Lesson Number Undefined

Standard

“The tale is told,” Isra said.  “Of how these figures came to be that we so thoughtlessly take for granted.”

She noted the shift in the kids’ expressions toward the end of that sentence.

“Or rather,” she continued, “how their values came to be.”

Isra stood before the whiteboard in the drawing room, or as she called it after repurposing, the lesson room.  Or as her three children had designated it, Room Seven.  Their friends always talked about what room they were in, or how so-and-so teacher was in charge of a particular room number.  So they wanted a room number.

Libby and Dom sat back in their seats, knowing what was to come because they recognized the prompt.  They kept their notebooks open, but set their pencils down.  But little Irene still sat in the “row” in front of her brother and sister, with her back straight and her pencil poised.

“There was a people who lived in our world who set themselves above all others,” Isra said.  “They lived so long that our people thought they were immortal.  They knew so much that our people thought they were wise.  And their rule was so severe that our people thought they were to be feared.”

“Who were they?” Irene asked.

And Isra began her story.

***

They had the power to possess mortal minds, and they seized people at random, at their pleasure.  When they tired of the mortal mind and body they possessed, they would leave it, and often left their host far from their homes, or in dire situations.  Sometimes, they would compel a person to do something harmful or harming.  Sometimes, they would simply trap a mortal mind and leave it trapped.  Sometimes they would possess a mortal mind and stay until the mortal body perished.

One day, at great risk to themselves and their people should they be found out by sudden possession, some mortals came together in secret to discuss what might be done about the possessors, for they had shown themselves to care little for the mortals under their rule.  Even without the power to possess mortal minds at will, the possessors held great power over mortal men and women.  But those who gathered in secret wondered how well mortals would fare in a conflict with the possessors, if the possessors could not possess minds.

“How can we keep them from possessing our minds?” one man said.

“There is no way,” said another.  “We know not their weaknesses, and we do not have the strength to face their strength.”

Yet another spoke up.  “Then let us discover their weakness.”

It was a dangerous plan, for at any time, those mortals who watched and waited and plotted might be possessed.  And though the possessors typically did not think enough of mortal minds to be curious about what was contained in them, they did sometimes peer into the minds of the mortals’ greatest thinkers, just to confirm their own superiority.

But when they possessed the minds of mortals, they left their own minds open to those mortals, so unthreatening did they deem the mortals.

In time, the mortals found one and only one weakness in the possessors.

The possessors considered themselves perfect beings with knowledge of the infinite, infinite knowledge, and the capability to accomplish any and every challenge posed to them.

No mortal ever posed any challenge to them of course.  It was unthinkable, terrifying.  But the possessors were constantly challenging each other, sometimes while they possessed a mortal mind.  And it was observed that when a possessor was in the midst of solving a challenge, no other thought could occupy that possessor’s mind or time.  And if the challenge were worthy, other possessor’s joined in the contemplation of the challenge, and the competition of being first to solve it.

If the mortals could pose a challenge that could occupy the possessors, even a good number of them, they would earn a respite from possession.  They might even have the time and peace of mind to think of a way to be free of the possessors for good.

The mortals thought of many challenges, but they also figured out the solutions to those challenges.  And they knew that if they could think of a solution, the possessors could too, and likely more quickly.

At last, a great thinker arose who aimed to devise a puzzle that the possessors could never solve.  A riddle they could never fathom.  A calculation they could never do.

He did indeed think of a problem that no one seemed able to solve.  No mortal, that is.

After some time, the challenge was deemed worthy, and it was delivered to the possessors.  One by one, the possessors took to contemplating it and trying to solve it.  They became so distracted that they did not interfere with the daily affairs of the mortals.  They stopped possessing mortal minds.

A time came when the possessors solved the calculation at last.  They woke, as if from a stupor, and found that it had been an eon-long stupor, for the mortals had advanced greatly.  They had advanced so greatly that they too had solved the problem.  The possessors were enraged.  And they sought to vent that rage.  The mortal who had tricked them was long dead.  The possessors could not trace any descendants.  So they named all mortals as descendants, and they proceeded to conquer and subdue as they had done before, without lifting a finger, without taking a single step.  For they began again to possess the minds of mortals.

***

Isra paused, giving the kids time to ask questions.

Libby asked the one that Isra had expected they would ask.

“What was it?  What was the problem that they were asked to solve that took them an eon?”

“They had boasted about having knowledge of the infinite,” Isra said.  “So the question that was posed to them was about the infinite.  What was the infinite divided by the infinite?  In other words, what is infinity divided by infinity?”

The two older children glanced at each other.  It wasn’t just a riddle.  It was a math problem.  Libby nodded to her brother and he turned his face to the front and said, somewhat hesitantly.

“The answer is one.  Right?  A number divided by itself is one.”

Their mother grinned and nodded.  “Yes, that is the answer.”

Libby frowned.  “I don’t get it.  Why did it take them so long to figure that out?  If we could do it in a few seconds?”

Their mother crossed her arms.  “Take a moment.  Think about it.  Why would it take a long time to solve a math problem?”

Dom sighed.  He lifted his pencil and tapped it on his paper.  “Hmm, they weren’t using math tricks?”

“You’re getting there,” Isra said.

She let the children think for a while, but they couldn’t really come up with anything.  And so, she uncrossed her arms, folded her fingers together in front of herself, and she told them.

“They were trying to do it manually.  Trying to consider the number itself, rather than using the concept.  Dom knew the concept of a number divided by itself being one.  So it was a snap for him.  But for the mortals in the story and even the possessors, that concept had not been figured out yet.  The possessors tried to do the actual calculation manually.  The way Irene divided the twenty-five pieces of candy by five a few Halloween’s ago.  By counting them.”

Libby’s eyes widened.  She blinked and shook her head slightly.  “But then, how did they even get to the end?  Because if they were really trying to count infinity…”

“…they would still be going,” Dom finished.

Isra shrugged.  “I would think so.  But, that’s how the story goes.”

“What’s ‘infindy’?” Irene asked.

Libby spelled out the word to help her sister pronounce it.  “It’s something that goes on forever, but it’s important in math and science…and maybe other subjects, to help us understand better.”

“Understand what better?”

“The world?  Everything?” Libby shrugged.  “I’m still learning too.”

Irene nodded and thanked her sister.

Dom brought their attention back to the story.  “Okay, so before the possessors woke up, the mortals already knew the answer because they had figured out the concept.  And that’s why we have it now and we kind of take it for granted.”

Isra nodded.

He continued.  “Okay, then…what happened next?  Obviously they didn’t destroy everything because we’re still here.”

Isra raised a brow.  “What do you mean?”

Libby sighed and mirrored her mother’s brow-raise.  “The mortals.  We’re the mortals in the story, right?”

Isra shrugged again.  “Maybe, but you are right about the possessors. They didn’t manage to destroy the mortals.  The mortals managed to save themselves again by presenting the possessors with another problem.”

Libby raised her hand.  She was frowning again.  Isra nodded to her.

“I mean, instead of just thinking of a riddle, were people trying to think of ways to block the possessors’ powers?  Block them from getting into mortals’ minds?”

“Yes, of course, to an extent.  But without possessed minds to study, all they had was the accounts of previous possessions.  It wasn’t enough to go on to produce any kind of protection.  It wasn’t until the possessor’s woke up and began possessing again that the mortals observed at least one change.  The mortals found that the more strength they had in their minds, to observe, to think, to contemplate, the more difficult it was for the possessors to possess their minds.  They could still do it.  It just took a little bit longer.”

***

While the possessors had been occupied, the mortals, without fear of their minds being possessed at any moment, knowing that their minds were their own, had flourished.  They had not known how much time they had, so they made the most of it.  They developed a rich culture, and part of that culture involved thinking of more riddles, more numbers, more equations, more figures.

They knew that if the possessor’s possessed the minds of the great thinkers, which is what they tended to do, they would find out that the mortals were devising another challenge.  So the mortals had to think of something that they themselves had not solved and could not solve.  They thought of many such equations and they spread them throughout the world.  Rather than hiding the knowledge, sequestering it, the best way to assure that the problem they chose would be challenging enough to confound the possessors was to teach knowledge of past problems and solutions to as many as possible, and to challenge all with the chosen problems to see if any could be solved.  So they did.

Over time, it so happened that mortal minds came up with many challenging problems, but someone kept solving them, either one brilliant mind working alone, or many brilliant minds working together.  So they had to come up with even more and even more.  And at last, when the possessors woke and threatened the mortals, they had the next problem ready, one that no mortal had yet solved.

They presented the problem to the possessors.  The possessors were once again compelled to solve it, and they once again went into contemplation.  And this time, several millennia passed, more eons.  By the time they solved this problem, the mortals had advanced even more.  To the point where the possessors were beginning to fear that their powers were being rivaled by the mortals.  For by the time the possessors woke again, the mortals had crafted intricate machines.  The mortals had created even more breathtaking art.  The mortals had gathered into larger cities.  And had traveled to ever more distant distances upon their world, below it, and even above it.

Part of the reason the possessors were so surprised and angered when they observed such advancement was that their lives were so long and mortal lives so comparatively short, that they lost track of time and didn’t realize that many generations of mortals had lived and died and contributed to further strengthening their people as a whole.

But the possessors found that they still had a hold on the minds of mortals.  Though some minds were more difficult to possess than others, mortals never did find a way to completely block the possessors from possessing their minds.  No shield, no helmet.  No meditation.  No powerful thoughts.  Nothing that could afford complete protection.  The only protection mortals had was their own minds.  But they had to focus and actively think to summon such protection, and even then, if they let their guard down, the possessors could possess their minds.

The mortals had not intended for the problem to be their first defense.  They had hoped for some permanent solution to block the possessors.  They didn’t believe the possessors would fall for the same trick.  They didn’t quite believe their ancestors’ accounts that it wasn’t quite a trick so far as the possessors were concerned.  The problem would work, because of a quality that the possessors possessed.  A compulsion for needing to solve a problem, to demonstrate to themselves that they did indeed have knowledge of the infinite.  And to then present the answer to the mortals to demonstrate their superiority.

***

“Anybody interested in knowing what that second problem was?” Isra asked.

“You know we are, Mom,” Libby said.  “I mean Professor…Mom.  Please continue.”

Isra paced back and forth, narrowing her eyes, and flourishing her hands.  “What is a number that both is and is not?”

This time she let the kids think about it and discuss it amongst themselves for a time, while she pondered how well she would have done if the question had been posed to her.  Maybe she wouldn’t have been able to come up with the answer.   Was it fair?  It was a riddle,  and not so much a math problem.  She deemed it was fair, because it was something they could figure out on their own.  The kids—even Irene—started writing numbers and equations on their papers.

Many times, Isra had advised them not to start with complex thoughts.  To start simple and build up from there.

Isra was close to deciding that she should give them the answer and move on with the story when Libby started to laugh as she stared at her paper.  And Dom said, “oh!” at almost the same moment.

“Zero,” Libby said.

Isra couldn’t help beaming at her children.  “Well done.  Can you explain your logic?”

“It is because we’re talking about it,” Dom said.  “And we can write it down.  It exists.”

Libby nodded.  “But it is not because, because it’s zero.”

Irene giggled.

“Because the value means ‘nothing,’ or ‘absence,’” Libby clarified.

“Dad told us something about how ‘zero’ didn’t always exist,” Dom said.  “Like, as a concept.  I can’t remember who he said invented it or discovered it.  But once they did…”

“…game-changer,” Libby said.

“So, it took the mortals and the possessors some time to figure out that concept,” Dom said.  “And once they did, they came out of their thought-trance again.  And wrecked everything for everybody…again.”

“Or did they?” Libby asked.

Irene giggled again.  It wasn’t a response to the story, which seemed to be mostly going over her head.  It was more about how her brother and sister were doing their fraternal twin thing.  She seemed to love it when they finished each other’s sentences and bounced off each other’s thoughts, even when she didn’t understand what those sentences and thoughts meant.

“Unfortunately,” Isra said.  “They did.”

Dom sighed.  “And still no way to block them?”

“I’m afraid not.  But this time, instead of just thinking about the possessor’s and their weakness for solving challenges, the mortals thought about themselves and their strengths.  They asked themselves if there was a strength they had that surpassed the strength of the possessors.  And if they could use that strength to devise the perfect problem.  One that would occupy the possessors for the rest of their lives, for eternity if that’s what it took.”

“For in-fin-ity?” Irene offered.

Isra smiled.  “We need to work on that definition, but you’re actually on the right track.”

“Good one, Irene,” Libby and Dom both said at the same time.

Irene’s eyes widened and her mouth dropped open in a gasp of delighted surprise.

Isra paused to enjoy the moment before she continued with her story and her lesson.

“Millenia passed.  Eons passed.  This time, the possessors were incapable of solving the problem that had been put before them.  Unlike the previous times, they even tried to work together to solve it.  But they did not quite know how to cooperate.  We had finally done it.  We had devised a puzzle that even they could not solve.  A riddle they could not fathom.  A calculation they could not do.  More eons passed.”

Isra turned away from her children, her students.  She looked at the whiteboard and clasped her hands.  They remained silent.  She turned back around.

“As far as I know,” she said, glancing deliberately from child to child, “they still haven’t solved it.”

***

Dom’s eyes were wide.  He was smiling from the corner of his mouth and nodding his head.  He stopped nodding and tipped his head to one side, then the other, stretching his neck.

Libby interlocked her fingers and extended her arms in front of her, palms facing out, stretching her arms.  “All right, Professor,” she said.  “Present the challenge.”

Isra peered at them.  “What is everything divided by nothing?  What is infinity divided by zero?”

After the last two times, Isra expected they wound hunker down over their papers again.  But right away, Dom began to think out loud.

“Divided by.  Okay, a simpler example.  Ten divided by five.  To figure it out, I have to find out what times five equals ten.”

Libby took over.  “So…what times zero equals infinity?”

The twins frowned at each other.

Dom bit his lip.  “But…that’s not.”

“There’s no way you can multiply anything by zero to get infinity,” Libby said.

“Or to get any number really.  Right?” Dom said.

They both looked at Isra.

Dom twisted his mouth in an unsure expression.  “Is it…an imaginary number?”

Isra shook her head.  “No, there are such things as imaginary numbers, but we’ll go over that in a future lesson.”

“What is the solution?” Libby asked.

“We call it an ‘undefined number’,” Isra said.

“So we haven’t really solved it…because there is no solution?”  Dom stared at the whiteboard.

“At least in this universe,” Libby said, and if it was possible to smirk sheepishly, she did it as she glanced her Isra.

“That’s weird.  That we didn’t call it something,” Dom said, still staring.

“But it kind of makes sense,” Libby said.  “Because it’s not something.  But it’s also not nothing.”

“But if it’s undefined and that’s that, why are the possessors still stressing over it?” Dom asked, breaking himself out of his daze and meeting Isra’s gaze.  “Why haven’t they figured out by now that they can come up with a concept to help them understand?”

“According to the myth,” Isra said, “it is because they lack a quality that mortals have.  A power that in some ways is greater than any power that the possessors ever had.”

“Imagination,” Dom said.

“That’s a good guess.  But it’s not what I’m getting at.”

“Working together,” Libby said.  “What’s the word?  Everyone working together, learning from each other and from people in the past, building on what they knew.”

“From simple to complex,” Dom said.

Isra grinned.  They had figured it out.  The not-so-hidden lesson beyond the myth and the math.  “Solidarity,” she said, in answer to Libby’s question.

“It seems like the possessors also need to learn to let go,” Dom said.

Libby nodded.  “They’re way too much into possessing.”

Isra chuckled.  “Absolutely they do.  And absolutely they are.”

“Does Dad know this myth?” Dom asked.

“He doesn’t,” Isra said, narrowing her eyes and pressing her lips in a suspicious smile.

“Can we run it past him later and see if he gets the answers?”

“Of course.”

She asked if they had any other thoughts or questions.  They didn’t at the moment, but they asked if they could revisit the story in the next few days, after they’d had some time to let their thoughts simmer.

Isra transitioned to a more straightforward math lesson and followed up with problems for Libby and Dom to solve.

Later that night, after reading Irene a story and before tucking her in, Isra entertained a question from her youngest daughter.

Irene had a serious expression on her face.  Before asking her question, she assured Isra that she was a good teacher.  Then, she asked.  “Can Libby and Dom teach me fractions?  Since they’re my and-sister, and also, brother.  Like in your story.”

Isra furrowed her brow for a moment, puzzling out what Irene had just said.  Then realization struck when she figured out that Irene had absorbed part of the lesson her elder siblings were taught that day.

“I think you mean ‘ancestor,’ hon.  We’ll work on that definition.  But that’s a wonderful idea.”  Isra had been planning it all along.  To have her older children teach their little sister, so they could further solidify their own knowledge and maybe find ways that Isra would never think of to help Irene understand concepts.

“I don’t want the possessors to get me,” Irene said.

Isra froze, suddenly panicked at the thought that she had created a new monster for her daughter to fear.  But Irene didn’t seem scared or even troubled.

She seemed…determined.

Isra put her youngest daughter to bed.  A few hours later, her husband stopped at each twin’s room and gave them the lights out warning.  Isra heard his voice as she checked the downstairs, calling the dog out of the room with the whiteboard and the desks, where he was expressly not allowed to go.  She mussed his head as he passed.  It was Friday night.

Isra glanced around the lesson room, Room Seven, before pulling the door closed.  “See you Monday,” she said.

 

Copyright © 2018  Nila L. Patel.

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