\chapter{Numbers} \section{Pie} When she woke up, Alice found herself back in the woods, lying on a bed of leaves and covered by a blanket of five-pointed stars. The Pig was sitting nearby mumbling to himself and scribbling notes on a pad. Noticing she was awake, he stopped scriblling and said, ``I'm sorry I got so carried away before. I was a bit irrational, I'm afraid. I could be more irrational, though. You know how I told you there were all of those irrational numbers? Well, what I didn't tell you is that some irrational numbers are more irrational than others? It's kind of like all pigs being equal.'' He chuckled. Alice wondered if he would be terribly upset if she interrupted him. His lectures so far had been interesting, but she hadn't had anything to eat since early that morning and was now very hungry. And he was terribly confusing. What did he mean about pigs being equal? About being more irrational? She thought he was already very irrational, though she dared not say so. The Pig looked back toward his notepad and continued, ``The first troubling irrational number that the Greeks discovered was $\sqrt{2}$, but there are many other irrational numbers that are even more interesting. Two of my favorite irrational numbers are known as $\pi$ and $e$. They are both very important numbers, especially in geometry and calculus.'' Alice sighed lightly and shifted her position, trying to ignore her growing hunger. Startled by the noise of her movement, the Pig looked up. ``I'm sorry, I'll stop now. You've had an awful lot of math for one day. And you must be hungry,'' he said. Surprised and slightly embarrassed by the Pig's perceptiveness, Alice felt she had to apologize. ``I really am enjoying the math. It's just that I haven't had anything to eat and that makes it hard to concentrate.'' ``Well, then,'' replied the Pig, ``let's get some food. I'll take you back to my cabin. It's not far from here, just back by that grove of trees.'' Alice saw a small clearing in the direction that he pointed. The Pig stood up and collected his belongings. Standing next to him, Alice guessed that he was about three feet tall. He had a funny way of walking, a fast somewhat bouncy skip. He had to take several little steps to keep ahead of Alice. The two walked in near silence, giving Alice time to examine the Pig more closely. His ears were now pointy and standing on the top of his head. Before they had been sort of floppy and drooped on either side of his head. His eyes, she noticed, were different colors. His left eye was bright blue while his right eye was a dark green. He had a curly little tail which Alice was very tempted to tug. She didn't though, because she thought that would be rather rude. The trees were getting more congested. The Pig led Alice on a small cobblestone path. The terrain became much hillier. ``It's just over this hill,'' he said. The path was slightly overgrown with bushes, and a canopy of taller trees shaded it from the sun. Alice saw the clearing ahead. There was a semi-circle of rocks in front of two very large trees. As they walked around the trees, Alice saw a large rock with a chimney sticking out of it. On the side of the rock was a small cabin. The cabin had a very small door, in front of which was a welcome mat and above which read the inscription ``Y. Pig''. Alice followed the Pig inside, ducking so she could fit through the door. ``This is my home,'' the Pig said almost timidly. ``I don't have many visitors.'' The cabin was not the least bit spacious. It was the sort of place Alice imagined a real estate agent describing as ``cozy'' because ``cramped and cluttered'' didn't sound nice enough. To be fair, Alice thought, it probably wouldn't seem nearly as small if she were as short as the Yellow Pig. The kitchen was big enough and had a barstool as there was no dining room. It looked like the Pig slept in the living room on a pile of hay. Surrounding the hay were piles of papers, jigsaw puzzles, and a Rubik's cube. The most impressive aspect of the cabin was the full wall of books. ``What would you like to eat?'' asked the Pig, interrupting Alice's thoughts. ``What do you have?'' Alice asked, afraid that the Pig might only have foods that would interest a pig, though she didn't know what exactly a pig, especially a yellow pig, would eat. ``I don't have very much food. I have some fruit pies: strawberry, blueberry, and key lime. I also have numbers, my favorite snack.'' ``Numbers? You eat numbers?'' asked Alice. ``Of course I eat numbers,'' the Pig replied. ``How do you think I learned so much math?'' Alice thought that he was serious for a moment, but his blue eye twinkled merrily and the corners of his mouth were twitching. ``So what are these edible numbers?'' ``They're crackers in the shape of numbers. They're especially yummy when dipped in numeral soup, but I don't think I have any of that.'' He took out a plate of the number cookies for Alice. They were small, and there were dozens of them. An awful lot of them were 17's, but Alice saw other whole numbers and even some decimals and fractions. ``Oh, they're like animal crackers!'' exclaimed Alice. ``I like animal crackers. My sisters and I often get them on the way home from school.'' Here Alice grew pensive for a moment, wondering when she would have animal crackers again. She could do without school and maybe even her sisters, she supposed, but she would like to go home. How would she ever get out of this strange land? She had fallen quite a long distance. ``Animal crackers come in all different shapes: elephants and cows and pigs. I like to eat them slowly, saving their heads for last.'' ``Pig heads?'' the Pig gasped. ``You eat pig heads?'' ``Oh no,'' Alice clarified. ``They aren't real pig heads. I would never eat pig heads. Well, I suppose I like bacon, but that's not from the head, is it?'' She could tell that she was only making things worse. The Pig had turned a very pale shade of white. ``I'm sorry,'' Alice apologized again. ``I would never eat yellow pigs.'' ``One of my brothers is a blue pig,'' said the Yellow Pig rather irritably. ``I would never eat blue pigs either,'' said Alice. ``Or orange pigs or purple pigs. Or even silver pigs. I won't eat pigs anymore. Please don't be angry with me,'' Alice pleaded, now almost close to tears. ``I'm not angry with you,'' said the Pig after a pause. ``Try one of the numbers.'' Alice gingerly picked up a number 3, afraid that eating a number 17 might be sacrilegious. She didn't want to offend the Pig again. The number was sugary and somehow crunchy and chewy at the same time. Alice helped herself to another. The Pig had one as well. He chose a number 17. Alice supposed she was allowed to eat them. After Alice and the Pig had eaten a sizeable portion of the number cookies, the Pig brought out a small blueberry pie. ``My pies are perfectly circular, or rather cylindrical,'' he said, ``and each pie has a diameter of 2 punits.'' ``Punits?'' Alice asked. ``What's a punit?'' ``Why, it's one pig unit, of course,'' said the Pig in a way that made it sound as if he found the entire matter perfectly obvious and was surprised that Alice would ask such a simple question. ``A punit, in this case, is between two and three inches long. So my pies are about 5 inches in diameter.'' ``What do you mean by `in this case'?'' Alice further inquired. ``Exactly that,'' said the Pig. ``What makes the punit such a wonderful unit of measurement is that it changes. Punits for pies may be different from punits for the height of ice cream cones or the shortest distance across a mud puddle. It's the most natural thing in the world to want to refer to completely different lengths as being one punit.'' ``If you say so,'' conceded Alice. It sounded horribly confusing to her, but she didn't want to argue with the Pig when he was being so illogical. ``Oh, you'll be glad we're dealing with punits soon,'' the Pig said. ``It's much easier to do arithmetic on punits than messier arbitrary units. My pies have a diameter of 2 punits, and the radius is half the diameter. So each of my pies has a radius of 1 punit. Try that with your inches.'' ``I guess you're right,'' Alice said, thinking it best to agree. ``Of course I am,'' said the Pig. ``Now, what's the circumference of this pie?'' he asked. ``That is, what is the distance around the outer crust? I mean, how does the distance around the crust compare to the distance of the diameter?'' Alice stared at the pie. ``It's certainly more than twice the diameter. Though I wouldn't think it's more than four times the diameter.'' ``It isn't,'' the Pig confirmed. He took out a piece of string. ``I can make a square with this string around the pie, so that the pie is just touching the square on the center of each side. The length of each of these sides is 2 punits, the same as the diameter. And there are four of them for a perimeter --- that's what we call the circumference of things which aren't round --- of 8 punits. And that's larger than the perimeter of the circle. \centerline{\epsfbox{images/2-circle.ps}} He took out a punit ruler. ``I could try measuring the circumference of the pie with this, but the pie doesn't have any straight edges, so it wouldn't work very well.'' ``I know!'' interrupted Alice excitedly. ``We can wrap the string tightly around the pie and mark the length of the circumference. Then we can straighten out the string and measure it against your ruler.'' ``Wonderful!'' the Yellow Pig exclaimed. And that's just what Alice proceeded to do. She held up the string to the ruler. ``It's a little over 6 punits,'' she announced triumphantly. ``That's not precise enough,'' said the Pig. ``Fortunately, I have a very special magnifying ruler and calculator. It will show quarter-markings and third-markings and so on. Why, it will divide your punit into hundreds and thousands and even quintillions of equal parts if you want. It does lots of other things too.'' He typed the number 3 on the small keypad on one end of the ruler, and third-markings appeared on the punit. ``The string doesn't reach the first mark, so it's less than $6 \frac{1}{3}$ punits,'' the Pig explained. He set the magical ruler to 4. ``It goes beyond the first marking. So it's more than $6 \frac{1}{4}$ punits,'' said Alice. ``More than $6 \frac{1}{4}$ and less than $6 \frac{1}{3}$. How about tenths?'' The Pig showed her how to reset the ruler, and they saw that it was less than $6 \frac{3}{10}$. ``$\frac{3}{10}$ is less than $\frac{1}{3}$,'' said the Pig. ``Which means we have lowered our upper bound for the length of the string. This ruler has a special button to compare two numbers. It turns the fractions into decimals and then displays them in order from smallest to largest.'' ``Neat,'' said Alice. ``You said I could break up a punit into as many pieces as I wish?'' The Pig nodded. ``What about 100 pieces?'' she asked, and he punched in 100. ``Oh my, that's hard to count,'' Alice exclaimed. ``You don't need to count it,'' the Pig said. ``I told you it was a very special ruler.'' He showed her a button on the ruler to display the number of the marking just to the left of the string and the number just to the right of it.'' Alice looked on amazedly. ``Oh, that doesn't work with every piece of string,'' explained the Pig. ``This is a magnetized string for use with this ruler.'' The ruler's LCD screen displayed 28 and 29. ``That means the length of the string, and therefore the circumference of the pie is between $6 \frac{28}{100}$ and $6 \frac{29}{100}$. Or between 6.28 and 6.29,'' said Alice, pleased to show off her knowledge of fractions and decimals. ``Right-o,'' said the Pig. ``You can set the ruler to thousandths to find the next decimal place.'' He did so and announced, ``The circumference of the pie is just over 6.283 punits.'' He and Alice worked out a few more decimal places. But each time Alice told the Pig a new number, he shook his head, which looked sort of funny because it made his ears, which were floppy again, swing back and forth, and said ``That's not precise enough.'' Alice was starting to get frustrated. ``If your ruler is so advanced,'' she asked, ``why does it take so long to measure the string?'' ``That's an excellent question,'' said the Pig. ``It actually can determine the length itself, but not as precisely as I would like.'' ``Well, I'm not going to measure it anymore,'' said Alice defiantly. ``It's one of those tricky irrational numbers and I'll never be able to tell you the exact length.'' The Pig smiled. ``You're right of course. The roundness of the pie makes the length of the string an irrational number. You can stop measuring it now. But you are also wrong. I can tell you the exact length of the string.'' ``How?'' Alice asked. ``The same way we dealt with the diagonal of a square. We can't write out that length as a decimal, but we know it is $\sqrt{2}$.'' ``You mean, this 6.283 \ldots number is the square root of some whole number?'' Alice asked. ``Unfortunately not,'' the Pig explained. ``If you square it, or cube it, or raise it to any power, it's still not going to equal any normal fraction. This number is different from $\sqrt{2}$ in that it is not the solution to a regular polynomial equation. We say it is a {\it transcendental} number.'' ``Transcendental?'' repeated Alice. ``That makes it sound all mythical or something.'' ``Well, it is in some ways. It's certainly mysterious. Actually, if you take our number and divide it by 2, you find a very special number: 3.14159 \ldots . This number is mysterious because it shows up all over the place in mathematics, especially with circles. It shows up so often that it has its own name. And that's how I can tell you the exact length of the circumference. Just as the punit is a sort of made up unit for convenience, so is this one. Mathematicians call this number pi after pies like my own of course. They write $\pi$, which is a Greek letter. So the circumference of the pie is $2\pi$,'' the Pig concluded simply. ``That's it?'' Alice asked. ``You expect me to be satisfied that we know the distance of the circumference of the pie just because we've given it some funny Greek name? That's a bunch of hogwash. No offense.'' ``None taken. It would be more accurate to call it a bunch of math-wash. You see, mathematicians are often much more concerned with concepts than with numbers. It's enough that they are able to communicate a complicated idea with a little symbol. Just like our punit. We used punits to simplify a problem and communicate.'' ``I guess that is sort of convenient,'' Alice agreed. ``Now that we've solved that problem, let us eat pie!'' squealed the Pig in delight. And he set out to cut the pie into two equal pieces. The pie was rich and moist, and it was so very small that they soon had finished off every last crumb. Alice was about to ask the Pig to get out another of his yummy pies, when he asked, ``What is the pie's area?'' By this point Alice was not at all surprised that the Yellow Pig had another riddle for her. He was full of questions. She thought for a moment. ``The pie fit inside that string square we made earlier. And the square has an area of $2^{2}$ or 4 punits, so the area of the pie is less than 4 punits.'' ``That it is,'' agreed the Pig. ``And I can tell you how much less than 4 punits. The area of the pie is precisely $\pi$ square punits!'' ``$\pi$ punits?'' Alice asked. ``How can that be?'' ``That's $\pi$ square punits. The square part is because we are talking about area. And I'll show you,'' the Pig said, taking out another pie. ``I'm going to cut this pie into very small pie wedges.'' And he cut and he cut until the pie was in dozens of itty-bitty pieces. ``Now, I'm going to put all of the pieces back together into what is almost a rectangle. Because we know how to calculate the area of a rectangle.'' ``The area of a rectangle is length times width,'' Alice interjected. ``Yuperdoodle. Now here's how I make a rectangle out of our circle.'' Alice watched. The pieces were very nearly triangles, with two of their sides the same length. He took two slices and put them touching so that the sharp point of one was next to the crust of the other. He did this again for each pair of slices, and then he put all of the pieces together in that same way. ``Ta-da!'' he exclaimed. Sure enough he had made a rather long strip of the curved triangles that looked a little like a rectangle. \centerline{\epsfbox{images/2-cheesy.ps}} ``It looks more like a cheesy-poof than a rectangle,'' said Alice. The Pig didn't seem to understand her comment. ``It's not a perfect rectangle,'' he explained, ``but that's only because I didn't cut small enough pieces. If I had cut each of these pieces in half, the curves would be less noticeable. And if I had cut each of those pieces in half, you would hardly see the curving at all. And so on and so on. In fact, if I had cut an infinite number of pieces, more than you could ever count, they would be infinitely thin, so small that you couldn't even make out the crust. And then I would have a perfect rectangle.'' ``I think I understand,'' said Alice, though she was not entirely sure that she did. There was a limit to the amount of the Pig's logic that she could take at one time. ``But anyway,'' said the Pig, ``let's just pretend or suppose, as mathematicians like to say, that we have a rectangle. What's the length of a rectangle?'' He paused. ``Well, what was the length or circumference of a circle?'' ``$2\pi$ times the radius punits,'' supplied Alice. ``Correct. Mathematicians call the radius $r$ and say $2\pi r$ punits. The circumference, or crust of the pie, borders the two long sides of our rectangle, half on each side. So the length of the rectangle is half the circumference of the pie, or $\pi r$ punits. Since the radius of our pie is 1 punit, that's exactly $\pi$ punits. Now what's the width of the rectangle?'' Alice studied the rectangle closely. Finally she saw the answer. ``It's the radius of our pie. Because the distance from the crust to the center of the pie is the radius. And all those sharp points on the slices are what was the center of the pie.'' The Pig beamed. ``Exactly right, Alice! So the area of the rectangle, which is the same as the area of the pie when it was in the shape of a circle, is the length $\pi \cdot r$ times the width $r$ punits. That's $\pi \cdot r \cdot r$ punits or $\pi \cdot r^{2}$ punits. So the area of a pie is always $\pi r^{2}$. That means the area of our pie with radius 1 punit is $\pi$ punits. That's how much pie we've eaten. Well, not really, but let's eat another pie before I explain.'' Alice was confused, but hungry, so she didn't question the Pig and his tricks. The two ate the second pie quietly. When they finished, the Pig continued, ``We calculated the area of the pie, which is like knowing what size plate we would need to put the pie on or how much frosting we would need to put a thin layer across the top. We could even calculate how much frosting we would need for the sides because we know the circumference. But the area we have is not quite what we want to know, because area is only two-dimensional. We want something three-dimensional. We want to know the volume of our cylindrical pie. A cylinder is a circle with height. Mathematicians use the letter $h$ to represent height.'' ``They aren't very original,'' interjected Alice. ``I suppose not,'' agreed the Pig. ``If we multiply area by the height, we get the general formula for the volume of a cylinder: $\pi r^{2}h$. The height of my pie is $\frac{1}{2}$ punit, so the volume of the pie is $\pi \cdot 1^{2} \cdot \frac{1}{2} = \frac{\pi}{2}$ cubic punits. That's the volume of one of my pies. Do you want more pie?'' he asked. Alice said she did, and the Pig took out the third pie. After they had finished that pie, they decided to go outside and lie in the sun while they digested their sugary meals. \section{Endless Numbers} Outside, the Pig helped Alice climb up onto the big rock, where they then lay resting for a few moments. The Pig took out his notebook and began to write again. Alice, afraid of interrupting him, finally peered over at his notebook. He had written: $$ 2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1} {1+\frac{1}{1+\frac{1}{6+\frac{1}{\ddots}}}}}}}}} $$ ``Egads!'' she thought. ``What a horribly long fraction. It looks as though it will never end,'' she said to the Pig. ``It's like those irrational numbers, going on and on forever.'' ``Yes, it is in some ways,'' said the Pig. ``But not all fractions that go on and on are irrational. It's the same with decimals. For instance, 0.33333 \ldots is a repeating decimal that is rational. It is equal to $\frac{1}{3}$. Even though it is endless, it is regular. Decimals with more complicated patterns are rational too, like 0.248248248248 \ldots. My fraction does go on forever, but I won't write it forever because there's a pattern. Do you see the pattern?'' he asked Alice. She thought about it for a moment and the Pig offered her his notebook and pencil. ``Well, between every fraction bar is something plus one over something. And there are an awful lot of 1's. The first number is a 2 and then there are larger even numbers and the 1's. It goes 1, 2, 1, 1, 4, 1, 1, 6, \ldots . Two 1's and then the next even number. So I would guess that the next three terms are 1, 1, and 8.'' ``Absolutely correct,'' the Pig said. ``I can stop writing out the fraction now that you understand it. Do you want to know what this fraction looks like as a decimal?'' ``Yes,'' said Alice. ``It must be awfully strange. How can we calculate it when the fraction never ends?'' ``We can approximate it, just as we did with $\pi$.'' explained the Pig. ``We'll compute the values of parts of the fraction and see what they look like.'' ``But how can we calculate even the part of the fraction that you wrote?'' asked Alice, more than slightly daunted by the large fraction looming before her and the Pig. ``There's no reason to be intimidated by that fraction,'' said the Pig, ``but we can start out by calculating a smaller fraction, such as $2+\frac{1}{1+1}$. That's the beginning of our continuous fraction. We start from the bottom of that, and work our way up and to the left. So, $2+\frac{1}{1+1}=2+\frac{1}{2}$. We can simplify that to the single fraction $\frac{5}{2}$, which is equal to 2.5. Do you understand?'' ``I do,'' said Alice, ``but that was a pretty short fraction.'' ``Are you ready for something longer?'' asked the Pig. Alice nodded. ``How about this one?'' He wrote: $$ 2+\frac{1}{1+\frac{1}{2+1}} $$ ``Now that looks a lot harder,'' said Alice. ``It isn't really harder,'' said the Pig, ``but I guess it does look more complicated. Just remember that we need to work our way up from the bottom of the fraction, simplifying it in several steps. What's at the very bottom?'' ``The fraction ends with $2+1$,'' answered Alice. ``Right. So our fraction is the same as $2+\frac{1}{1+1/3}$. Next we consider the $1+\frac{1}{3}$ part. That's $\frac{4}{3}$.'' ``I see,'' said Alice. ``The fraction is $2+\frac{1}{4/3}$. Now what?'' ``Do you know what $\frac{1}{4/3}$ is?'' Alice looked puzzled. The Pig continued, ``That's $1 \div \frac{4}{3}$. Dividing by a fraction is the same as multiplying by its inverse. The means we flip the $\frac{4}{3}$ to get $1 \cdot \frac{3}{4}$. And that's just $\frac{3}{4}$, so our fraction is $2 + \frac{3}{4}$, or $\frac{11}{4}$. Written as a decimal that is 2.75. That wasn't so bad, was it?'' Alice agreed that it wasn't. ``Good,'' said the Pig, ``because I have a longer fraction for you to simplify.'' He wrote: $$ 2+\frac{1}{1+\frac{1}{2+\frac{1}{1+1}}} $$ ``Oh my,'' said Alice. ``Just start at the bottom,'' advised the Pig, handing Alice his pencil. Slowly, Alice added together fractions: $$ 2+\frac{1}{1+\frac{1}{2+\frac{1}{1+1}}} = 2+\frac{1}{1+\frac{1}{2+\frac{1}{2}}} = 2+\frac{1}{1+\frac{1}{\frac{5}{2}}} $$ She paused. ``Try flipping the fraction,'' the Pig suggested. $$ 2+\frac{1}{1+\frac{1}{\frac{5}{2}}} = 2+\frac{1}{1+\frac{2}{5}} = 2+\frac{1}{\frac{7}{5}} = 2+\frac{5}{7} = \frac{19}{7} $$ ``Whew,'' said Alice, letting out her breath. ``What is that as a decimal?'' The Pig reached for his calculator. ``It's about 2.714. You did an excellent job with that fraction, by the way. Since it would take an awfully long time to simplify the whole fraction I wrote before and even longer to simplify that fraction with the new terms you suggested, I'll work those out on my calculator.'' He rapidly punched buttons for a minute or two and then announced his results: ``$\frac{1257}{463}$ or about 2.7149 and $\frac{23225}{8544}$ or 2.718281835.'' ``Those numbers are awfully similar,'' observed Alice. ``They look like they are approximating another special endless number. Is there a name for this number?'' she inquired. ``As a matter of fact, there is,'' the Pig replied. ``The number 2.718281828 \ldots is called $e$. It was named after Leonhard Euler, a famous mathematician.'' ``Oiler?'' repeated Alice. ``Yup,'' said the Pig. ``We'll come across more of his math later. But back to $e$. It's extremely important in calculus for limits and for computing continuously compounded interest.'' The Pig could see that he was losing Alice again. ``We can derive $e$ as a limit in another way,'' he said, writing: $$ \lim _{n\to \infty }\biggr{(}1+\frac{1}{n}\biggr{)}^{n} $$ ``That looks like some horrible mathematical expression,'' said Alice. ``How am I ever going to understand that?'' ``It's not as bad as it looks. Just ignore the `lim' part and think of it as the value of $(1+\frac{1}{n})^{n}$ for a really large integer $n$. Let's try some computations with different values of $n$,'' said the Pig. And so they did: \vbox{ \ssp \begin{eqnarray*} (1+\frac{1}{10})^{10} &\approx& 2.5937 \\ (1+\frac{1}{20})^{20} &\approx& 2.6533 \\ (1+\frac{1}{80})^{80} &\approx& 2.7015 \\ (1+\frac{1}{600})^{600} &\approx& 2.7160 \\ (1+\frac{1}{10000})^{10000} &\approx& 2.7181 \\ (1+\frac{1}{1000000})^{1000000} &\approx& 2.7183 \\ \end{eqnarray*} \dsp } ``Why, it is that very same number,'' Alice exclaimed. ``How does that number keep showing up? Just like $\pi$ did!'' ``Both numbers are very important in different branches of mathematics: $\pi$ is in some sense a basis of geometry and $e$ is a basis of calculus, which is the study of limits. Limits are pretty neat. ``Here's an old riddle known as Zeno's paradox. Let's say I'm running from here to that tree,'' said the Pig, pointing at a tree in the distance. ``I can run very quickly and accurately. So in the first second, I run half the distance to that tree. Then, in the second instant, I run half the remaining distance, or one-fourth of the original distance. At the third moment, I run half of the now remaining distance which is only one-eighth the original distance. I continue doing this advancing $\frac{1}{16}$, $\frac{1}{32}$, and $\frac{1}{64}$ of the total distance in the next three steps. Each time I go half the distance that I had gone the time before. Mathematicians say that on the $n$th turn, I will have $(\frac{1}{2})^{n}$ of the total distance left. The paradox is that I will never reach the tree. I can keep taking steps forever, but they are so small that I will never get to the tree.'' Alice thought about the paradox. In order to get to the tree, first the Pig would need to get halfway to the tree. After he got halfway to the tree, he would have to cover half of the distance remaining between him and the tree. And after that, the Pig would have to traverse half the still remaining distance. It seemed that he would never reach the tree, but Alice knew that in reality the Pig could get to any tree that he wanted to. ``How odd,'' she exclaimed. The Pig continued, ``The total distance that I have covered is the sum of all the individual distances. Mathematicians like talking about sums. They like talking about sums so much that they have a special notation for dealing with endless sums. Instead of writing $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots + (\frac{1}{n})^{2} + \cdots$, mathematicians use the Greek letter Sigma, written $\Sigma$.'' ``Sigma?'' Alice repeated. ``Like that guy Sigma Freud?'' ``No,'' said the Pig patiently. ``That's different. This $\Sigma$ is just a letter to the Greeks, as is $\pi$. And mathematicians love to use Greek letters. They like writing confusing things like this: $$ \sum _{n=1}^\infty \frac{1}{2^{n}}. $$ ``The limit that I wrote before reads `the limit as $n$ approaches infinity \ldots'. Similarly, we read this as `the sum where $n$ goes from 1 to infinity\ldots .' ``Now, if we add up all of those numbers, we'll find that we get really close but don't quite reach 1. That's the paradox. Mathematicians go even further and talk about an infinite number of steps and limits and the sequence created by partial sums as converging to 1. They say things like series and least upper bound and Cauchy. Sometimes they even say sequentially compact, totally bounded, and clopen. Something is clopen if it is closed and open at the same time. Isn't that silly?'' asked the Pig. Alice agreed that it was very silly. It didn't make much sense for something to be both open and closed. She was still trying to digest what the Pig had told her, that $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16} +\cdots $ was 1. She thought maybe the sum would be less than 1 because the terms were so small, but then she thought it would be greater than 1 because there were infinitely many terms. Neither was true; the Pig said that the infinite sum was exactly 1. It sort of made sense. She could see by adding the first few terms together that the sum was close to 1. Adding more terms didn't make too much difference because each term was smaller than the one before it. The Pig continued, ``Here's another sequence $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \ldots $. It's formed by the numbers $\frac{1}{n}$ as $n$ goes from 1 to infinity. Now, let's look at the sum of all those terms. How would you write that sum using sigma?'' Alice looked at what the Pig had written before and wrote: $$ \sum _{n=1}^\infty \frac{1}{n}. $$ ``Correctomundo. Now, what do you think this sum is equal to?'' he asked. ``Something not too large, I would guess. The terms are all getting smaller and smaller.'' The Pig didn't say anything. Alice thought about it more. ``Wait, it would have to be larger than 1 because our old sum is contained in this sum.'' Now the Pig nodded. ``Watch this,'' he said, and he proceeded to write out the sum, grouping some of the terms with parentheses. $$ 1+\biggr{(}\frac{1}{2}\biggr{)} + \biggr{(}\frac{1}{3}+\frac{1}{4}\biggr{)} + \biggr{(}\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\biggr{)} + \biggr{(}\frac{1}{9}+ \cdots +\frac{1}{16}\biggr{)} + \cdots. $$ ``I've put only $\frac{1}{2}$ in the first set of parentheses,'' the Pig explained. ``The next set of parentheses contains the numbers up to $\frac{1}{4}$, our next power of 2. What I'm going to do is add up the groups of numbers within the parentheses. Then I will have infinitely many partial sums to add together. Look at the $(\frac{1}{3}+\frac{1}{4})$ part. Instead of adding those two fractions together, I am going to approximate them with something that I know is less than their sum. Listen carefully: $\frac{1}{3}$ is greater than $\frac{1}{4}$ and $\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$, so we know that $\frac{1}{3}+\frac{1}{4} > \frac{1}{2}$. The next partial sum we have is $\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}$. There are four numbers and they are all at least as big as $\frac{1}{8}$. That means there sum is greater than $4 \cdot \frac{1}{8}$ or $\frac{1}{2}$. We can do the same thing again for $\frac{1}{9}+ \cdots +\frac{1}{16}$. There are eight numbers, each at least $\frac{1}{16}$ in value for a total that is bigger than $8 \cdot \frac{1}{16}$ or $\frac{1}{2}$ again. Each mini-sum grouped by parentheses represents a number that is greater than or equal to $\frac{1}{2}$. So the sum of the part that I have written out is larger than 2.'' He continued, ``What is really neat is that there are infinitely many such partial sums. I can always group together subsequences that add up to values of at least $\frac{1}{2}$. And since there are infinitely many such subsequences, the total sum will not stop at one number like our last series did. Since these partial sums are not getting smaller, the total sum will always get larger. Mathematicians say that this sequence, known as the harmonic series, diverges. Unlike the sum from Zeno's paradox which converged at 1, this one doesn't converge to any value. When you add more terms to the sum, it will always get larger. So you see, these two sums are fundamentally different.'' Alice was quite impressed with the Pig's little proof. ``So the first sum never actually equals 1, does it?'' she asked. ``That's right,'' confirmed the Pig, ``but it converges to 1. It's like Euler's limit $(1+\frac{1}{n})^{n}$ thing. It keeps getting closer and closer to $e$. That's one way we can deal with irrational numbers. They are just limits. We can't write out an exact decimal representation, but we know what the number is approaching. All this talk about limits is making me thirsty,'' he said abruptly. And he picked up his notebook and pencil, and the two went back inside. \section{The Golden Garden} Inside, the Pig offered drinks. Alice had grape juice, and the Pig had orange. The Pig picked up his glasses and a deep blue cape which he wrapped over his shoulders. ``Shall we go into the heart of the garden?'' he asked. ``It's a most beautiful place.'' Alice, delighted by the scenery so far, was eager to see the garden. So the two of them set off back down the path toward the garden. On the way, the Pig told Alice of another irrational number. It was, the Pig told her, not a transcendental number, but an algebraic one because it was the solution to a polynomial, not polymer, equation. The Pig began, ``My most favorite irrational number is often represented by another Greek letter, the letter $\phi$ (phi). It is also known by a bunch of different names including the golden mean and the golden ratio. It's another of those infinitely many numbers that cannot be expressed as the ratio of two whole numbers, but like $\sqrt{2}$, $\pi$, and $e$, it's another very useful number. The value of $\phi$ is $\frac{\sqrt{5}+1}{2}$. That's the solution to the polynomial equation $x^{2}-x-1=0$. It is approximately equal to 1.61803398875 \ldots . ``The number $\phi$ has lots of exciting algebraic properties. For example, I'll bet you would be surprised if you calculated the value of $\phi^{2}$ or $\frac{1}{\phi}$.'' Alice made a note to try those two calculations sometime. ``The number $\phi$ also shows up in geometry. Take a look at this star,'' the Pig said, stopping for a moment to draw a five pointed star in his notebook. ``This pentagram was the sign of Pythagorean brotherhood.'' \centerline{\epsfbox{images/2-pentagram.ps}} ``The ratio of the length of a side of the star to the length of a side of the bordering pentagon is precisely $\phi$. Furthermore, each segment in the star can be broken into two segments at its intersection with another segment of the star. Then, the ratio of the longer segment $a$ to the shorter segment $b$ is also $\phi$. I can keep drawing stars within pentagons within stars, and the ratio will always hold. This magically proportionate number $\phi$ abounds in the pentagram.'' He continued, ``For some reason, this ratio is just a wonderfully pleasing proportion to see, especially in art and architecture. The Greeks used the value of $\phi$ in designing the Parthenon. The divine ratio shows up in the art of the Renaissance. What I find impressive about $\phi$, is how frequently it occurs in art and nature. But,'' said the Pig, ``I won't tell you about that yet. Instead, I will let my garden show you.'' The trees were becoming less dense again, so Alice figured they were near the garden. ``Your garden knows about this number?'' Alice asked. ``How can that be?'' ``That's the mystery,'' the Pig said. ``Nature, artists, and mathematics. All are founded on beauty. And $\phi$ is the most beautiful number there is. Except for maybe 17, of course.'' ``Of course,'' agreed Alice, since 17 seemed so important to the Pig. ``We're almost at the golden garden,'' said the Pig. ``But first, I want to tell you about another sequence of numbers, known as the Fibonacci sequence. The numbers in this sequence start with 1 and 1. Successive numbers can be found by adding the previous two numbers. So the next number is 1+1=2. The number after that is 1+2=3.'' He continued generating a list in his notebook: \vbox{ \ssp \begin{eqnarray*} 1 \\ 1 \\ 1+1 &=& 2 \\ 1+2 &=& 3 \\ 2+3 &=& 5 \\ 3+5 &=& 8 \\ 5+8 &=& 13 \\ 8+13 &=& 21 \\ 13+21 &=& 34 \\ 21+34 &=& 55 \\ 34+55 &=& 89 \\ \end{eqnarray*} \dsp } He motioned for Alice to sit down on the grass. She did so and he stood next to her. ``So the Fibonacci sequence begins $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89$,'' explained the Pig. ``What's the next number?'' ``The next number would be the sum of 55 and 89,'' said Alice, ``which is 144.'' ``Right. The Fibonacci sequence is a neat sequence. Like our special irrational numbers, it shows up all over the place. As an example, let me explain to you the White Rabbit problem.'' ``The White Rabbit problem?'' asked Alice. ``I had a dream about a white rabbit with a problem once. He was always late.'' ``Well, this problem doesn't have to do with being late, but it does have to do with time and an awful lot of rabbits. Suppose at the beginning of the year there is 1 white rabbit couple, one boy and one girl. At the end of the January, they give birth to a boy bunny and a girl bunny. There are now 2 rabbit couples. At the end of February, the younger couple isn't old enough to have bunnies yet, but the original pair has another set of twins, another couple. Now there are 3 sets of rabbits. At the end of March, the first couple has two more babies. Additionally, the next couple is now two months old which is old enough for bunny reproduction. So that couple has two bunny babies as well, for a total of 5 couples of March hares. Each rabbit couple gives birth to a boy bunny and a girl bunny every month. At the end of April, there are three sets of rabbits to have bunnies, and they bring three new bunny pairs into the world. Now there are 8 pairs of rabbits. By the end of May, all except the three new pairs of rabbits can have babies and they give birth to one pair each of course. That's five new rabbits so there are 13 rabbit pairs altogether. Things get very hairy very quickly. At the end of June there are 21 pairs of rabbits. How many pairs of rabbits are there at the end of the year if the rabbits keep reproducing in the same way?'' \centerline{\epsfbox{images/2-rabbits.ps}} ``Those numbers of rabbit pairs are the same as the Fibonacci numbers you just wrote down!'' exclaimed Alice. ``At the end of July there will be 34 pairs. At the end of August there will be 55 pairs. At the end of September 89, and there will be 144 pairs of rabbits for Halloween. That's an awful lot of rabbits. There will be 89+144 which is,'' Alice paused, ``233 rabbit pairs. And finally, at the end of the year, there will be 144+233 or 377 pairs of white rabbits. Whew. Even though they only have two bunnies at a time, they sure end up with a lot of rabbits.'' ``The Fibonacci sequence grows fairly quickly. What is neat about it is the rate at which it grows. Let's look at fractions formed by successive Fibonacci numbers,'' the Pig said, writing: $$\frac{1}{1}, \frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{13}{8}, \frac{21}{13}, \frac{34}{21}, \frac{55}{34}, \frac{89}{55}, \frac{144}{89} \ldots $$ He gave Alice his calculator. ``Here, compute the decimal values for these fractions.'' She did and wrote them into his notebook so that it read: \vbox{ \ssp \begin{eqnarray*} 1/1&=&1 \\ 2/1&=&2 \\ 3/2&=&1.5 \\ 5/3&=&1.6666 \\ 8/5&=&1.6 \\ 13/8&=&1.625 \\ 21/13&=&1.61538 \\ 34/21&=&1.61905 \\ 55/34&=&1.61765 \\ 89/55&=&1.61818 \\ 144/89&=&1.61798 \\ \end{eqnarray*} \dsp } ``I get it,'' said Alice. ``Those fractions are getting closer and closer to each other. They look like they are \ldots what's that word? \ldots converging to a number. And they look as though they are converging to your special number $\phi$.'' ``That they are,'' the Pig said. ``The golden ratio and the Fibonacci numbers are closely tied together. And now that you know that, I think you are ready to fully appreciate my garden. I worked in this garden for many summers as a younger pig,'' he told Alice. About thirty feet in front of them was a large, well-sculptured hedge which seemed to surround the garden. The Pig led Alice around the side toward a wrought-iron arched gate. The gate had a complicated combination lock on it. ``This is to keep uninvited people out of the garden,'' he explained. ``The combination is 1-1-2-3-5-8. You are welcome any time.'' The gate to the garden opened, and the Pig ushered Alice inside. None of the incredible things Alice had seen and experienced that day compared to the golden garden. ``You should feel very honored that you have been allowed into this garden,'' the Pig told her, and she did feel very special to know that the Pig was sharing something this wonderful with her. The garden was like another world. Alice didn't see anyone else in the garden, though it would not have surprised her if there were several other pigs romping about or perhaps an extended family of bunnies frolicking in some shrubbery. The garden was positively teeming with life. It was sunny, but there was no bright sun overhead. Instead, the light seemed to be coming from within the garden in almost the same sort of way that light reflects off freshly fallen snow. The garden was its own world, and Alice found it hard to remember that there was anything outside the garden. She could feel the energy in the air. Alice looked up and was astonished by the glorious sight of the sky, which was almost dripping in light. Instead of being the usual blue or gray, it was swirled with tints of colors that were just slightly brighter than pastels. Pinks and purples spun around one another. Blues and greens filled in their gaps, and bits of yellows and oranges shone through, all somehow twinkling as if, instead of having clouds, someone had sprinkled iridescent glitter all over the sky. Parts of the sky seemed to be winking at her, somehow inducting her into this awesome world with a private display of beauty. ``It's --- it's wonderful,'' Alice whispered breathlessly. The Pig reached over and held her hand in his hoof. ``Watch as the colors change,'' he said. ``They change slowly enough so that you cannot tell where one color ends and another begins. Yet they fade into one another in swirling spirals that are almost dizzying.'' Sure enough, the colors began to move inward so that what was once blue had become purple and the greens had been replaced by the blue. When Alice thought they could wind themselves no tighter, the colors began to move back, expanding until Alice was certain that the outer pinks had escaped the garden entirely. The sky was like a blazing fire, only much more soothing. The air was cool. Alice somehow managed to turn her attention away from the mesmerizing sky. In the center of the garden was a circular fountain. Spiraling out around it were dozens of different types of flowers, all growing in perfect health. ``Everything in this garden is beautiful,'' said the Pig. ``This garden has no place for ugly mathematics. ``I'll take you around the outer path, on what I like to call the Fibonacci tour,'' he said. ``We'll start with threes and fives. Lilies, irises, and trilliums are all flowers with three petals,'' he said, pointing them out as they walked around the garden's perimeter. ``Columbines, buttercups, hibiscuses, and larkspurs have five petals on their flowers.'' They stopped in front of a large red and white rosebush. ``Wild roses also have petals in multiples of five. Three and five makes eight, another Fibonacci number. Delphiniums and bloodroot have eight-petaled flowers. Over here we have corn marigolds which have thirteen petals.'' ``The Fibonacci numbers run this garden, don't they?'' Alice said questioningly. ``You could say that,'' responded the Pig. ``Or maybe the garden runs the Fibonacci numbers. Personally, I think it more likely that they share a common sense of aesthetics.'' The Pig continued his tour. ``Asters have 21 petal parts. They aren't really petals, you see. And daisies behave as if they know of even larger Fibonacci numbers. Their parts frequently occur in 34's and 55's.'' Alice was completely in awe of the garden. She was impressed by the mathematics that the Pig was sharing with her, but even more so she was overwhelmed by the beautiful flowers. It was an ideal garden for a tea party. Her teddy bear! Why, she had almost forgotten. She wondered where he had gone off to. ``Fibonacci numbers don't stop at the flowers, though,'' continued her guide. ``They apply to all parts of the plant, including stems and leaves.'' He picked off a branch from a small pear tree. ``Look at the bottom-most leaf. The next highest one is not directly above it, but a slight twist away. Then there is another about the same distance up and the same distance around the stem. Let's keep going until we get to a leaf that is in the same position as the first leaf.'' He counted the leaves aloud to Alice. ``One \ldots two \ldots three \ldots four \ldots five \ldots six \ldots seven \ldots eight.'' \centerline{\epsfbox{images/2-leaves.ps}} ``A Fibonacci number,'' said Alice. ``I'm not at all surprised. What about the leaves on this stem?'' she asked, pointing at a cherry tree. They looked carefully at the stem and this time Alice counted. ``One \ldots two \ldots three \ldots four \ldots five. But why do they do that?'' she inquired. ``I can't explain it entirely,'' said the Pig. ``It's just one of those mysterious things about nature. The term for the leaf arrangement that we have been studying is the {\it phyllotactic} ratio. My guess is that the plant has evolved to make use of the most effective way for its leaves to get sunlight without blocking each other. The plant doesn't actually know about Fibonacci numbers; it is just that having a Fibonacci number of leaves is optimal. We can learn a lot from nature if we study it. We can learn a lot from numbers if we study them too.'' The Yellow Pig led Alice down a path that shone gold from fallen pine needles. The air smelled strongly of pine sap, and Alice caught the occasional whiff of perfume from the surrounding flowers. The Pig picked up a pine cone. ``Pine cones also have Fibonacci numbers nested in their spirals. ``There are two sets of spirals in the pine cones. There are the ones that go out clockwise and the ones that go out counter-clockwise. Both of these have spirals with different, successive Fibonacci numbers. Different pine cones may have different Fibonacci numbers depending on the tightness of the spiral.'' The Pig carefully labeled the pine cone so Alice could see for herself. ``Again, they do that because at the top of the pine cone, their kernels are so tightly packed together. When it unwraps around itself, that's just how it ends up.'' Alice looked at the pine cone. \centerline{\epsfbox{images/2-pinecone.ps}} The Pig continued, ``Spirals very often display properties of Fibonacci numbers and $\phi$. My tail does, though it is sort of hard to notice. Seashells are another good example. When small sea creatures are very young, they start developing a protective calcium layer. It grows around their bodies. Then, it spirals around, growing over itself again and again. Each time it gets thicker. If you were to cut open a seashell so you could see the cross section of the spiral, you would notice that a lot of the time the shells have the same pleasing spiral. It's known as the golden spiral because it follows the golden ratio.'' He drew a few rectangles and sketched in a spiral. ``The ratio between the lengths of sides in each rectangle is $\phi$.'' \centerline{\epsfbox{images/2-gspiral.ps}} They were approaching a huge row of sunflowers. ``Sunflowers are a wonderful example of the golden ratio. Look at the florets in its head. They also spiral outward in Fibonacci numbers.'' The Pig took out a magnifying glass and a protractor. ``Look closely and measure the angle between the center of one floret and its neighbor.'' Alice did so for several pairs of florets, and each time she came up with a number between $130^{\circ}$ and $140^{\circ}$. She was very pleased with herself for knowing how to use a protractor. ``The angles are actually about $137.5^{\circ}$. That's a very special angle which is known as the golden angle.'' ``Golden mean, golden ratio, golden angle. I see why you call this the golden garden,'' said Alice. ``Everything is golden. It's amazing.'' ``Sometimes I sit in this garden for hours working on mathematics or just staring at the flowers,'' the Pig confided. ``I like to sit over there under the golden tree. It gives me inspiration. When I was younger, my friends and I would camp out under the tree, staying awake talking until dawn. I'm just an amateur mathematician, but some of my friends are quite accomplished now.'' He paused. ``Would you like to meet a few of them?'' he asked Alice. ``Two of them live nearby.'' Alice, curious to meet other residents of this magical world, said she would, and hesitantly the Pig and Alice exited from the garden to which Alice knew she would one day return. \section{The Pig's Friends} Alice and the Pig walked across a small meadow. ``Thank you for showing me the garden,'' Alice said to the Yellow Pig. ``It's one of the most beautiful things I have ever seen.'' ``You're welcome,'' said the Pig. ``I'm glad you liked it. My mathematician friends live right over here. They are named Isabel and Gus the Rascal.'' They approached a door to a cabin that looked much like the Pig's from the outside, only it was considerably larger. The Pig knocked twice. A lamb answered the door. She was wearing two pieces of jewelry around her neck: a cross and a triangle. ``How nice to see you,'' she addressed the Yellow Pig. ``It has been a long time since I have seen you and Gus. And I was just over in the garden, and I thought I would stop by. I have a friend that I would like to introduce to you. Isabel, meet Alice. Alice, this is Isabel.'' ``Hello,'' said Alice shyly. Isabel shook her hand warmly. Alice found shaking hands with a lamb funny, but didn't want to laugh. She looked at her jewelry instead. ``I'm afraid Gus is out,'' Isabel told the Pig, ``but he will be back shortly.'' She turned to Alice. ``Would you like to know why I am wearing a triangle?'' Alice nodded. ``Let's go into the living room where we can sit down. Would either of you care for drinks?'' The Pig asked for two glasses of water. Isabel disappeared momentarily into the kitchen and returned with them. ``Isabel, can I look through your books?'' asked the Pig. ``Certainly,'' said Isabel. ``You know where my study is.'' The Pig left the room. ``Now,'' said Isabel, ``I will tell you about this triangle. It's a very special triangle.'' She produced some pencils and a stack of paper. ``It all starts with 1,'' she said, writing the number 1 centered at the top of the paper. ``That's the top row of the triangle. The next row contains just two 1's.'' These she wrote as well. ``Now you can continue filling in the triangle, if I tell you the rule.'' ``What's the rule?'' asked Alice. ``Each row starts and ends with a 1,'' said Isabel. ``Each number in the triangle is the sum of the two staggered numbers in the row just above it. So the next row starts with a 1, ends with a 1, and has a 2 in the middle because $1+1=2$. The row after that starts and ends with a 1, and has two 3's in between. That's because $1+2=2+1=3$.'' She wrote: \centerline{\epsfbox{images/2-pascal1.ps}} ``What comes next?'' she asked Alice. Alice thought about her question. It started and ended with a 1. That left three other numbers of fill in. The first one was between the 1 and the 3, so that was 4. The second was sandwiched between two 3's, so it was 6. And the third one was between a 3 and a 1, so it was 4 again. Alice recited, ``1, 4, 6, 4, 1. Are they always the same forwards and backwards?'' ``Why yes,'' said Isabel, almost surprised. ``That's a good observation. The numbers in my triangle are symmetric. Because it starts with symmetry, it must always preserve that symmetry. The next row, for instance, is 1, 5, 10, 10, 5, 1.'' ``And the one after that,'' said Alice, giving the matter some thought, ``must be 1, 6, 15, 20, 15, 6, 1.'' ``Exactly,'' said Isabel. I'll write out a bunch of the triangle. \centerline{\epsfbox{images/2-pascal2.ps}} ``The numbers get large awfully quickly,'' said Alice. ``They do,'' said Isabel. ``It would take a long time to find a certain number, like the seventh number in the seventeenth row.'' ``Yes,'' Isabel said, ``if you had to write out the whole triangle. Fortunately, you don't have to. There's a complicated formula to find a number in the triangle. It's used in probability.'' ``Neat,'' said Alice. ``That formula must save a lot of time.'' ``It does,'' said Isabel. ``But I like drawing out the whole triangle because it has such neat patterns. Look at the diagonal columns, if you can call them that. The first and last column are all 1's. The second and next to last column are just the counting numbers in order. The next row is one that you should ask Gus about when he returns. It contains the triangular numbers.'' ``Triangular numbers?'' repeated Alice. ``But isn't this whole thing a triangle?'' ``It is,'' said Isabel. ``But look at that sequence of numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45. It starts with 1 of course. Then $1+2=3$. And $1+2+3=6$, $1+2+3+4=10$, $1+2+3+4+5=15$, $1+2+3+4+5+6=21$, and $1+2+3+4+5+6+7=28$. Think about arranging coins with those numbers. One coin in the first row, two in the second, three in the third, and so on. The arrangement of coins looks like a triangle, just as the arrangement of coins in a square would give you the square numbers.'' \centerline{\epsfbox{images/2-triangle.ps}} ``I see,'' said Alice. ``If you add 8 to that you get 36, and if you add 9 to 36, you get 45. Are there any other special numbers hidden in the triangle?'' ``Most definitely,'' Isabel said. ``Take the sum of each row.'' Alice began taking sums. The first row was just one. That didn't really count. The second row was $1+1=2$. The third row was $1+2+1=4$. The fourth row was $1+3+3+1=8$. The next row was $1+4+6+4+1=16$. The numbers were getting harder to calculate, so Isabel wrote them out for Alice. \begin{center} 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048 \end{center} ``Each sum is twice the sum before it,'' said Alice. ``Those are the powers of 2.'' ``Yup,'' said Isabel. ``That's why the triangle works out so well for probability. There's another set of incredible numbers in the triangle. Do you know the Fibonacci numbers?'' she asked. ``I do,'' said Alice. ``The Pig just taught them to me. They are 1, 1, 2, 3, 5, 8, 13 \ldots . Each number is created by the sum of the previous two. Don't tell me they are in your triangle too?'' ``They are,'' said the lamb. You just have to add up the numbers along some specific diagonals. It's hard to visualize the diagonals, so I'll draw you a picture.'' \centerline{\epsfbox{images/2-pascal3.ps}} ``The first diagonal contains a single 1 as does the second diagonal. The third diagonal contains two 1's so it sums to 2. I'll write out the numbers in each diagonal and you can add them up.'' She wrote: \vbox{ \ssp \begin{eqnarray*} 1 \\ 1 \\ 1+1 \\ 2+1 \\ 1+3+1 \\ 3+4+1 \\ 1+6+5+1 \\ 4+10+6+1 \\ 1+10+15+7+1 \\ 5+20+21+8+1 \\ 1+15+35+28+9+1 \\ \end{eqnarray*} \dsp } Alice computed the sums out loud. ``The fourth sum is $2+1$. That's 3. The fifth sum is $1+3+1$ which is 5. Then $3+4+1=8$. The next one is $1+6+5+1$ which is 13. Then comes $4+10+6+1$ or 21. After that should be 34. $1+10+15+7+1$ is 34. And $5+20+21+8+1$ is 55, and the Fibonacci number after 21 and 34 is $21+34$ or 55. The last one should be $55+34=89$. Let's see, $1+15+35$ is 51 and $51+28+9+1$ is 89. So they are the Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.'' Alice was now even more impressed by Fibonacci numbers and Isabel's triangle. ``Amazing, isn't it?'' said Isabel. ``It's such a simple triangle of sums, and yet it has so many special number sequences.'' ``Hello,'' called out a voice. ``Oh, good,'' said Isabel. ``Gus is home.'' She called out, ``We're in the living room, Gus. The Yellow Pig brought over a friend.'' A green turtle, standing upright, walked into the room. The Yellow Pig came back. ``Gus, how good to see you.'' They greeted each other enthusiastically. He did the introductions. ``Alice, this is Gus. Gus, this is Alice.'' ``How do you do?'' asked Gus. ``Fine,'' said Alice. ``Isabel said I should ask you about the triangular numbers.'' ``Ah,'' said Gus. ``Yes, I was quite a rascal in my youth. Why, I was about your age I would imagine.'' ``When what?'' asked Alice. ``When I was in a very boring class at school. The teacher loved to give us busy work so we wouldn't bother him. It infuriated me.'' ``I'll bet,'' said Alice, who was no fan of busy work herself. ``I didn't like the teacher, and he didn't like me. He would give us long lists of numbers to sum. One day he told us to sum the numbers from one to one hundred.'' ``Egads,'' said Alice. ``That's a lot of numbers.'' ``It is,'' said Gus. ``And I certainly didn't feel like adding them up. My classmates began diligently summing, but I stared off into space. Suddenly, it came to me. A simple way to add up the numbers.'' ``How?'' asked Alice, intrigued. ``I wrote down the numbers 1, 2, 3, 4, 5, 6, \ldots in a row, but only up to 50. And below them I wrote down the numbers 100, 99, 98, 97, 96, 95, \ldots . Then I looked at the columns I had created. Each column sums to 101.'' ``You're right,'' said Alice. ``How many such sums are there?'' ``There are 50 of them. Half of 100 because there are 100 numbers to add up. So the sum is the same as $50 \cdot 101$. That's 5050. And that was the answer. If you are adding up the numbers from 1 to $n$, the sum is just $\frac{n(n+1)}{2}$.'' ``So what happened?'' Alice asked. ``Did you tell the teacher you had the answer?'' ``I did,'' said Gus. ``At first he didn't believe me. How could I have possibly finished the sums so quickly? Everyone else was still adding up the first ten numbers. But finally he looked at my answer, and after I explained how I had gotten it, he conceded that I was right. He also gave me a bunch of math books to read. He turned out to be an okay teacher after all.'' ``You made his job fairly difficult,'' interjected the Yellow Pig. ``Well, yes,'' admitted Gus. ``That's how I got the nickname `rascal'.'' ``You've done so much math,'' said the Pig. ``Algebra, Diophantine equations, differential geometry, and my favorite, the construction of the regular 17-sided polygon. You've done a lot of math, too, Isabel: numbers, cycloids, and all sorts of interesting things.'' Isabel, Gus, and the Pig talked for awhile, catching up on old times. Alice half-listened, and half-studied the numbers on the triangle that she and Isabel had drawn. Finally, the Pig rose, and they all shook hands. ``It was wonderful meeting you Alice,'' said Isabel. ``You, too,'' Alice said. ``And thank you for your story, Gus.'' They walked Alice and the Pig to the door, welcoming her back any time. \section{Primes} Outside, the Pig said, ``I realized that there's a lot about numbers that I have to tell you.'' ``Like what?'' asked Alice. ``Let's find a place where we can talk,'' the Pig said. They walked away from the garden. ``It's hot,'' Alice said. The Pig led Alice to a gazebo where they could sit in the shade. ``One very important set of numbers is the set of prime numbers,'' he said. ``Prime numbers are natural numbers --- remember, they're the counting numbers --- with exactly two divisors. The only even prime is 2, being divisible by 1 and itself. The next several primes are 3, 5, 7, 11, 13, and 17.'' ``17 again!'' Alice exclaimed. ``It does seem to show up all over the place.'' ``That it does. But there are a lot of primes. I'll show you how you can find lots more primes,'' he said, writing the numbers in order from 1 to 100 on 10 lines. ``We are going to strike out all of the numbers which are not primes and circle the numbers which are.'' ``1 is a special case; it isn't a prime because it is only divisible by 1, so we'll strike that out. We circle 2 because it is a prime. And we can strike out all multiples of 2. Now we do the same for the next number. We circle 3 and strike out all of its multiples. The next number is 5 which is prime. Again, we circle it and striking out multiples.'' He continued circling and striking out numbers. \centerline{\epsfbox{images/2-primes.ps}} ``So the primes less than 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.'' ``All numbers can be written as the product of primes. If a number is prime, it is already factored into primes. And if it isn't, just start factoring it. If you don't consider the order of the factors, every number has exactly one such factorization,'' the Pig told Alice. ``That's an extremely critical fact in number theory. It is known as the Fundamental Theorem of Arithmetic. It may seem fairly basic, but it's essential for an awful lot of proofs about prime numbers. What do you notice about our primes?'' he asked. ``None of them end in even numbers or 5's. We crossed out a whole bunch of columns in our table. And it seems like there are getting to be fewer and fewer primes as we go along,'' remarked Alice. ``There are a lot more primes less than 50 than there are between 50 and 100.'' ``That's very true,'' said the Pig. ``Mathematicians even know approximately how rapidly they are growing apart.'' ``Do you know if the prime numbers end? Or are there always more?'' Alice asked. ``There are always more,'' the Pig said in a tone of authority. ``There are infinitely many primes. In fact, Euler came up with one of my favorite proofs of this. Would you like me to demonstrate it for you?'' Alice wasn't sure she would be able to understand it, but the Pig seemed eager to show her, so she agreed. ``First let me state what it is I want to show. I want to show that there are infinitely many primes. Instead of proving this directly, I'll try to prove that there are finitely many primes. When I reach a contradiction, there's my proof. Contradiction is one of the many tricks mathematicians have up their sleeves when they try to prove something. Mathematicians are like magicians that way.'' The Pig waved his hooves extravagantly, as if clearing the air for his proof. ``The proof goes something like this: Suppose there are only finitely many primes. This `suppose' part is important. It means that we just assume the statement is true. Then we'll see what conclusions follow from it and whether or not they make sense. If there are finitely many primes, no matter how many there are, I can write an ordered list that contains them all. Now we name the primes, from smallest to largest, using boring names: $p_{1}$, $p_{2}$, $p_{3}$, \ldots , all the way up to $p_{n}$. Then $p_{n}$ is the largest prime on our list; that is, it is the largest prime. ``But,'' he continued. ``The `but' part is the key to contradiction, you see. But,'' he repeated ``I will now construct another prime, which isn't on this list. Let me outline the procedure for generating a new prime. First, I multiply all of the primes on the list together. The resulting number is definitely not prime. It's divisible by every prime. But, what happens when we add 1 to that number? I claim this new number $(p_{1} \cdot \cdots \cdot p_{n})+1$ is a prime.'' ``How can you be so sure?'' asked Alice. ``You don't even know what those $p$ numbers are.'' ``Well, I'll show you. If a number isn't prime, that means it has prime divisors. So, to show that this large number isn't prime, we just have to find one number in our list which divides it. If none of the numbers on our list divide the new number, then it must be prime. It's not too hard to see what happens when I divide this big old $(p_{1} \cdot \cdots \cdot p_{n})+1$ by $p_{1}$ and $p_{2}$ and $p_{3}$ and so on. When I divide $(p_{1} \cdot \cdots \cdot p_{n})$ by $p_{1}$ I'm going to find that it divides evenly. That's how I constructed my number. But when I divide $p_{1}$ into $(p_{1} \cdot \cdots \cdot p_{n})+1$, I'm left with a remainder of 1. That means my conjectured prime is not divisible by $p_{1}$.'' ``That's just one divisor,'' pointed out Alice. ``You're right,'' said the Pig. ``But there's absolutely nothing that makes $p_{1}$ any different from $p_{2}$ or any of those other enumerated primes. By the same reasoning, $p_{2}$ evenly divides $(p_{1} \cdot \cdots \cdot p_{n})$, so it can't evenly divide $(p_{1} \cdot \cdots \cdot p_{n})+1$. The same is true when I divide by the other primes. The remainder will always be 1. None of the primes in my list are divisors of my new number because that's precisely the way I generated my number. And since we can't factor $(p_{1} \cdot \cdots \cdot p_{n})+1$, it follows that it must be prime,'' he concluded. ``At the beginning I said --- or supposed --- that we had a finite list of all the primes. And now we know that is false. I couldn't have had a finite list of all the prime numbers because I have shown you a prime number that isn't on such a list. And what's even neater is that I can never have a complete finite list of primes. Even if I add my new prime number to the list, I can repeat my algorithm again to create a new prime. No matter how many primes you list, I can always find a prime that isn't on the list. Therefore, there must be infinitely many primes!'' He paused to catch his breath, and Alice had a chance to think about what he had said. ``I'm not sure about all that contradiction stuff, but I see how you got a new prime. It's almost simple, really,'' she said. ``It's one of my favorite proofs because it is so simple. Euler was very clever to come up with it. He designed the bridges here, too. They are near the art gallery, which is one of my favorite places to be. I like it almost as much as the garden. Have you had enough of primes, or shall I tell you more about them?'' ``Oh more. They sound interesting,'' said Alice, who was really starting to get interested in all of this math the Pig kept spouting. It was much more fun than math was in her school. She thought that when she returned home, she would have to sit her stuffed animals down and teach them everything she had learned. The Pig pondered his next topic. ``Another question is how many twin primes there are. Twin primes are two prime numbers whose difference is 2. Like 3 and 5, or 5 and 7, or 11 and 13. Just like the primes, the larger the number, the fewer twin primes you will find. Unlike the question of how many primes there are, the answer is still unknown. A man by the name of Goldbach conjectured that there were an infinite number of twin prime pairs, but no one knows for sure. If you can figure it out, why you'll be famous. There are lots of open problems in mathematics like that. It's very exciting. There was one problem, sort of like the Pythagorean theorem only much more complicated, that was unsolved for hundreds of years before its proof was discovered. There are lots of exciting things like that happening in number theory. ``Prime numbers are the basis of numbers and number theory. Another important concept is that of modular arithmetic.'' ``Modular arithmetic?'' repeated Alice. ``What's that?'' ``It's a way of adding that only deals with remainders. Maybe the easiest way to explain is to give an example.'' He pointed at his watch. ``When we tell time we often use modular arithmetic. It's 1:00, then 2:00, then 3:00, 4:00, 5:00, 6:00, 7:00, 8:00, 9:00, 10:00, 11:00, and 12:00. And then after that it's 1:00 again and 2:00 again and so on. And 1:00 is really the same as 13:00 and 2:00 is the same as 14:00. We can go around again. Then, 24 hours after 12:00, it will be 12:00 again. And 25 hours after 12:00, it will be 1:00. We don't really care that it's 97 hours later at 1:00 PM on Friday than it was at noon that Monday. We just care that it's 1:00. That means one hour after the most recent noon. And that's a quick example of arithmetic modulo 12, or mod 12 for short,'' the Pig explained. ``If I ask you what time it will be 30 hours from now, I expect an answer between 1 and 12. In other words, I expect you to do the addition and then subtract off the closest lower multiple of 12 to find the remainder when the number is divided by 12. ``Here's a problem for you,'' said the Pig. ``Find the remainder when 30 is divided by 12. We'll call that number Rudolph since you like creative names. Solve for Rudolph the remainder.'' Alice wasn't sure if she understood. She wondered how long it had been since she met the Yellow Pig. She wondered how long it would be until she found her teddy bear. She wondered if it would like some honey. She could try to find some honey for it. She began uncertainly, ``I subtract 12 from 30. That leaves me 18.'' There was a long pause before the Pig suggested, ``18 is still bigger than 12. Try subtracting 12 again.'' Alice did. ``So 18 minus 12 is 6. Is that the answer? Is Rudolph 6?'' ``Rudolph is 6,'' the Pig said. He plunged onward, ``Now instead of 12, let's study a different modulus. Number theorists like to consider prime moduli. Prime numbers are good building blocks in modular arithmetic. With the watch, we considered addition. Now let's look at multiplication in, say, mod 7. Because it's mod 7 and we are interested only in remainders, we just look at the whole numbers from 0 to 6 inclusive. For example, 9 is the same as 2 because $9-7=2$. And $6 \cdot 2$ isn't 12 as it usually is. Instead, it's $12-7$ or 5. Get it?'' the Pig asked. Alice began to nod, and then asked, ``What does 17 become in this new system? When we subtract off 7, we are left with 10 which is still too large. So we subtract off 7 again. We just keep subtracting off 7 until the remainder is less than 7. So 17 becomes $17-7-7=3$.'' ``Exactly,'' said the Pig, beaming at Alice. ``Let's look at some powers in mod 7, like squares and cubes and numbers raised to the fourth and fifth. We don't need to think about 1 very much because 1 raised to any power is 1. But what about 2? We see that $2^{1}=2$, and $2^{2}$ is $2 \cdot 2$ which is 4. Next, $2^{3}$ is $4 \cdot 2$ or 8, but our numbering system only goes up to 7. Since 8 is larger than 7, we have to subtract 7 from it to find the remainder. So $2^{3}$ is $8-7$ or 1 in our system. Actually, we say that 8 is congruent to 1 mod 7. Mathematicians write congruence using an equal sign with an extra line. Like this,'' he said, writing: $$ 8 \equiv 1 \pmod{7} $$ ``So $2^{4}=2$, because $1 \cdot 2$ is 2. And we're in a loop. Again, $2 \cdot 2=4$, $4 \cdot 2$ is 1 again. I'll write out the first six powers of 2 in mod 7: 2, 4, 1, 2, 4, 1. Now you try the powers of 3,'' the Pig instructed. ``The powers of 3 are 3, then $3 \cdot 3$ or 9,'' she paused. ``And 9 is larger than 7 so I have to reduce it.'' She frowned. ``Subtract 7 from 9,'' prompted the Yellow Pig. ``So $9-7$ is 2,'' finished Alice. ``The powers of 3 are 3, 2, and then $2 \cdot 3$ which is 6. Then $6 \cdot 3$ which is 18. That's a lot larger than 7. If I subtract 7, it's 11 which is still larger than 7. So we subtract again: $11-7$ is 4. Then $4 \cdot 3$. It's not repeating this time.'' ``It will eventually,'' assured the Pig. ``Okay, $4 \cdot 3$ is 12 and $12-7$ is 5; $5 \cdot 3$ is 15. And $15-7$ is 8 and $8-7$ is 1, and $1 \cdot 3$ is 3. Now it's repeating.'' ``Now write down the first six powers,'' said the Pig, and Alice wrote: 3, 2, 6, 4, 5, 1. ``I'll do the powers of 4 and 5.'' Quickly, he recited: ``We start with 4, then $4 \cdot 4=16 \equiv 2$, $2 \cdot 4=8 \equiv 1$, $1 \cdot 4=4$, 2, 1.'' ``Now for 5: $5 \cdot 5=25 \equiv 4$, $4 \cdot 5=20 \equiv 6$, $6 \cdot 5=30 \equiv 2$, $2 \cdot 5=10 \equiv 3$, $3 \cdot 5=15 \equiv 1$. You can do powers of 6.'' ``First is 6, then $6 \cdot 6=36$. That's 1 more than 35 which is a multiple of 7,'' observed Alice, pleased that she didn't need to keep subtracting 7's and could just subtract 35 instead. ``So 36 is congruent to 1. And $1 \cdot 6$ is 6. And $6 \cdot 6$ is 1 again. Also, $1 \cdot 6$ is 6 again. And $6 \cdot 6$ is congruent to 1 again. That one looped quickly.'' While she was talking, the Pig had turned to his notebook again and had written: \ssp \begin{center} $ \begin{array}{cccccc} {\bf n} & {\bf n^{2}} & {\bf n^{3}} & {\bf n^{4}} & {\bf n^{5}} & {\bf n^{6}} \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 4 & 1 & 2 & 4 & 1 \\ 3 & 2 & 6 & 4 & 5 & 1 \\ 4 & 2 & 1 & 4 & 2 & 1 \\ 5 & 4 & 6 & 2 & 3 & 1 \\ 6 & 1 & 6 & 1 & 6 & 1 \\ \end{array} $ \end{center} \dsp ``They all end in 1,'' remarked Alice, looking over at the Pig's notebook. ``And that's no coincidence,'' assured the Pig. ``If we had chosen mod 17 and written out the first 16 powers, we would have noticed the same thing. It works because 17 and 7 are both prime numbers, and primes are very special numbers. If you look at $n^{3}$ you will see that we always got 1 or 6. In mod 17 we would have always gotten 1 or 16. That's another important result in number theory.'' Alice decided to take the Pig's word on this. He seemed to know so much. She told him so. ``In fact,'' she continued, twirling her hair anxiously, ``you know so much that maybe you have an idea where I might find my bear.'' The Yellow Pig paused. Alice couldn't tell if he was gazing off in space ignoring or if he was focusing on her question. At last he said, ``If I were a lost bear, I'd probably look around for awhile, explore the area. I might end up near the water to catch some fish. Bears like fish don't they?'' ``Yes,'' said Alice. ``And berries. Though my bear isn't much of an eater.'' ``Hmmm \ldots . If I weren't hungry, I might go somewhere indoors after my exploration. Like to the art gallery. We can go there.'' ``I'd like that very much,'' said Alice. ``Okay,'' agreed the Pig. ``I have one last riddle about modular arithmetic. It's a Chinese riddle about elephants, but you can pretend it's about bears. A commander wanted to organize his elephants in rows so that there were the same number of elephants in each row. He tried organizing them in rows of 4, but there was 1 left over. He tried organizing them in rows of 3, but there was one left over again. He tried rows of 2, but there was still one elephant left out. Finally he tried rows of 5 and that worked. Later, he was telling a fellow commander about his herd of elephants and couldn't remember how many of the beasts he had taken to battle. He knew it was between 50 and 100 elephants. It bothered him that he didn't know how many elephants there were, but the other commander told him not to worry about it. He could figure it out from the information he had been given. Can you?'' challenged the Pig. Alice was fairly certain she could given enough time, but she didn't have the faintest idea how to approach the problem mathematically. ``It must have something to do with modular arithmetic. Mods 2, 3, 4, and 5; all of those except 4 are prime numbers,'' she added. ``That's exactly right, and the fact that they are prime is an excellent observation. Here's how the commander solved the problem. He didn't know how many elephants there were, so he decided to say that there were $x$. Now, $x$ is just some number which happens to be our solution.'' ``I think Rudolph was a much more creative name than $x$,'' interrupted Alice. ``Maybe,'' said the Pig, ``but $x$ is the answer to our mystery, like an X that marks the spot of buried treasure. We know that when $x$ is divided by 4, the remainder is 1. That's the same as saying $x \equiv 1 \pmod{4}$. We know that when $x$ is divided by 3 and 2, the remainder is also 1, so $x \equiv 1 \pmod{3}$ and $x \equiv 1 \pmod{2}$. And finally, we know that $x$ is evenly divisible by 5, so $x \equiv 0 \pmod{5}$.'' He licked his hooves and continued, ``Now we just need to find a number $x$ that satisfies these properties. If a number leaves a remainder of 1 when divided by 2, 3, and 4, it will leave a remainder of 1 when divided by 12. It's like that thing we did to make a large prime from our list. Because 2, 3, and 4 all divide 12, any number that is 1 more than a multiple of 12 has to be 1 more than a multiple of 2, 1 more than a multiple of 3, and 1 more than a multiple of 4. Like 13, which is $2 \cdot 6 + 1$ and $2 \cdot 3 \cdot 4 + 1$, or 25. It has a remainder of 1 when divided by 2. And it also has a remainder of 1 when divided by 3 or 4. This 12 is extremely useful in simplifying our problem. It enables us to combine three equations into just one. All because 12 is a common multiple of 2, 3, and 4. It's also the least common multiple; that is, 12 is the smallest number that all three of those numbers divide evenly. Now we only have to look at numbers that are congruent to 1 modulo 12. And, in fact, we only have to look at numbers from 50 to 100.'' The Pig thought quickly, ``Those happen to be 61, 73, 85, and 97. We have one more piece of information. The commander also remembered that the number of elephants was divisible by 5.'' ``Only one of those numbers is divisible by 5,'' interrupted Alice. `85. There must have been 85 ephelants.'' The Pig smiled. ``There were 85 elephants. Neat, isn't it?'' Alice agreed that it was. ``The first commander thought it was so neat that he tried to generalize his solution for other similar problems. Mathematicians do that an awful lot. They say `this works in these cases, now I can generalize it for any $n$'. Often they use the word `induction'. Induction is another one of those tricks mathematicians use when tackling proofs. This problem has been generalized, but there's some pretty messy mathematics involved. Why, just the statement of the result is complicated. It goes something like this: If $x \equiv a_{1} \pmod{m_{1}}$ and $x \equiv a_{2} \pmod{m_{2}}$, then $x \equiv (a_{2}-a_{1})pm_{1}+a_{1} \equiv (a_{2}-a_{1})qm_{2}+a_{2} \pmod{m_{1}m_{2}}$, where $pm_{1}-qm_{2}=1$.'' ``Egads!'' exclaimed Alice. ``That sounds horribly complicated.'' ``It is. I think we're ready to move onto something else. Number theory is the queen of mathematics, but there is so much more to math.'' ``Like what?'' Alice asked. ``Well, combinatorics for one,'' answered the Pig, and Alice settled down for another of the Pig's lectures.