“Lenora,” whispered Lucy, pressing close against her, “you were right about the creeps. Are they going to come after us?”
Lenora shook her head. “Not here, I don’t think. The Forces of Darkness seem afraid of this place.”
“Why do you call them the Forces of Darkness?” said Lucy. “That sounds evil.”
“Lucy,” said Lenora as gently as possible, “it’s not something I invented. They have been known by that name for a very, very, very, very long time.”
“Daddy must not know that!” exclaimed Lucy. “He can’t possibly.”
“I don’t know,” said Lenora. And that was true, for she had never met the Director and couldn’t say. “Anyway, if they are afraid of this place, then they can’t have fired anyone here.”
And then she stopped, as she began to get the first glimmerings of an idea … if the Forces were afraid of this place, then maybe she could …
But her thoughts were interrupted when Lucy broke in. “So there must still be a librarian around here somewhere, right?”
“Oh, there is,” said a voice, and once again Lenora and Lucy leapt about a foot in the air. And now she’d completely forgotten the idea she’d almost had. Lenora was getting rather tired of this, and felt that these invisible librarians really needed to cut it out.
But this librarian was not invisible. A small boy stepped from behind the sign. He was fresh-faced and smiling, and dressed in very old-fashioned clothes. He had a librarian’s badge, which read:
MILTON SIROTTA
ONE GOOGOL
“You really didn’t need to scare us like that,” said Lenora crossly.
“I am deeply sorry,” replied Milton in a soft voice. “I sensed the Forces were near, and even though they are afraid to enter this section, I thought it best to hide nevertheless.”
“What exactly are they scared of, anyway?” asked Lucy.
“There are beings here that they cannot understand or control. And that frightens them more than anything else, as it should. They could destroy all books of history, and history would be lost to us forever. They could destroy all poetry, and that would be lost forever, too. But these beings—the numbers, math itself—even if you destroyed all books of math, it could be rediscovered. So here is something the Forces of Darkness are powerless against.”
That sounded to Lenora as though she and Lucy had nothing to fear, either, and Milton himself was utterly at ease—but she was determined to keep her guard up regardless. Of the many unexpected dilemmas she’d faced in the Library, this was one of the most unsettling. She needed more information. “Are you really Milton Sirotta?” she asked.
The boy shook his head. “No, not really. I am a googol, which is—”
“One followed by a hundred zeros,” interrupted Lucy. “We know.”
“Ah,” replied the boy. “Well, then you also know that I was named by nine-year-old Milton Sirotta. And so, for purposes of helping anyone who might enter this section (very few do), I have chosen to take on his form. You may address me as Milton, which I rather prefer to ‘Googol.’”
“I can relate,” said Lucy sympathetically. “I hate my real name, too.”
“All right, Milton,” said Lenora, a bit impatiently. “I need to know what the world’s largest number is.”
“Hmm,” said Milton. “That really depends on a number of factors. And they have their own opinions, of course.”
“They?” said Lenora.
“Yes,” said Milton. “Can’t you hear them?”
“The whispering,” said Lenora, whispering, too. Shadowy numbers whispering things she could not understand—she did not like this at all.
“Correct. They live out there, far off in the dark, at distances one can scarcely imagine.”
“What are they whispering about?”
“I’m not always sure,” said Milton. “I’m unsure most of the time, really.”
“But aren’t you a large number?” asked Lenora.
“Oh, yes. Very large. In fact, the total number of electrons, protons, and neutrons in the entire observable universe is less than me.” Lenora began scribbling furiously in her notebook. “Quite a lot less, in fact. But that’s really nothing. Milton also invented the term googolplex, which is one followed by a googol zeros. Next to that, I am nothing. If you wrote one digit of a googolplex on every Planck volume in the universe—”
“A Planck volume is basically the smallest thing that isn’t nothing,” Lenora explained to Lucy, who had just opened her mouth to ask.
“—you would run out of space long before you finished writing the number. It simply can’t be done. Even if you use regular English to write the number’s actual name, it’s ten tremilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliatrecentretrigin
milliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliatrecentretriginmilliamilliamilliatrecentretriginmilliamilliatrecentretriginmilliatrecenduotrigintillion. So you can see it’s rather long no matter how you look at it.”
At the beginning, Lenora had tried to write that down, but had quickly given up as Milton went along. “So a googolplex is the largest number?” she asked.
“Oh, no,” laughed Milton. “Not even close.”
“Well then, what is it?”
Milton said. “I don’t know. As I said, the great numbers are far beyond me.”
Lenora’s head was spinning at the thought of all this, but Lucy had her chin in her hand, deep in thought. “Why don’t we just go talk to them, then? They could tell us.”
“Talk to them?” said Lenora and Milton together. Lenora wasn’t sure about this, and Milton looked alarmed as well. These large numbers were sounding more dangerous all the time. Lenora wondered if the Forces of Darkness hadn’t been right to stay away. Still, she had no choice but to get an answer for her patron. She’d already let one patron down on Plato’s Republic, and she didn’t plan on that happening again. “Good idea, Lucy. Let’s do it. Is that all right with you, Milton? Can you take us to them?”
Milton looked uncertain, but he nodded. “I can. But you must proceed carefully. You are to the great numbers as a top quark is to you.”
“A top what?” said Lucy.
Lenora was very glad she had read Your Friends, the Subatomic Particles over the summer. “A top quark is an extremely tiny particle that lives for less than a trillionth of a trillionth of a second.”
“Yes,” said Milton. “And to the large numbers, you are little different from that. So to meet them, we must get large ourselves. Very large indeed. Give me your hands.”
They did. And then they grew—very, very, very, very large indeed.
CHAPTER NINE
Lenora and the Numbers
Lenora had once seen the glory of the stars from outer space when she had been tossed through the open door of a rocket ship. She recognized them again, but saw them so briefly it hardly counted, before she was staring down at a double-armed disc of stars disappearing beneath her feet—the Milky Way galaxy, dwindling to a speck. Around her, Lucy, and Milton were clusters of other galaxies, which in turn became smaller blurs of light until they couldn’t be seen at all.
Lenora could scarcely believe it, but she, Lucy, and Milton were zooming up, larger and larger, the trillions of galaxies streaking past like shooting stars, and then they were beyond even those, and now below her Lenora could see one collection of lights, all the universe in something like a bubble, but … different. She thought, but couldn’t be sure, that there were other bubblish things nearby, but her mind was too rattled to be sure of any of it. And then all that vanished away into nothing, also.
Lenora looked over at Lucy, whose eyes and mouth were wide with wonder. It seemed she, too, could understand what had happened, that they were larger, far larger now, than the universe, or perhaps many universes. Lenora’s brain trembled at the thought.
“Here we are,” said Milton. “This is where the large numbers live.” He was perfectly calm, though Lenora was on the verge of telling him to take them back to normal size. For something in her bones told her: Humans are not supposed to be here.
She held her tongue. Lucy looked thrilled, Milton probably did this all the time, and Lenora had a patron who needed an answer. She did want to get this over with as quickly as possible, though, and so she asked, “What now?”
Milton thought for a moment. “Let’s meet Graham’s number. He might be willing to speak with us. Not all of them are.” And then, though she could not tell how, they began moving, flying really, toward one of the lights that still surrounded them.
As they flew, Lenora remembered the third task Malachi had given her. And so she asked, “By the way—do you know who Zenodotus is?”
“Of course!” said Milton. “He was the first head librarian of the Library of Alexandria.”
“Oh!” said Lenora.
“What’s that?” said Lucy.
“It was considered the greatest library of the ancient world, founded in Egypt long ago,” said Lenora. And then, to Milton: “Do you know where Zenodotus is? I have to find him.”
“No,” said Milton, “I have not seem him in quite some time. But I remember him as a dashing man with a long, curled mustache, full of energy and life.”
Lucy spoke up. “Wouldn’t it be easy to find him? Let’s just go to this Library of Alex-whatever-ia and see if he’s there.”
“It’s not there anymore,” said Milton. “I’m not sure what happened, but I heard it was accidentally burned down in 48 BC by Julius Caesar.”
“What a dummy,” said Lucy.
Lenora scribbled L of A destroyed by Caesar’s fire—maybe in her notebook. Then she saw they were coming swiftly toward one of the lights, and soon Lenora could make out a man sitting on a chair amid a brilliant gleam. He wore a flat-brimmed straw hat and an elegant old striped suit. He sat with one leg crossed over the other and took no notice of them at all. He seemed deep in thought, and though his lips did not move, Lenora could hear a whispering coming from him.
The three stopped in front of the man, and Milton bowed low. Lenora and Lucy looked at each other, and then Lenora bowed deeply, too, while Lucy executed an elegant curtsy.
“Sir,” said Milton, “may we please speak with you for a moment? This librarian has a question.”
The man did not respond, still gazing at something faraway, still whispering.
Lenora whispered, too. “Who is this?”
Milton spoke low, as though he did not want to disturb the man. “This is Graham’s number. He is the result of certain calculations concerning the edges of many-dimensional cubes. He was once the largest number ever produced by a mathematical proof. But that record has since been broken.”
At this, the man’s head turned slowly toward them, and Milton immediately went silent. Graham’s number regarded each of them in turn for what seemed to Lenora like a terribly long time. First Lucy, then Lenora, and then his gaze, which was not a friendly one, landed on Milton.
“Why have you disturbed me?” he asked in a soft voice with just a hint of menace. “I am engaged in several very important conversations.”
“I apologize, sir,” said Milton, bowing again. “But this librarian is searching for the largest number. I thought you might be able to direct us.”
The man uncrossed his legs and looked hard at them. “Am I not large enough for you? What use could you possibly have for anyone larger?”
Lenora spoke up. “Sir, if I may ask—how large are you, exactly?”
Graham’s number looked at her, and his gaze softened slightly. “A librarian. I would like to tell you how large I am. However, there is nothing your mind can understand, nothing I can be compared to, that will aid you in understanding my vast size. I am far beyond the realm of your imagination.”
Lenora wondered how bold she dared be. “Thank you, sir.” And then she hesitated before saying, “Milton has told us there are larger numbers than even you. Perhaps you can tell me the largest of you?” She hoped this would not offend.
Graham’s number turned his gaze back to whatever he had been doing before. “Ask TREE(3), if it will speak to you. It is the next number that humans—you are human, are you not?—were able to discover beyond me.” And then he went back to his whispering. Lenora had a feeling he would not speak to them again.
The three silently withdrew, Milton still holding both girls’ hands. Lenora sensed hesitation from the boy. “Is something wrong?” she asked.
“It’s TREE(3),” replied Milton with a tremor in his voice. “I have never encountered it before. But I have heard it is not friendly, not friendly at all. Perhaps we should stop here.”
Lenora was worried, but firm. “I must have an answer for my patron. If you will not go on, please show me how to go myself, and yo
u and Lucy can go back and wait for me.”
“No way!” announced Lucy, who, against all odds, seemed to be having the time of her life. “I’m going with Lenora!”
“Very well,” said Milton. “It is better that I go with you, then. A mere googol may or may not be beneath notice in this realm, but two humans certainly will. No offense meant.”
“None taken,” said Lenora.
The children squeezed each other’s hands tightly as they flew off again, this time toward a brilliant light farther above. As they went, Milton said, “I might as well tell you that TREE(3) is just a way of counting up ‘trees’ you can draw like this.” He released Lenora’s hand long enough to borrow her notebook.
He continued, “They’re drawn according to certain rules. It’s quite a bit simpler to understand than Graham’s number, but … well, we shall see what TREE(3) has to say.”
Now they were nearing something absolutely enormous—Lenora was not surprised to see it was a tree, but of such massive size that its branches extended in all directions farther than the eye could see, and the light that poured from it was nearly blinding. The three of them were as small before it as a tardigrade would be in front of Devils Tower. It reminded Lenora of what she had read of Yggdrasil, the world tree of Norse mythology that connects nine different worlds.
She had plenty of time to ponder this connection, because TREE(3) did not react to them in any way whatsoever, no matter how many bows and curtsies and polite entreaties they produced. But Lenora was not about to give up (how could she possibly fail two patrons in a row?). She decided she’d had enough bowing and curtsying and begging. She was a librarian, after all, and she would not be ignored.
She walked up to TREE(3) and raised a fist.
“Uh…” said Milton.
Lenora gave TREE(3) a firm knock. “Hello? Hey? Listen to us. I’ve got a question I have to answer!”
“Lenora, I—” said Milton, backing away and pulling Lucy with him.
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