x-1 = 1/x

Herb and Stan met again at the coffee shop at 4:30 the next afternoon. The shop had several tables sitting outside under a couple of small city-beautification trees planted on a widened, brick-covered part of the sidewalk. The coffee shop side of the street was in the shade, and the air was cool enough, barely, for them to sit at the only unoccupied outside table.

Before they sat down, Stan and Herb both briefly noticed a good-looking couple sitting at one of the tables drinking iced coffee drinks. The young woman was reading a book and the young man was reading the Austin Chronicle—or, rather, flipping through it and not reading it. The two had a movie-star look about them, but neither Stan nor Herb was willing to look at them long enough to make a positive celebrity ID.

Across the street sitting in the sun by himself on a green bus-stop-style bench, a middle-aged homeless man in a heavy, dirty, black overcoat, dirty black pants and a big black ski cap smoked a cigar stub, and looked straight ahead, occasionally glancing at the folded newspaper in his lap. Stan and Herb saw the man every time they came downtown. Usually he was sitting on the same bench, but sometimes he was searching through one of the Congress Avenue garbage cans.

“I don’t see how that guy stands the heat of the sun in those heavy clothes,” Stan said as he and Herb sat down.

“I don’t see how that guy stands his own body odor,” Herb replied, just before taking a bite of cranberry bread.

“But have you noticed the cigar smell? He’s always smoking good cigars-maybe that helps negate the body odor,” said Stan, after his first swallow of hot tea.

“Maybe the cigar store owner down the street gives him free samples,” Herb said after swallowing his second sip of coffee and putting the large white china cup back in its saucer.

“Yeh-probably pre-smoked free samples,” said Stan, smiling one of his tight little smiles and squinting his eyes.

“Probably,” Herb agreed, grinning a little and emitting a slight chuckle. Taking a bite of cranberry bread and then a quick sip of coffee, he thought for a moment and then said, “You know, I was reminded of something I couldn’t quite put my finger on when we were talking about your affection for x =1/x last night. I just realized what it is.”

“You mean your affection for x =1/x. I was partial toward x squared equals one, myself. Something related to my unrequited affection for you-know-who, and how it should be obvious there’s only one solution-okay two solutions, negative and positive, but only one number. A song by Three Dog Night comes to mind.” Stan glanced toward the homeless man and took a bite of his walnut-apricot scone then a sip of tea.

“Well,” Herb replied testily, “they’re the same equation.” He paused for a moment and thought about changing the subject and depriving Stan of the thought he’d wanted to share with him. He decided to share it. “I suppose you’ve heard of the golden ratio.”

“Sure, I’ve heard a lot about that,” Stan said, putting his teacup in its saucer and sitting back in his chair. “In a visual context, it’s an eye-pleasing proportionality used by the Greeks in their architecture and also found in nature.”

Pausing and leaning back in his chair like Stan, Herb said, “Yeah, that’s part of the story.” Stan thought Herb was beginning to smirk a little, and indeed Herb looked away from Stan (to hide his smirk, Stan thought) as he continued, “But what is the equation for the golden ratio?” Herb looked back at Stan, waiting for a response.

In no hurry to provide a response, Stan took another bite of his scone, then a sip of his tea. He found the mixture to be very flavorful. “I know there’s a quadratic equation,” he replied after swallowing his tea and scone mixture, “that has a solution that’s a never-repeating decimal expression like 1.618... or something like that.”

“Right!” Herb said quickly, in a sort of enthusiastic whisper. In his more analytical, normal tone of voice he continued, “I’m impressed with your memory of the first digits of that decimal number. But the quadratic equation is very much like your beloved x squared equals one. I thought you’d like that. Care to take a guess what it is?”

“Okay,” Stan said, sitting up and taking another sip of tea. After thoughtfully swallowing, he said, “Is the equation x over 1-x or something similar?”

“Very dang close, Stanley old bean,” Herb said, not smirking but instead involuntarily smiling.  He enjoyed quizzing Stan and getting him to think, and he mostly enjoyed the superior feeling he got when he posed a question Stan couldn’t answer quickly. Sitting up straight, and picking up his coffee cup, Herb said just before he took a sip: “But that’s not an equation.”

“You are an ass,” Stan said in a good-natured tone of voice.

“That’s irrelevant,” Herb immediately said, as if mimicking a judge overruling an objection.

“Well, it’s more like irrevelant, which probably isn’t even a word,” said Stan in response. “But I can come up with an equation.” Stan takes a pen out of his pocket and does some scribbling on his napkin. “Okay, here is the equation,” he said, looking Herb in the eye and sliding the napkin over to him.

Herb took a sip of his coffee, put the cup down, and picked up the napkin. On the it was written “x²/(1-x) = 1”.

Herb pulled a pen from his shirt pocket and scribbled on the napkin. “Not quite, but actually only off by a minus sign,” he said. “Changing the minus to a plus and kicking the 1+x over to the right-hand side gives the right quadratic equation. But this is what I thought was so cool.” He handed the napkin back to Stan. “Check out the rearrangement I did.”

Stan looked at the scribbled equations on the napkin. The last equation on the napkin, written on the back of the napkin, was “x-1 = 1/x”.

Sitting up straight in his chair, his back not supported by the back of the chair, Stan held the napkin in front of him and stared at it intently. He kept staring, lost in thought. Herb let him think without bothering him.

He noticed the young couple they’d seen earlier get up from their table, talking to each other now that she’d put away her book. Apparently, they’d arranged to meet at the coffee shop, and were about to go their separate ways. Hearing the girl’s voice, Herb realized she was daughter of his ex-girlfriend Denise. Her name was Rachel.

“Did you see anything in there about us?” Rachel said as the guy put the Austin Chronicle into his army-green knapsack.

“I wasn’t really looking for anything about us,” said the guy, glancing up at Rachel in a distracted sort of way that was nevertheless intense. Herb recognized him then as Jacob Josephson, the movie star son of Jason Josephson, a local film director who had recently made it big. In the issue of the Chronicle that Jacob had been looking through so quickly, Herb had earlier read a short article about a movie with Jacob in it that was being filmed in Austin.

“Us?” Herb said quietly to himself.

“What?” Stan said quickly, his reverie with the napkin now broken.

Herb leaned over the table toward Stan. “It’s Rachel,” he said in a low whisper, nodding toward the young couple’s table.

Stan turned around to look at Rachel as she was shouldering her backpack. For the first time, she looked toward Stan and Herb’s table. At first she looked puzzled, then she smiled and unselfconsciously said, “Hi Herb, hi Stan!”

“Hey, Rachel,” Herb and Stan said simultaneously, causing the three of them to smile briefly. Then Herb added, “How are you?”

“Great. Good to see you guys.” As Rachel spoke, Jacob looked at Stan and Herb, then back at Rachel. Before Jacob could speak, Rachel stepped toward Stan and Herb’s table. “Jacob, these are some friends of mine."

“How ya doin’” Jacob said, with a friendly nod and a quick smile. He and Rachel stepped over to Stan and Herb's table.

“This is Herb, and this is Stan,” Rachel said, motioning with her left hand toward Herb and then Stan. Looking at Jacob, she added, “And this is Jacob.”

Being the closest to Jacob, Stan put his hand out. “I thought you looked familiar,” he said to Jacob as they shook hands. “How’s the shooting going?”  Stan had read that "shooting" was the proper word among film people, not "filming".

“Well, we don’t start shooting till tomorrow,” said Jacob. After a brief pause, he added quickly, “At sunrise.”

Herb stood up and shook hands with Jacob across the table. “Good to meet you,” Herb said. Rachel spoke next. “It’s been, like, four or five years since we’ve seen each other.”

“Yep, five years.” Herb said quickly. “You look really great. How’re your mom and Luke doin’?”

“Oh, about the same, doing okay,” Rachel said, moving a loose strand of blond hair out of her face and tucking it behind her ear. “They’re gonna be in the movie, too.”

“So you’re in it with Jacob,” Herb said, realizing why Rachel had used the word “us” earlier. “Congratulations on that!”

“Thanks. I’m a little nervous. I’ve been in lots of plays, but never a movie”

Jacob looked at Herb and Stan and said, “Well, I better get going. Good to meet you guys.” He shook hands with Herb, who was still standing, then with Stan, who was still seated.

“Break a leg!” Stan said as Jacob turned to leave. After he said it, he immediately wondered if it was appropriate to movie making.

“Yeh, thanks, I’ll do my best—it’s actually part of the story!” Jacob said with a quick smile as he turned to walk away.  Stan, Herb and Rachel all laughed together, then Herb sat down in his chair, picked up his coffee cup and leaned back.

Seizing the moment to keep the conversation going, and thinking Herb and Rachel probably felt awkward after not having seen each other for so long, Stan said to Rachel, “Herb and I are exploring the romantic implications of the golden ratio.”

Rachel giggled a comfortable and confident sort of giggle, then asked, “What romantic implications?”

Herb grinned, remembering how good Rachel was at asking direct questions.

“Well, we haven’t figured that out yet,” Herb said, looking at Rachel then at Stan. “The philosopher here is working on a theory.”

“Actually, I just now worked out the theory,” said Stan, looking from Rachel to Herb and raising his eyebrows while smiling a bigger than normal tight-lipped smile. “This equation for the golden ratio, x minus one equals one over x, says when you become an ex, you have to let go of the other person—that's the minus one part—if you are going to turn your life around—that's the reciprocal part. And I re-wrote the reciprocal to make the equation have a more symmetric appearance.”

Stan pushed his scribbled-on napkin to the center of the table where Rachel and Herb could both see it. The final equation on the napkin was x-1 = x^-1.

“Stan, you’re a genius!” Herb exclaimed as he picked up the napkin. “That’s a beautiful equation!”  Stan looked up at Rachel, who shifted her weight and adjusted her backpack on her shoulder, smiling at Stan as she did so.

“You guys ought to write a book.” Rachel said, looking back at Herb. “A book about the algebra of breaking up.”

“If only it were that simple,” said Herb, taking a sip of his coffee but finding it too cold to be enjoyable.

“Well, I better get going,” Rachel said, looking at her watch. “Mom wants me to get some stuff at the store for dinner. It’s going to be a crazy night around our house—everybody’s nerves are jangled.”  

“Yeh, I’d be nervous too,” Herb said, standing up and looking Rachel.. “Congratulations again on being in the movie. Tell your mom and Luke I said hi.”

Stan then stood up, and said, “Tell them hi for me, too.”

“I will!” Rachel said as she smiled and hugged Herb. “Good to see you guys.”

“Good to see you,” said Herb as Rachel stepped over to hug Stan.

“Take care,” Stan said as he and Rachel hugged.

“You too,” said Rachel.  Then as she was turning away, she said “Bye guys.”

“Bye!” Stan said as Herb simultaneously said, “Bye-bye!”

 “She’s grown into quite a beautiful young lady,” Stan said as he and Herb sat down.

“She always was one,” Herb said, before taking his last bite of cranberry bread and washing it down with a sip of cold coffee. “Now I need to get going,” he said, looking at his watch. “I’m teaching a class at 6 tonight, you know. But first, I’ll show you something. “

Herb picked up a fresh napkin and began writing on it. "You have x, which is really x to the first power," he said as he briefly glanced up at Stan. "Then you have one, the loneliest number, which can be written as x to the zeroth power, right? Then, lastly, you have x to the minus one power."

Herb pushed the napkin over to Stan, who looked at it and saw: “x¹- x⁰- x^-1= 0.”

“Well…” Stan said, pausing as he looked down at the napkin. “I still like my way of writing it better."

"Well, naturally," said Herb as he stood up and picked up his book bag. "And it is definitely a beautiful way of writing it. But having dealt with sequences and series in math, I couldn't resist the temptation to write it as consecutive powers of x, like it's part of an infinite series. Now I gotta go, bro, so take care and get in touch about that thought experiment we’ve been talking about."

"Sorry to say I'm losing interest in it," Stan said, looking in Herb's direction, but looking past him rather than at him. "But, yeah, I'd like to come up with an answer to the original question, about the time of the streetlight flash in the two rest frames."

"Let's do that, then," Herb said, starting to walk toward his nearby bicycle.

"Hey!" Stan said suddenly, as if having a delayed insight into the thought experiment. 

“What?” said Herb, turning around quickly.

"Remember, stay away from the Golden Arches."

Herb laughed and said, "I'll do my golden best." His bike had been leaning unlocked against one of the city beautification trees on the sidewalk. He swung himself onto the seat and rode into the bike lane, heading south on Congress Avenue.

One of the café staff was cleaning off Jacob and Rachel's table. Stan figured he’d go ahead and leave so his table could be cleaned and he wouldn't have to worry about whether to take his and Herb's cups and saucers inside.

After walking about a block south on Congress he passed the homeless man dressed in dirty black clothes who'd been sitting on the bench across the street earlier. Stan was about to nod and say hi to him, but the man glared so ferociously at him that Stan just looked away.

As he was about to turn the corner to head west on 8^th Street, Stan looked back and saw the homeless man holding up the hinged top to a green metal trash container while poking his head inside and reaching down into it.  After turning the corner, Stan didn’t see anyone on the sidewalks nearby. Instead of whistling as he often did in such situations, he began to sing the theme from George of the Jungle in a quiet but energetic voice.