reciprocals should keep multiplication commutative and associative. If we think of m*(1/n) as separating m things into n even groups, we can think of (1/n)*m = 1*(1/n)*m as cutting one thing into n pieces, then taking m of those pieces. Suppose you have two apples to share between three people, each person should get 2(1/3) of an apple. But on easy way to make sure that this happens is that you could share each apple between all three people individually, then each person gets 1*(1/3)*2 of an apple. While the pieces may appear different, the amount of apple they represent will be the same.

Suppose you have two cartons of a dozen eggs each and you use all the eggs in one of the cartons to make a cake. Now suppose you have the same two cartons, but you decide to take half the eggs from each carton for your cake, either way you use 12 eggs. The first way you have (2*1/2)*12 eggs and the second you have 2*(1/2*12) eggs.

Fractions:

Because writing m(1/n) all the time seems like a waste of space, we condense this to m/n. This is what we call a fraction. It has a numerator, which is the number before the slash, and a denominator, which is the number after the slash. If you write a fraction vertically the numerator is on top and the denominator is on bottom.

Suppose you want to multiply to fractions, what is (m/n)*(a/b)? We know we may rewrite this as m(1/n)*a(1/b) because this is how fractions are defined. Since multiplication remains commutative, this is certainly ma(1/n)*(1/b) so we need to consider what it means to separate something into n pieces then further separate into b pieces. Suppose we try to put something back together after it has been separated into n pieces then each piece into b further pieces. Each of the n pieces is in b smaller pieces, so the total number of pieces is n*b. Thus separating something into n pieces then each piece into b pieces is the same as separating the whole thing into n*b pieces. This leads us to the rather convenient conclusion that (1/n)*(1/b) ought to be 1/(nb). So (m/n)*(a/b) = (ma)/(nb). Unfortunately, addition will not work out so conveniently as we will soon see.

Lesson Eight: In Which We Multiply With Negatives

So far our exploration into multiplication has been restricted to the non-negative numbers, that is, the whole numbers along with zero. However, it certainly seems plausible that we would want to multiply using negative numbers.

Multiplying a Non-Negative by a Negative:

For example, suppose HBGary Federal is losing fifty-thousand dollars every month, how much has it lost in a year? Since losing money can be represented as having negative money, the amount that is lost turns out to be 12*(-50,000). From how we defined multiplication using addition, we know that this is what we get if we add -50,000 to itself 12 times. So far, so good.

It turns out that we can keep our special rules for 0 and 1 even when we consider negative numbers. Since 1*n can be thought of as just one "thing" containing n, for example one moth containing -50,000 dollars, 1*n=n even if n happens to be negative. Since 0*n is still no things containing n, 0*n=0 is also true when n is negative.

Another interesting rule we can obtain is that -1*n=-n. For example, -1*7 is the same thing as 7*-1, which we know is -1 added to itself seven times, or -7. We may conclude that n+(-1)*n=0 for any integer n.

A Negative Times a Negative:

Using the previously introduced rules of multiplication, we can establish that -1*-1=1. Because anything times 0 is 0, -1*0=0. We can rewrite 0 as 1+-1, so -1*(1+(-1))=0. Using distribution we obtain that -1*1+-1*-1=0, or that -1+-1*-1=0, since anything times 1 is itself. If we add one to both sides we get 1+(-1)+(-1)*(-1)=1+0, or 0+(-1)*(-1)=1. Thus we conclude that (-1)*(-1)=1, as desired. This is further confirmation that -(-1)=1, as -1*-1 is supposed to be -(-1).

Now, if we wish to multiply together two arbitrary negative numbers, say (-7)*(-3), we may first rewrite as (-1)*7*(-1)*3 and do the positive multiplication separate from the multiplication of negative ones. Thus (-7)*(-3)=(-1)*(-1)*7*3=1*21=21. You might note that two negatives multiply together to get a positive, a fact that we are often told as children. The fundamental reason for this is the fact that (-1)*(-1)=1. We also saw that a negative times a positive was a negative, because adding together negative numbers must necessarily stay negative. It turns out that three negatives multiply together to a negative. In fact, whether you get a negative or positive depends on whether the number of negatives you multiply together is odd or even, unless you multiply by 0, in which case you will get 0 of course.

Lesson Seven: Multiplication and Addition

Now we know how to combine two numbers with multiplication, and we know how to combine two numbers with addition, but how do multiplication and addition interact?

Multiplication First:

It is an unfortunate truth that multiplication and addition do not get along as well with each other as addition gets along with addition and multiplication gets along with multiplication. Since both are associative 2+3+4 is unambiguously 9, and 2*3*4 is unambiguously 24. However 2+3*4 takes different values depending on if you interpret it as (2+3)*4, which is 20, or 2+(3*4), which is 14. So, to resolve this ambiguity, we decree that multiplication occurs before addition. This is a common convention upon which we agree, rather than a mathematical truth which we discover, but we should stick with it nonetheless. So, things within parenthesis happen before things outside of them and multiplication happens before addition, these are conventions by which we must agree to live.

Distribution

The other interesting way in which addition and multiplication interact is called distribution. Consider 2(3+4), which is 2*7 or 14. This is the same value as 2*3+2*4 which, since we know to multiply before we add, is 6+8 or still 14. This is not a coincidence, when we switch from doing addition first to doing multiplication first, we must make sure that we multiply both values in the sum by 2. Consider if you had two bank accounts, into one you deposited a certain amount of money each month to save up for a vacation, and into the other you deposited a different amount of money to save for a rainy day. If, after three months, you wished to know how much money was in the accounts total, you could find out in two different ways. You could first figure out how much you deposited each month, by adding the amounts going into each account together, then multiply this monthly deposit size by 3, this way you add first. You could also figure out how much money was in each account individually after three months, by multiplying the deposit to each account by 3 separately, then add these account totals together to obtain the overall total, this way you multiply first. Notice that in the second case, the amount of money you put into the first account must be multiplied by 3 and the amount you put into the second account must be multiplied by 3, we say that the 3 is distributed to each of them.

Lesson Six: In Which We Go Forth and Multiply

Now that we are familiar with addition, we might amuse ourselves by repeatedly adding a number to itself. We do this when counting by numbers larger than 1, such as listing even numbers, or counting by two, 2, 4, 6..., or when items come in packages of a fixed amount, one carton of eggs is 12, two cartons is 24, three 36, and so on. It would be convenient if there were a way to quickly add together a specific number multiple times, which is exactly the role multiplication plays.

Multiplication:

While we are often taught to represent multiplication with the 'x' symbol, the way we represent addition with '+,' this becomes confusing when 'x' eventually gets a different meaning. Instead of learning one thing then changing halfway through, let us agree to write multiplication with an asterisk, n*m is n multiplied by m, or just by writing two things next to each other, nm is also n multiplied by m. When there would otherwise be ambiguity I shall use an asterisk or parentheses to clarify, so 34 is always thirty-four, if I mean 3 times 4 I shall either write 3*4 or 3(4) . That said, what is 3*4?

If one says it out, 3*4 is 3 multiplied by 4, which means you will be adding three to itself until you have 4 of them. Thus, 3*4 = 3+3+3+3 = 12. Because multiplication can be thought of in terms of addition, the whole numbers are closed under multiplication, by which I mean two whole numbers always multiply to another whole number. It turns out that multiplication is also commutative, that is, n*m = m*n, which is something you may be able to convince yourself of by thinking of n*m as counting up m groups each with n things in them, then rearranging the things into groups of size m. It turns out that multiplication is also associative, so m(n*l) = (m*n)*l, feel free to try to convince yourself why this must be true, but I think it is a harder property to intuit than commutativity.

Multiplication With One:

If you only have one set with n things in it, then you have n things in total, so it seems reasonable that n*1 = 1*n = n. In this sense, 1 is providing the same service for multiplication that 0 did for addition, it is the multiplicative identity, that is the number which leaves every number alone when they are combined using multiplication.

Multiplication With Zero:

If you have no sets, then you have no things. If I get 100 dollars every time I win the lottery, but I never win the lottery, then I get no dollars. In fact, no matter how much the lottery pays, if I do not win, I get 0 dollars. Thus, it should not be too surprising that n*0 = 0*n = 0.

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!