“You’ll soon wear it out using it for everything,” replied the Dodecahedron. “Now I have one for smiling, one for laughing, one for crying, one for frowning, one for thinking, one for pouting, and six more besides. Is everyone with one face called a Milo?”
“Oh no,” Milo replied; “some are called Henry or George or Robert or John or lots of other things.”
“How terribly confusing,” he cried. “Everything here is called exactly what it is. The triangles are called triangles, the circles are called circles, and even the same numbers have the same name. Why, can you imagine what would happen if we named all the twos Henry or George or Robert or John or lots of other things? You’d have to say Robert plus John equals four, and if the four’s name were Albert, things would be hopeless.”
“I never thought of it that way,” Milo admitted.
“Then I suggest you begin at once,” admonished the Dodecahedron from his admonishing face, “for here in Digitopolis everything is quite precise.”
“Then perhaps you can help us decide which road to take,” said Milo.
“By all means,” he replied happily. “There’s nothing to it. If a small car carrying three people at thirty miles an hour for ten minutes along a road five miles long at 11:35 in the morning starts at the same time as three people who have been traveling in a little automobile at twenty miles an hour for fifteen minutes on another road exactly twice as long as one half the distance of the other, while a dog, a bug, and a boy travel an equal distance in the same time or the same distance in an equal time along a third road in mid-October, then which one arrives first and which is the best way to go?”
“Seventeen!” shouted the Humbug, scribbling furiously on a piece of paper.
“Well, I’m not sure, but——” Milo stammered after several minutes of frantic figuring.
“You’ll have to do better than that,” scolded the Dodecahedron, “or you’ll never know how far you’ve gone or whether or not you’ve ever gotten there.”
“I’m not very good at problems,” admitted Milo.
“What a shame,” sighed the Dodecahedron. “They’re so very useful. Why, did you know that if a beaver two feet long with a tail a foot and a half long can build a dam twelve feet high and six feet wide in two days, all you would need to build Boulder Dam is a beaver sixty-eight feet long with a fifty-one-foot tail?”
“Where would you find a beaver that big?” grumbled the Humbug as his pencil point snapped.
“I’m sure I don’t know,” he replied, “but if you did, you’d certainly know what to do with him.”
“That’s absurd,” objected Milo, whose head was spinning from all the numbers and questions.
“That may be true,” he acknowledged, “but it’s completely accurate, and as long as the answer is right, who cares if the question is wrong? If you want sense, you’ll have to make it yourself.”
“All three roads arrive at the same place at the same time,” interrupted Tock, who had patiently been doing the first problem.
“Correct!” shouted the Dodecahedron. “And I’ll take you there myself. Now you can see how important problems are. If you hadn’t done this one properly, you might have gone the wrong way.”
“I can’t see where I made my mistake,” said the Humbug, frantically rechecking his figures.
“But if all the roads arrive at the same place at the same time, then aren’t they all the right way?” asked Milo.
“Certainly not!” he shouted, glaring from his most upset face. “They’re all the wrong way. Just because you have a choice, it doesn’t mean that any of them has to be right.”
He walked to the sign and quickly spun it around three times. As he did, the three roads vanished and a new one suddenly appeared, heading in the direction that the sign now pointed.
“Is every road five miles from Digitopolis?” asked Milo.
“I’m afraid it has to be,” the Dodecahedron replied, leaping onto the back of the car. “It’s the only sign we’ve got.”
The new road was quite bumpy and full of stones, and each time they hit one, the Dodecahedron bounced into the air and landed on one of his faces, with a sulk or a smile or a laugh or a frown, depending upon which one it was.
“We’ll soon be there,” he announced happily, after one of his short flights. “Welcome to the land of numbers.”
“It doesn’t look very inviting,” the bug remarked, for, as they climbed higher and higher, not a tree or a blade of grass could be seen anywhere. Only the rocks remained.
“Is this the place where numbers are made?” asked Milo as the car lurched again, and this time the Dodecahedron sailed off down the mountainside, head over heels and grunt over grimace, until he landed sad side up at what looked like the entrance to a cave.
“They’re not made,” he replied, as if nothing had happened. “You have to dig for them. Don’t you know anything at all about numbers?”
“Well, I don’t think they’re very important,” snapped Milo, too embarrassed to admit the truth.
“NOT IMPORTANT!” roared the Dodecahedron, turning red with fury. “Could you have tea for two without the two—or three blind mice without the three? Would there be four corners of the earth if there weren’t a four? And how would you sail the seven seas without a seven?”
“All I meant was——” began Milo, but the Dodecahedron, overcome with emotion and shouting furiously, carried right on.
“If you had high hopes, how would you know how high they were? And did you know that narrow escapes come in all different widths? Would you travel the whole wide world without ever knowing how wide it was? And how could you do anything at long last,” he concluded, waving his arms over his head, “without knowing how long the last was? Why, numbers are the most beautiful and valuable things in the world. Just follow me and I’ll show you.” He turned on his heel and stalked off into the cave.
“Come along, come along,” he shouted from the dark hole. “I can’t wait for you all day.” And in a moment they’d followed him into the mountain.
It took several minutes for their eyes to become accustomed to the dim light, and during that time strange scratching, scraping, tapping, scuffling noises could be heard all around them.
“Put these on,” instructed the Dodecahedron, handing each of them a helmet with a flashlight attached to the top.
“Where are we going?” whispered Milo, for it seemed like the kind of place in which you whispered.
“We’re here,” he replied with a sweeping gesture. “This is the numbers mine.”
Milo squinted into the darkness and saw for the first time that they had entered a vast cavern lit only by a soft, eerie glow from the great stalactites which hung ominously from the ceiling. Passages and corridors honeycombed the walls and wound their way from floor to ceiling, up and down the sides of the cave. And, everywhere he looked, Milo saw little men no bigger than himself busy digging and chopping, shoveling and scraping, pulling and tugging carts full of stone from one place to another.
“Right this way,” instructed the Dodecahedron, “and watch where you step.”
As he spoke, his voice echoed and re-echoed and re-echoed again, mixing its sound with the buzz of activity all around them. Tock trotted along next to Milo, and the Humbug, stepping daintily, followed behind.
“Whose mine is it?” asked Milo, stepping around two of the loaded wagons.
“BY THE FOUR MILLION EIGHT HUNDRED AND TWENTY-SEVEN THOUSAND SIX HUNDRED AND FIFTY-NINE HAIRS ON MY HEAD, IT’S MINE, OF COURSE!” bellowed a voice from across the cavern. And striding toward them came a figure who could only have been the Mathemagician.
He was dressed in a long flowing robe covered entirely with complex mathematical equations and a tall pointed cap that made him look very wise. In his left hand he carried a long staff with a pencil point at one end and a large rubber eraser at the other.
“It’s a lovely mine,” apologized the Humbug, who was always intimidated by loud noises.
/>
“The biggest number mine in the kingdom,” said the Mathemagician proudly.
“Are there any precious stones in it?” asked Milo excitedly.
“PRECIOUS STONES!” he roared, even louder than before. And then he leaned over toward Milo and whispered softly, “By the eight million two hundred and forty-seven thousand three hundred and twelve threads in my robe, I’ll say there are. Look here.”
He reached into one of the carts and pulled out a small object, which he polished vigorously on his robe. When he held it up to the light, it sparkled brightly.
“But that’s a five,” objected Milo, for that was certainly what it was.
“Exactly,” agreed the Mathemagician; “as valuable a jewel as you’ll find anywhere. Look at some of the others.”
He scooped up a great handful of stones and poured them into Milo’s arms. They included all the numbers from one to nine, and even an assortment of zeros.
“We dig them and polish them right here,” volunteered the Dodecahedron, pointing to a group of workers busily employed at the buffing wheels; “and then we send them all over the world. Marvelous, aren’t they?”
“They are exceptional,” said Tock, who had a special fondness for numbers.
“So that’s where they come from,” said Milo, looking in awe at the glittering collection of numbers. He returned them to the Dodecahedron as carefully as possible but, as he did, one dropped to the floor with a smash and broke in two. The Humbug winced and Milo looked terribly concerned.
“Oh, don’t worry about that,” said the Mathemagician as he scooped up the pieces. “We use the broken ones for fractions.”
“Haven’t you any diamonds or emeralds or rubies?” asked the bug irritably, for he was quite disappointed in what he’d seen so far.
“Yes, indeed,” the Mathemagician replied, leading them to the rear of the cave; “right this way.”
There, piled into enormous mounds that reached almost to the ceiling, were not only diamonds and emeralds and rubies but also sapphires, amethysts, topazes, moonstones, and garnets. It was the most amazing mass of wealth that any of them had ever seen.
“They’re such a terrible nuisance,” sighed the Mathemagician, “and no one can think of what to do with them. So we just keep digging them up and throwing them out. Now,” he said, taking a silver whistle from his pocket and blowing it loudly, “let’s have some lunch.”
And for the first time in his life the astonished bug couldn’t think of a thing to say.
15. This Way to Infinity
Into the cavern rushed eight of the strongest miners carrying an immense caldron which bubbled and sizzled and sent great clouds of savory steam spiraling slowly to the ceiling. A sweet yet pungent aroma hung in the air and drifted easily from one anxious nose to the other, stopping only long enough to make several mouths water and a few stomachs growl. Milo, Tock, and the Humbug watched eagerly as the rest of the workers put down their tools and gathered around the big pot to help themselves.
“Perhaps you’d care for something to eat?” said the Mathemagician, offering each of them a heaping bowlful.
“Yes, sir,” said Milo, who was beside himself with hunger.
“Thank you,” added Tock.
The Humbug made no reply, for he was already too busy eating, and in a moment the three of them had finished absolutely everything they’d been given.
“Please have another portion,” said the Mathemagician, filling their bowls once more; and as quickly as they’d finished the first one the second was emptied, too.
“Don’t stop now,” he insisted, serving them again,
and again,
and again,
and again,
and again,
“How very strange,” thought Milo as he finished his seventh helping. “Each one I eat makes me a little hungrier than the one before.”
“Do have some more,” suggested the Mathemagician, and they continued to eat just as fast as he filled the plates.
After Milo had eaten nine portions, Tock eleven, and the Humbug, without once stopping to look up, twenty-three, the Mathemagician blew his whistle for a second time and immediately the pot was removed and the miners returned to work.
“U-g-g-g-h-h-h,” gasped the bug, suddenly realizing that he was twenty-three times hungrier than when he started, “I think I’m starving.”
“Me, too,” complained Milo, whose stomach felt as empty as he could ever remember; “and I ate so much.”
“Yes, it was delicious, wasn’t it?” agreed the pleased Dodecahedron, wiping the gravy from several of his mouths. “It’s the specialty of the kingdom—subtraction stew.”
“I have more of an appetite than when I began,” said Tock, leaning weakly against one of the larger rocks.
“Certainly,” replied the Mathemagician; “what did you expect? The more you eat, the hungrier you get. Everyone knows that.”
“They do?” said Milo doubtfully. “Then how do you ever get enough?”
“Enough?” he said impatiently. “Here in Digitopolis we have our meals when we’re full and eat until we’re hungry. That way, when you don’t have anything at all, you have more than enough. It’s a very economical system. You must have been quite stuffed to have eaten so much.”
“It’s completely logical,” explained the Dodecahedron. “The more you want, the less you get, and the less you get, the more you have. Simple arithmetic, that’s all. Suppose you had something and added something to it. What would that make?”
“More,” said Milo quickly.
“Quite correct,” he nodded. “Now suppose you had something and added nothing to it. What would you have?”
“The same,” he answered again, without much conviction.
“Splendid,” cried the Dodecahedron. “And suppose you had something and added less than nothing to it. What would you have then?”
“FAMINE!” roared the anguished Humbug, who suddenly realized that that was exactly what he’d eaten twenty-three bowls of.
“It’s not as bad as all that,” said the Dodecahedron from his most sympathetic face. “In a few hours you’ll be nice and full again—just in time for dinner.”
“Oh dear,” said Milo sadly and softly. “I only eat when I’m hungry.”
“What a curious idea,” said the Mathemagician, raising his staff over his head and scrubbing the rubber end back and forth several times on the ceiling. “The next thing you’ll have us believe is that you only sleep when you’re tired.” And by the time he’d finished the sentence, the cavern, the miners, and the Dodecahedron had vanished, leaving just the four of them standing in the Mathemagician’s workshop.
“I often find,” he casually explained to his dazed visitors, “that the best way to get from one place to another is to erase everything and begin again. Please make yourself at home.”
“Do you always travel that way?” asked Milo as he glanced curiously at the strange circular room, whose sixteen tiny arched windows corresponded exactly to the sixteen points of the compass. Around the entire circumference were numbers from zero to three hundred and sixty, marking the degrees of the circle, and on the floor, walls, tables, chairs, desks, cabinets, and ceiling were labels showing their heights, widths, depths, and distances to and from each other. To one side was a gigantic note pad set on an artist’s easel, and from hooks and strings hung a collection of scales, rulers, measures, weights, tapes, and all sorts of other devices for measuring any number of things in every possible way.
“No indeed,” replied the Mathemagician, and this time he raised the sharpened end of his staff, drew a thin straight line in the air, and then walked gracefully across it from one side of the room to the other. “Most of the time I take the shortest distance between any two points. And, of course, when I should be in several places at once,” he remarked, writing 7 × 1 = 7 carefully on the note pad, “I simply multiply.”
Suddenly there were seven Mathemagicians standing side by side, and each one
looked exactly like the other.
“How did you do that?” gasped Milo.
“There’s nothing to it,” they all said in chorus, “if you have a magic staff.” Then six of them canceled themselves out and simply disappeared.
“But it’s only a big pencil,” the Humbug objected, tapping at it with his cane.
“True enough,” agreed the Mathemagician; “but once you learn to use it, there’s no end to what you can do.”
“Can you make things disappear?” asked Milo excitedly.
“Why, certainly,” he said, striding over to the easel. “Just step a little closer and watch carefully.”
After demonstrating that there was nothing up his sleeves, in his hat, or behind his back, he wrote quickly:
4 + 9 − 2 × 16 + 1 ÷ 3 × 6 − 67 + 8 × 2 − 3 + 26 − 1 ÷ 34 + 3 ÷ 7 + 2 − 5 =
Then he looked up expectantly.
“Seventeen!” shouted the bug, who always managed to be first with the wrong answer.
“It all comes to zero,” corrected Milo.
“Precisely,” said the Mathemagician, making a very theatrical bow, and the entire line of numbers vanished before their eyes. “Now is there anything else you’d like to see?”
“Yes, please,” said Milo. “Can you show me the biggest number there is?”
“I’d be delighted,” he replied, opening one of the closet doors. “We keep it right here. It took four miners just to dig it out.”
Inside was the biggest Milo had ever seen. It was fully twice as high as the Mathemagician.
“No, that’s not what I mean,” objected Milo. “Can you show me the longest number there is?”
“Surely,” said the Mathemagician, opening another door. “Here it is. It took three carts to carry it here.”
Inside this closet was the longest imaginable. It was just about as wide as the three was high.
“No, no, no, that’s not what I mean either,” said Milo, looking helplessly at Tock.
“I think what you would like to see,” said the dog, scratching himself just under half-past four, “is the number of greatest possible magnitude.”