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.