\chapter{A Strange Creature}
\section{Down the Hole}
It was an unusually warm day, and Alice had taken the opportunity to
walk to a field not far from her house. She had just prepared a tea
party for a select few of her stuffed animals. They were finishing
their tea, though she never actually saw them drinking. Alice
was sitting on a comfortable patch of grass, chaperoning to
make sure their napkins didn't blow away. She detected some
motion out of the corner of her eye and got up to investigate. Alice
hoped it wasn't her neighbors' dog. The last time she had had a
tea party, he had nearly made off with her stuffed llama. Fortunately there
was no dog this time. Instead she saw a group of small animals
scampering by in a pink and yellow speckled flurry. Snatching up her
stuffed animals, she chased after them. They led her around in circles
a few times and finally stopped just near the picnic blanket where they
disappeared down a surprisingly large hole.
Cautiously, Alice approached the hole. She knelt down beside it,
leaned over, and peered in. It appeared to be a long tunnel, but she
couldn't see how long it was or where it went because of the darkness.
She stood up and inspected her dress to
make sure it hadn't gotten dirty. Mother would throw a fit if it
had. Another little yellow creature leaped from behind her and
jumped down the hole. Startled, Alice jumped too, losing her grip on
her favorite teddy bear. ``Oh no,'' she cried out, as she watched it
fall down the hole. She thought for a moment. Then, she carefully
wrapped up her other animals in the blanket and put them in the basket
she had used to carry them to the field. ``I'll be back soon,'' she
said, kissing them each once. ``Do not worry.'' And in another
moment, down went Alice after her bear and the pink and yellow
creatures, never once considering where the hole would lead her and
how in the world she was to get out again.
The hole went straight on for some way. It was like being on a roller
coaster or a slide. Alice held her skirt to keep it from blowing. She
supposed she was going
downward, though she really couldn't tell. ``I'm falling, so it must be
downward,'' she rationalized. Down, down, down, with nothing but the whoosh
of the wind. As she slid, she wondered if
she would ever stop falling. She wondered how far she had traveled and
where she was. She tried to calculate how far she must have fallen and how fast she
was falling, but she found it difficult to remember her sister's physics
lessons while falling. She found that fact to be rather inconvenient.
``What's the point of such learning,'' she thought ``if I can't use it when
I might need it?''
She wondered if she would end up in the center of the earth. Or perhaps
she would come out the other side. Or, even stranger still, she could
return to where she started, only to find everything changed. Or maybe
she would find everything the same but that she had changed. She thought
she wouldn't like that very much at all, but then she wouldn't be herself
anymore so maybe she would. And here Alice began to get rather
sleepy, until suddenly she landed --- thump --- on a pile of yellow books,
and the fall was over. There was nothing except for the books. Alice had
fallen into a dark cave with cold stone walls. Elaborate torches lined the
walls.
She was not a bit hurt, but she was slightly disoriented, and she thought
she saw a yellow pig. Or, it is more accurate
to say, she thought what she saw couldn't have possibly been a yellow pig
even though she was certain that was what she was seeing. Things became
still curiouser as she suddenly found herself singing under her breath a
song she had never heard before.
\vbox{
\ssp
\begin{center}
\it{Mine eyes have seen the glory of the coming of the pig, \\
She is trampling on the series where the terms have grown too big, \\
She's unleashed the boring lectures of geometry and trig, \\
Her proofs go marching on!}
\end{center}
\dsp
}
Alice looked around for her teddy. On the ground beside her lay a
thick blue velvet ribbon. This she recognized as the
ribbon she had tied around the bear's neck. The bear was nowhere
in sight. ``But what was that pig-like animal?'' she asked herself. ``Perhaps
it can help me find my bear.''
Alice ran off in the direction of the peculiar animal, but it seemed to
get smaller and then it vanished before her eyes. She stopped running
just before a wall which did not appear to be a wall at all. It was a
mirror that reflected Alice
and what was behind her, causing her to see an infinite tunnel of Alices.
She thought it very strange that a pig had been there
and was no longer, but as it wasn't much odder than seeing a yellow pig in the
first place, she tried to dismiss it. It was, after all, a Thursday.
She looked to her left and right. On either side of her was a stone wall.
There was no source of light, but somehow Alice was
able to see. The corridor looked as though it were frequently traveled, as
it didn't seem either dusty or lonely. She briefly considered asking it if
it were lonely, but didn't for fear that the girl in front of her who looked
just like her would think she was a ridiculous child, talking to hallways as
though they could answer. ``That's silly,'' she said aloud. ``It's just my
reflection, and it won't think anything of me talking to the hall.''
Sure enough, as she spoke, so did the other girl. She would have said
more, but it occurred to her that someone might arrive, and then wouldn't she
look even more ridiculous, talking to her own image!
Alice turned around and walked back down the corridor. All the way at the
other end was a red door. Above it was a sign that read ``Enter''
and below it another sign that said ``Exit''. Poor Alice, knowing not
whether to enter or exit, sat in front of the door considering her
predicament for quite some time. At last she decided to open the door
without either entering or exiting. ``After all, I'm only entering if I
think I am going somewhere. And I'm only exiting if I think I'm leaving
somewhere. But I don't know where I would go, and I don't know where I am,
and I'm certainly not thinking very clearly at all today.'' And so she
opened the door slowly and cautiously.
\section{Inside Out}
Alice had expected to find herself in another room, but instead she found
herself outside. She must have been outside because there was sunlight.
Unless she had been outside and in this strange land the sun was only
inside. But that didn't seem right. Alice stepped out, or rather,
through the door into one of the most magnificent meadows she had ever
seen. Flowers dotted the grass as far as she could see. A bubbling
creek wound its way through the flower beds. To one side was a grove of
trees; above Alice was an expansive bright blue sky, a
backdrop on which wispy white clouds had been painted. The aroma of the
flowers was stronger than anything Alice thought she had smelled before,
and though it was entirely pleasant, it made her dizzy.
Always drawn to water, Alice walked to the small pond which the creek had
formed. The water was clear and it sparkled like liquid diamonds. She
cupped her hands and dipped them in the pond and then drew the water to
her mouth. The water was colder than she expected, but it felt good as the sun
was quite warm. The water tasted clean, and immediately Alice felt
refreshed. She lay down on a patch of flowers and dipped her long hair
into the pond. Staring upward, she watched the clouds change shape.
``That one looks like a bunny,'' she thought, ``and that one a dragon.'' The
odd shapes and their boundaries entertained her until she detected a
slight motion out of the corner of her eye.
It was a yellow pig; she was now quite certain that it was a yellow pig,
as it appeared to be both yellow and a pig. She called out to it, ``Hello,
Pig.'' Startled, the Pig turned around and ran toward the trees. For
the second time that day, Alice took off after him.
She ran and ran through the green blur of trees. The trees were very
thin, but they seemed to be laid out in a square grid as if to trap
her, and as she ran through the spaces between the trunks, she had to be
careful so as not to run into them. ``Running into trees would not be
good,'' she thought to herself.
She was gaining on him. Suddenly he stopped and turned around. ``Do
you know why I will be able to outrun you?'' he asked. Without waiting
for an answer, he chortled, ``Because I am running irrationally!'' And with
that, he resumed running.
``Wait,'' Alice called out after it. ``What do you mean?'' Again, he
stopped and turned around. He stared at Alice for a very long moment.
In this moment Alice was finally able to observe the mysterious pig. He
looked like a normal pig, though perhaps he was a bit larger than most
pigs. He had a yellow pencil tucked behind his right ear and a
calculator tucked behind his left ear. He was a deep yellow in color,
somewhere between a golden orange and the color of lemons, with some darker
spots on his belly.
``Would you really like to know?'' he asked.
``Oh, yes, very muchly,'' replied Alice, who was intrigued by the talking
pig and didn't want it to run off again. Surely
this pig could help her find her bear and tell her how to return
home. She had fallen such a long way.
And so the Pig bounded over to Alice and motioned for her to sit down.
He proceeded to stand upright on his two hind legs, remove the pencil
from behind his ear, and speak, in a manner that wasn't much different
from preaching.
\section{What The Pig Said}
The Pig began: ``The trees in this forest are laid out in a most regular
pattern, as I'm sure you have already noticed. Consider not the trees, but
the center of each tree trunk. If you look at all of these points, they
make up a rectangular lattice.''
``A rectangular lettuce?'' interrupted Alice.
``Not a lettuce, a {\it lattice},'' responded the Yellow Pig. ``A rectangular
lettuce would be unproductive, not to mention silly. A lattice is just a
grid, like the corners of squares. Or, like the intersections of
streets.'' And so saying, he picked up a stick
and drew a series of evenly spaced parallel lines in the dirt. Then he
drew more evenly spaced lines that intersected those at right angles.
``All of these points of intersection are lattice points.'' Indeed, an
aerial view of the forest would have looked very much like a square grid
of evenly spaced points.
``In a unit square lattice, points are separated by one unit from their
horizontal and vertical neighbors. It doesn't matter what this one unit
is, but it's the same distance.''
Again Alice interrupted, ``But those two points,'' she said pointing,
``are further apart than those two.''
\centerline{\epsfbox{images/1-lorina.ps}}
``This is exactly correct,'' said the Pig. ``That's because instead of
being right next to each other, they are on a diagonal. How far do you
think those two points are from each other?'' he asked. ``First, let's
name the points. It's important to name them so we can talk about
them.''
``Let's call them Lorina and Edith,'' Alice suggested.
``Well, I was thinking of simpler names than that,'' the Pig explained.
``Let's call this point at the bottom $(0,0)$. And that one just to the
right of it $(1,0)$, then $(2,0)$, and so on. And the ones going up in the left
column $(0,0)$, $(0,1)$, $(0,2)$, \ldots . The first number in the pair
refers to how
far to the right the point is, and the second number refers to how far up
it is. The two points you mentioned are $(0,0)$ and $(3,4)$. They are
in both different rows and columns. You see how the naming works? The
name of a point identifies its location.''
``But those are boring names. Lorina and Edith would be much better.''
``Perhaps, but my naming is more logical, because with numerical names,
we can calculate the distance
between any two points. The distance between $(0,0)$ and $(0,1)$ is 1.
I have described the points using what are known as ordered pairs.
The first number in both pairs is the same: 0. It is only the
second number that changes. To determine the distance between
them, we need only to subtract 0 from 1. Similarly, the distance
between $(1,0)$ and $(3,0)$ is 2. This time the second number in the
pairs is the same so we subtract 1 from 3. Now do you understand?
It would make no sense at all to subtract Lorina from Edith.''
``Yes, but you haven't attempted to subtract Lorina from Edith either,''
pointed out an indignant Alice, who didn't see why the Pig wouldn't take
her naming suggestions. ``You picked two points that are in the
same row or two points that are in the same column. Lorina and Edith aren't
in the same row or column, so they wouldn't have either the first or
second number in their pairs in common. How could you subtract them?''
``Well, it would be quite a bit more complicated to do,'' admitted the
Pig, ``but I assure you I can. I want to explain a little bit
about how I was running before I answer your question. There
are all kinds of diagonals in this lattice. You pointed out one of them
to me already. The diagonal through $(0,0)$ and $(1,1)$ goes through $(2,2)$ and
$(3,3)$ and $(n,n)$, for any $n$. There's another diagonal parallel to it that
goes through
$(0,1)$, $(1,2)$, $(2,3)$, and so on. And there are diagonals
perpendicular ---
that is, at a ninety degree angle --- to these, such as the one that goes
through $(0,5)$, $(1,4)$, and $(2,3)$. Or the one through $(3,4)$ and
$(4,3)$.
I only need to specify two points to determine the line. That's a
very important fact. I can refer to each line by an equation or just in
terms of how steeply it slopes.''
``Oh, you are making my head hurt,'' said Alice. ``I feel as though I am in
a math class with all of this talk about equations.'' Alice found math class
to be confusing.
``Equations are just a way of expressing something, just as my point-pairs
are a way of identifying points with two values, an $x$ and a $y$.
Pictures are another way of expressing concepts,'' he said, pausing
to sketch some diagrams in his notebook. Alice watched as he drew a
grid of points and labeled several lines. She was impressed at how well he
could draw, being a pig and thus not having the advantage of a thumb.
\centerline{\epsfbox{images/1-lines.ps}}
``Then, in the diagonal you pointed out to me, we have all of the points where
$x$ and $y$ are the same. That's your equation: $y=x$. The second line I
pointed out is that line shifted, or translated, a bit. Its equation
is $y=x+1$ because it is translated up 1. And the first perpendicular line
is $x+y=5$. The first two lines have a slope of 1 and the third of -1.
When I talk about the slope of a line on a graph, I mean the change or
difference in height divided by the horizontal change. This means
in the first two graphs, as the $x$ number, or {\it coordinate}, increases by 1, the $y$
coordinate increases by 1 as well. And in the third one, as the $x$
increases by 1 \ldots .''
``The $y$ decreases by 1! So if I chose an entirely different two
points, like your $(0,0)$ and $(1,3)$, then as the $x$ increases by 1,
the $y$ increases by 3.'' Alice's head still hurt, but not as much anymore.
``Absolutely correct,'' praised the Pig, and Alice beamed. ``The slope in
that case would be 3. Now maybe you'll see what slope has to do with
running. I was running in a straight line with a certain slope.
Only I couldn't run in a straight line with slope 1. Look at
our graph. The line from my origin --- that's the point $(0,0)$ on the graph
--- with a slope of 1 intersects a point. That point represents a tree,
and I certainly didn't want to run into a tree. I couldn't run in a
straight line with a slope of 2 or 3 or 5 because I would eventually hit
a tree then, too. Nor could I run in a line
with a slope of $\frac{1}{2}$. That is, whenever the $x$ changes by 2, the
$y$ changes
by 1. Slope is just the change in $y$ divided by the change in $x$.
Running at a slope of $\frac{1}{2}$ is not very different from running at a
slope of 2,
and so I would hit a tree in the same amount of time. Running at a slope
of $\frac{3}{2}$ or $\frac{4}{3}$ isn't any better. Because if you look at
lines starting from that bottom left corner with those slopes, they all
intersect points.''
\centerline{\epsfbox{images/1-slopes.ps}}
``Do you follow?'' he asked.
``I think so,'' said Alice, hoping that maybe the Pig would make more sense if
he continued.
``Good, because here is the tricky part. I wanted to run in such a way that
I would never hit a tree, but running at any slope $\frac{x}{y}$ would cause me to hit
a tree. I didn't want to just run so I wouldn't hit a tree for a long
long time, but so that I would never ever hit a tree. So I just picked a
number that isn't $\frac{x}{y}$. That way, I will never intersect one of those
$(x,y)$ tree-points,'' the Pig said, waving his pencil gloriously.
``A number that isn't $\frac{x}{y}$?'' repeated Alice.
``There are a lot of numbers, surely you can't expect to be able to
represent them all so easily.'' He paused and scratched his forehead.
``Oh dear, I have to explain to you
about numbers. Well, think about a number line. And pay close attention
because I have a lot of terms to define here. On a number
line are {\it integers}, numbers like -17, -16, -15, 0, 1, 2, 3, and 1729,
to name a few. The positive integers are all the ones greater than 0;
that's 1, 2, 3,
and so on. They are called the {\it natural numbers} or counting numbers. And
between these natural numbers are fractions like $\frac{1}{2}$, $\frac{1}{17}$,
$\frac{17}{42}$. These are
called {\it rational numbers} because they express ratios between two
integer numbers. Even if we draw all of these rational numbers on a
line, we would
still be missing most of the points on the line. Because in between the
rational numbers, there are so many more non-rational numbers. The number
line contains the rational and {\it irrational numbers}; that's all of the
{\it real numbers}.''
\centerline{\epsfbox{images/1-nline.ps}}
``I just picked one of those irrational numbers, and then there was no
way I could hit a
tree,'' concluded the Pig, as if this cleared up everything. ``It was
pretty easy to pick an irrational number because there are
so many of them. But I haven't answered your question about how far we
ran, that is, about the distance between the two points. For that, I will
explain to you the Pythagorean theorem, if you are still interested. Then I
think irrational numbers will make more sense to you, and you will be able
to find the distance between Lorina and Edith ''
``Please do explain. But first, tell me a few things,'' requested Alice,
who was always full of questions. ``What are irrational numbers? What did
you mean by there being more irrational numbers than rational ones? Have you
seen a teddy bear? Are you a yellow pig? Oh,'' she said, remembering her
manners, ``My name is Alice.''
``I am a yellow pig, and I'm pleased to meet you. Welcome to Mathland,''
said the Pig.
``Mathland?'' repeated Alice. ``Is that where I am?''
``It is,'' said the Pig, ``so we should do math. You asked about irrational
numbers. Irrational
numbers are simply those that aren't rational. It's hard to
understand this without any examples, so if you wait a little while
I'll tell you about the first irrational number to be discovered.
It's also difficult to explain why there are more irrational numbers than
rationals. It has to do with
being able to list numbers. How many natural numbers are there? Remember,
that's the numbers 1, 2, 3, \ldots .''
``A lot,'' said Alice. ``More than I count even if I count for my entire
life.''
``Right,'' the Yellow Pig said. ``But for mathematicians, that's not always
precise enough. There are an infinite number of natural numbers. But this
doesn't bother them, because they can still describe the natural numbers.
That's because they
are ordered. If you give me a natural number, any natural number, I can
give you the next natural number by adding one to it. We can list all of
the natural numbers: 1, 2, 3, \ldots , $n$, $n+1$, $n+2$, \ldots .''
``I see,'' said Alice.
``Now, here's where it starts to get tricky,'' warned the Pig. ``How many
integers are there? Integers are all of the natural numbers, all of
the natural numbers with negative signs in front of them, and zero.''
``Infinitely many,'' said Alice.
``Right,'' agreed the Pig. ``But compare that `infinitely many' with the
number of natural numbers. Are there more integers than natural numbers?
Less?''
``There must be more,'' concluded Alice, ``because all of the natural numbers
are integers. There are twice as many integers as natural numbers because
there are the negative numbers as well. And there's a zero, so that's even
more than twice as many.''
``It seems so, doesn't it?'' the Pig replied. ``But think about that number
zero. How much difference does one number make? You already have an infinite
number of numbers. Why consider it at all? Isn't it enough to say you have
twice as many integers as natural numbers?
``It seems so. One number is meaningless when compared with so many,''
Alice agreed.
``Then by the same logic,'' taunted the Pig, ``what difference does twice
as many numbers make? Twice something infinite is just something infinite.''
``I guess so,'' said Alice. ``I hadn't thought of that. All infinity is
infinity. There's nothing bigger.''
``Not quite,'' said the Pig. ``There are different kinds of
infinities, but we don't need to get into that. The number of elements in
the set of natural numbers is the same as the number of elements in the set of the
integers. Or rather, their {\it order} or {\it cardinality} is the same.
That's just a fancy way of saying that there is a one to one correspondence
between the
natural numbers and the integers. So, we can list all of the integers. Any
idea how?''
``We can list all of the positive integers first, then zero, and then the
negative integers,'' Alice suggested.
``We could,'' said the Pig, ``but I'm afraid we would never get past the
positive numbers to list the negative ones. We are much better off writing
them like this: 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, \ldots . You see, it
is just like our listing of natural numbers, only we have stuck a zero at the
beginning and inserted negative numbers after each positive one. This
makes it clear how the integers are related to the natural numbers.''
``You mean to tell me that both of these lists have the same number
of elements, even though one list is contained in the other?'' asked Alice
incredulously.
``More or less. They have the same cardinality. I know that sounds
fishy,'' said the Pig.
Alice wasn't sure she believed him, but it did seem that one infinity
shouldn't be bigger than another. Certainly not significantly bigger, so
she let it go. She didn't quite understand what the Pig meant by
cardinality, but it sounded interesting. She wondered if he would ever get
back to the distance between Lorina and Edith.
``Now we have another set of numbers,'' the Pig said, ``known as the rational
numbers.
Those are the numbers in the form $\frac{x}{y}$, where $x$ and $y$ are both
integers. Thinking of how to list all the rational numbers is much
trickier, but there is a way. We make a table with the positive rational
numbers. It could begin like this.'' He wrote:
\begin{center} $ \begin{array}{ccccc}
\frac{1}{1} & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\
\frac{2}{1} & \frac{2}{2} & \frac{2}{3} & \frac{2}{4} & \frac{2}{5} \\
\frac{3}{1} & \frac{3}{2} & \frac{3}{3} & \frac{3}{4} & \frac{3}{5} \\
\frac{4}{1} & \frac{4}{2} & \frac{4}{3} & \frac{4}{4} & \frac{4}{5} \\
\frac{5}{1} & \frac{5}{2} & \frac{5}{3} & \frac{5}{4} & \frac{5}{5} \\
\end{array} $ \end{center}
``In the top row we have a bunch of
fractions with 1 as their numerator. That means $\frac{1}{1}$,
$\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{5}$, {\it et cetera}. In
the second row we have fractions where the numerator is 2. Like $\frac{2}{1}$,
$\frac{2}{2}$, $\frac{2}{3}$, $\frac{2}{4}$, $\frac{2}{5}$. What do the
columns look like?'' he asked.
``Each column has the same whatdoyoucallit? The bottom number in the
fraction.''
``Denominator,'' supplied the Pig.
``Thank you,'' said Alice politely.
``The first column has $\frac{1}{1}$, $\frac{2}{1}$, $\frac{3}{1}$,
$\frac{4}{1}$, $\frac{5}{1}$. The second column has $\frac{1}{2}$,
$\frac{2}{2}$, $\frac{3}{2}$, $\frac{4}{2}$, $\frac{5}{2}$.''
``Right,'' said the Pig. ``And these numbers go on and on in two directions.
The rows and the columns. There are infinitely many rational numbers in the
table.''
``Wait,'' interrupted Alice. ``Some of the numbers on your table occur
twice, like $\frac{1}{2}$ and $\frac{2}{4}$. How can you be sure that there
are infinitely many rationals when you have the same numbers so many
times?''
``Excellent question,'' said the Pig. ``If we consider only fractions
reduced to their lowest terms, like $\frac{1}{2}$ instead of $\frac{2}{4}$,
we avoid that problem. At any rate, every single positive rational number
is contained in our table. We may have duplicates, but we aren't missing
any numbers. And because our rationals are derived from the natural
numbers, there are the same number of rational numbers as natural numbers.
We can order them, too, though that is a bit more complicated.''
``But aren't they already in order?'' asked Alice, confused.
``I suppose they are,'' said the Pig, ``but I meant something more specific.
If I were to write a list in this order, it would start $\frac{1}{1},
\frac{2}{1}, \frac{3}{1}, \frac{4}{1}, \frac{5}{1}, \ldots , \frac{1}{2},
\ldots$. But that's no good, because it would have lots of huge gaps. When
I say I want to list the rational numbers, I mean that there shouldn't be
leaps in the list. To accomplish this we need to look at some diagonals in
our table.'' Alice watched him draw arrows:
\centerline{\epsfbox{images/1-rationals.ps}}
``That's how we order the rational numbers. Our table contains the
rational numbers and our diagonals make sure that we list them all.''
``Wow,'' Alice exclaimed, looking over the Pig's shoulder, ``that's neat.
We've ordered an awful lot of numbers.''
``We have,'' the Pig agreed. ``But you've gotten me off on a tangent. I
meant to be telling you about distance, not cardinality.''
``Well, then do that, too,'' instructed Alice. ``But,'' she asked
again, ``have you seen a stuffed teddy bear?''
``I'm afraid I haven't,'' he replied. ``Why?''
``Because I have lost my bear,'' said Alice sadly. ``I dropped it
down a hole, and I climbed in after it to try to find it. But I found
only its ribbon.''
The Pig looked at her consolingly. ``Don't worry. You'll find your
bear. I'll help you. But first, I'll explain about triangles so we can
subtract Lorina from Edith.''
\section{A Pig and A Greek}
``I'll explain to you how to find the distance between the two points you
chose in the lattice.
Think about your two points as being opposite corners of a square,'' he
said drawing a square.
``The question is to find the length of the diagonal of the
square with side length one. Let's look first at the area of the square.
It is 1 square unit because the area of a square is just the length of the
side squared. Now I'm going to add another square to
our diagram, this one with the diagonal as a side. This new square has a
center at $(1,1)$. And each of its corners is one unit away from the
center. Do you know what its corners are?'' asked the Pig.
``I think so,'' said Alice. ``The point above $(1,1)$ is $(1,2)$. The
point below $(1,1)$ is $(1,0)$. The point to the left of $(1,1)$ is
$(0,1)$, and
the point to the right of it is $(2,1)$. So those are the four corners.''
\centerline{\epsfbox{images/1-roottwo.ps}}
``Correct,'' said the Pig. ``I can draw those points and the lines connecting
opposite corners. Then we see that the square is made up of four right
triangles --- triangles with $90^{\circ}$ angles --- with length and height
of 1. Two of these triangles put together
have the same area as the smaller square, so the area of our new larger
square is equal to the area of two unit squares, or 2 square units. The area
of a square is the square of the length of its side. In fact, that's why we
talk about `squaring' a number. The square of a number is just the area of a
square with that side length. Similarly, cubing a number results in the
volume of a cube with that side length.
``The Greeks were just concerned with
geometry; what we think of as algebra was, for them, just another way of
representing geometry. We can rewrite many problems of algebra in terms of
geometry. We can also rewrite problems in geometry as problems in algebra.
From our picture, we get the equation $side^{2}=2$. Do you know what a
square root is?'' he asked.
Alice nodded. ``A square root is a funny sign that looks like a cross
between the long division sign and a check mark. It means the opposite of
squaring a number. So since $3^{2}$ is 9, the square root of 9 is 3.''
``Right,'' said the Pig. ``Actually, the square root of 9 is also $-3$, but
we don't want to worry about negative numbers. They don't make any sense
geometrically. To solve for the length of the side, we need to take the
square root of 2. That is, the length of each side of the larger square is
$\sqrt{2}$.''
``That's not equal to 1, or 2, or 3, or any such number,'' remarked Alice,
puzzled.
``It isn't,'' said the Pig. ``It isn't even equal to a fraction. It's an
irrational number.''
``I don't think it's nice to call a number irrational,'' said Alice.
``Perhaps not, but that's how the Greeks thought of them, as illogical
numbers, as numbers that didn't fit into their way of thinking of ratios.
And the square root of two was the first number to puzzle them in this way.
``Now I'm ready to tell you about the distance between any two points. This
is where we use the
Pythagorean theorem. The Pythagorean theorem says that
$a^{2}+b^{2}=c^{2}$, where $a$ and $b$ are the lengths of the
sides of a right triangle and $c$ is the length of the hypotenuse.''
``Hippopotamus?'' asked Alice.
``No, hypotenuse.''
``A hippo in a noose? What would you do with a hippo in a noose?''
``No, not a `hippo in a noose' either. A high-pot-en-oose. The {\it
hypotenuse} of a right triangle is the longest side, the one across from the
right angle.''
``Oh. That's a really silly name for it. Is that always the longest
side?'' asked Alice.
``Always,'' replied the Yellow Pig. ``And not only is it the longest, but
we have an explicit formula for finding its length, given the lengths
of the other two sides. That's our Pythagorean theorem,'' he said, writing:
$$
a^{2}+b^{2}=c^{2}.
$$
``So this hypotenuse thing is always the same length?''
``In relation to the other sides, yes. I'll show you. Let's see
\ldots today is a Thursday, so I'll give you the Thursday proof.''
``There's more than one proof?'' Alice asked.
``Why, pig-heavens, I bet there are over thirty-seven proofs, and they all
explain the same thing in a different way. I'll show you a few of the proofs.
This first one is a proof by picture. Our algebraic expression
$a^{2}+b^{2}=c^{2}$ is represented geometrically by this picture.'' Alice
studied the picture carefully.
\centerline{\epsfbox{images/1-pythag1.ps}}
``In the picture the sides of the outer square have length $(a+b)$ and the
inner slanted square has sides of length $c$. So, there are four right
triangles in the diagram with sides, or legs, of lengths $a$ and $b$ and hypotenuse
of length $c$. Geometrically, $c^{2}$ refers to the area of a square with
sides of length $c$. And similarly, $(a+b)^{2}$ is \ldots''
``The area of the outer square with sides of length $(a+b)$,'' supplied Alice.
``Right. So when we say that in a right triangle $a^{2}+b^{2}=c^{2}$,
what we mean is that the sum of the area of two
squares with side lengths $a$ and $b$ is equal to the area of a
larger square with side length $c$. I can draw a square on
side $a$ and a square on side $b$ and their combined area will equal the area
of the square on side $c$. Does that make sense?''
``Yes, I think so,'' said Alice.
``Good. Now, for some algebra,'' continued the Pig. ``The area of the larger
square is $(a+b)^{2}$. And that has to equal the area of the small inner
square, which is $c^{2}$, plus the area of the four triangles surrounding it.
Each of these right triangles has sides of lengths $a$ and $b$. So the
area of each triangle is $\frac{1}{2}ab$, and since there are four of them,
the combined area of the triangles is $2ab$. Now we need to find the value
of $(a+b)^{2}$.''
``Isn't that $a^{2}+b^{2}$?'' Alice inquired.
``No,'' said the Pig, ``try an example.''
Alice thought aloud, ``$(1+2)^{2}=3^{2}=9$, and $1^{2}+2^{2}=1+4=5$. I
guess it isn't,'' she concluded. ``So what is $(a+b)^{2}$?''
``You have to be careful when multiplying polynomials --- expressions like
$a+b$. It's like when you learned to multiply numbers. Think about
squaring 17, which is really the same as squaring $(10+7)$. First, we
multiply 7 by 7. Then, we multiply 7 by 1, which is really 10. This gives
us 17 times 7. Next, we multiply 1, or 10, by 7, and then by 1. This gives
us 17 times 10. We add these two results together to get 17 times 17.''
He continued, ``The same principle applies to squaring $(a+b)$, that is,
calculating $(a+b)(a+b)$. We use a method known as FOIL. That stands for
first, outer, inner, last. We multiply the first two terms, the outer two
terms, the inner two terms, and the last two terms. Then we add all of
those together. We can represent this with a diagram.'' He wrote:
\centerline{\epsfbox{images/1-foil1.ps}}
Alice giggled. ``That looks like a smiley face.''
``I guess it does,'' agreed the Pig. ``And it shows us how to obtain the
product $(a+b)(a+b)$ using the FOIL method.
The first two terms are $a$ and $a$, so we get $a^{2}$.
The outer terms are $a$ and $b$; we multiply those to get $ab$. The inner
terms are $b$ and $a$, yielding $ba$, and the last terms are $b$ and $b$, or
$b^{2}$. We add all of the terms together to get $a^{2}+ab+ba+b^{2}$ or
$a^{2}+2ab+b^{2}$.
``So, the area of the whole square is $a^{2}+b^{2}+2ab$,'' said the Pig.
``And it is also
$c^{2}+2ab$. This gives us the equation $a^{2}+b^{2}+2ab=c^{2}+2ab$.
Both sides have a $2ab$ so we can cancel them out. We are left
with precisely what we wanted to prove: $a^{2}+b^{2}=c^{2}$.''
``That's the first logical thing I've seen all day,'' Alice remarked.
``Yes, it's very logical. Most of what you will see here is logical,
although it might not appear to be so at first. This is a world ruled by
mystery and logic. Let me show you a second picture.''
\centerline{\epsfbox{images/1-pythag2.ps}}
``It looks slightly different from the first picture,'' Alice observed.
``Yup. In this diagram it's the outer square that has side length $c$. And
the inner square has side length $(b-a)$. Again, there are four triangles
with legs of lengths $a$ and $b$.''
``So, the area of those four triangles is still $2ab$,'' Alice thought out
loud.
``Right. And the area of the larger square is $c^{2}$, and the area of the
smaller square is $(b-a)^{2}$.'' The Pig looked at Alice. She remained
silent. He continued, ``$b-a$ is really just $b+(-a)$, so we can use the
FOIL method again.'' He drew:
\centerline{\epsfbox{images/1-foil2.ps}}
``Oh,'' said Alice. ``The first part is $(a)(a)$ or $a^{2}$. The outer
part is $(a)(-b)$ or $-ab$. The inner part is $(-b)(a)$ or $-ba$, and the
last part is $(-b)(-b)$ or $b^{2}$. So the sum is $a^{2}-ab-ba+b^{2}$.''
``Correct,'' said the Pig, ``and that's just $a^{2}-2ab+b^{2}$.''
``Right,'' agreed Alice. ``I think I can finish the proof now.
The area of the small square and the four triangles has to equal the area
of the large square. So $a^{2}+b^{2}-2ab+2ab=c^{2}$. And the
$-2ab+2ab$ part goes away, leaving us with $a^{2}+b^{2}=c^{2}$.''
``Absopositivelutely correct,'' praised the Pig. ``Those are my two favorite
proofs of the Pythagorean theorem, but there are many others that are quite
different. One, credited to a mathematician named Legendre, is based on the
idea of similar triangles. Similar triangles are triangles with different
side lengths but the same angle measures. I'll show you another geometric
proof that you should be able to understand if you analyze this picture
carefully,'' he said, sketching a right triangle and forming squares on all
three sides. He then split the largest square into a small square, which
was the same size as the small square on one of the sides of the triangle,
and four
equally shaped quadrilaterals. He also split the medium sized square into
the same four equally shaped quadrilaterals. ``Now you can see that the
combined area of the small square and the medium sized square on the sides
of the triangle is equal to the area of the larger square.''
\centerline{\epsfbox{images/1-pythag3.ps}}
``You're right,'' said Alice, after a moment of contemplation.
``Back to Lorina and Edith.
Lorina was the point $(0,0)$ and Edith was the point $(3,4)$ in my naming
scheme. We can make a right triangle from Lorina and Edith, like so,'' he
said, drawing:
\centerline{\epsfbox{images/1-lortriangle.ps}}
``So, the distance between Lorina and Edith is the hypotenuse of a triangle
with sides of lengths 3 and 4. Now that we've proven the Pythagorean
theorem, we can apply the formula.''
``I see,'' said Alice. ``The square of the distance is the square of the
hypotenuse which is equal to $3^{2}+4^{2}$. That's $9+16$ or 25. Then the
distance between Lorina and Edith is $\sqrt{25}$ which is 5,'' she
concluded. ``Is that right? Is the distance 5 units?''
``It is,'' said the Pig. ``I'm very pleased that you were able to calculate
the distance between Lorina and Edith. Now you know how to calculate the
distance between any two points.'' Alice was pretty pleased herself. She
thought that the Pythagorean formula was quite useful.
``Let's use our formula to calculate some more distances,'' the Pig
continued. ``If we
know the horizontal and vertical distances $a$ and $b$, we can calculate
the diagonal distance $c$. Most of the time $c$ is an irrational number,
like $\sqrt{2}$.''
``But for 3 and 4, we got 5,'' remarked Alice.
``We did,'' said the Pig. ``There are an
infinite number of such integer solutions to $a^{2}+b^{2}=c^{2}$.
Even though there are infinitely many integer solutions,
it's not very clear how to find them. The Greeks knew of a few triples with
integer values for side lengths. The smallest of these is our $(3,4,5)$.
Two more triples are $(6,8,10)$ and $(9,12,15)$. They work because they are
larger versions of $(3,4,5)$. We say they are multiples of $(3,4,5)$.''
The Pig continued, ``It turns out that all such triples can be written
in the form $(p^{2}-q^{2},2pq,p^{2}+q^{2})$.''
``What do you mean?'' asked Alice.
``Just pick two whole numbers, $p$ and $q$, with $p$ greater than $q$.''
``Like 2 and 1?'' asked Alice.
``Good example,'' said the Pig. ``Then $p^{2}-q^{2}$ is $2^{2}-1^{2}=3$.
And $2pq$ is $2 \cdot 2 \cdot 1$.''
``That's 4,'' said Alice.
``Right,'' said the Pig. ``And do you know what $p^{2}+q^{2}$ is?''
``Let's see,'' started Alice, ``it must be $2^{2}+1^{2}$ which is
$4+1$ or 5.''
``Exactly,'' said the Pig. ``See? That's our triple: 3, 4, and 5.''
``Neat,'' said Alice. ``Can I make another triple?''
``You sure can,'' said the Pig.
``I'll try 2 and 3,'' Alice said. ``That makes the first number in the
triple is $3^{2}-2^{2}$ or 5. The second number in the triple is $2
\cdot 3 \cdot 2$ which is 12. And the third number is
$3^{2}+2^{2}=9+4$ or 13. Is that a Pythagorean triple?''
``We can check: $5^{2}$ is 25, $12^{2}$ is 144, and $13^{2}$ is
169. What's $25+144$?''
``That's 169,'' answered Alice. ``And $5^{2}+12^{2}=13^{2}$, so it works.
But,'' she paused, ``why does it work?''
``I'm glad you asked. In mathematics it's important not to accept
everything, but to try to understand why things are true. It's fairly
difficult to find that magical formula,'' said the Pig, ``but
fortunately, we already know it, so it's fairly easy to see that it will
produce Pythagorean triples. We can verify that it works in the same way
we checked that $(5,12,13)$ was a triple. We just use substitution. The
Pythagorean theorem says that $a^{2}+b^{2}=c^{2}$. We let
$a=p^{2}-q^{2}$, $b=2pq$, and $c=p^{2}+q^{2}$. That's a lot of numbers
and variables, so hold on to your hat,'' he warned.
``I'm not wearing a hat,'' said Alice, slightly confused.
``It's just a figure of speech,'' the Pig said with a smile. ``It
means that you should pay close attention. Here's what we want
to show: $(p^{2}-q^{2})^{2}+(2pq)^{2}=(p^{2}+q^{2})^{2}$.''
``Well, $(p^{2}-q^{2})^{2}$ is $p^{4} - 2p^{2}q^{2} +
q^{4}$. And $(2pq)^{2}=4p^{2}q^{2}$. So when you add those two
together you get $p^{4} + 2p^{2}q^{2} + q^{4}$. That's $a^{2}+b^{2}$.
Our $c^{2}$ is $(p^{2}+q^{2})^{2}$ or $p^{4} + 2p^{2}q^{2} + q^{4}$, which
is the same thing we got for $a^{2}+b^{2}$. Sure enough,
$a^{2}+b^{2}=c^{2}$. Did you follow that?''
Alice had to admit that while it seemed logical, she would have to
look over the details before she could really be convinced. She borrowed
the Pig's notebook and slowly worked out the algebra for herself. ``It sure
is neat,'' she said.
``It is,'' agreed the Pig. ``And there's even a Pythagorean triple
with the number 17. That's my favorite number.''
``What's so special about seventeen?'' Alice asked.
``A lot,'' said the Yellow Pig. The Pig continued talking, but Alice was
having trouble
following him, for he had suddenly become almost frighteningly excited.
Instead, she dozed off and took a short nap, filled with dreams of hippos
and $a$'s and $b$'s.