Work Text:
Nene had one mission on her mind: to pass the next math exam with flying colors and boost her grade to the next letter.
In theory, it sounded easy. In reality, however, Nene was this close to giving up on life.
Her desk was a battlefield—loose papers scattered everywhere, eraser shavings dusting the surface like snow, and her textbook lying open to a page that might as well have been written in an ancient, cursed language.
“What the heck does this even mean?” Nene groaned, glaring at the equation as if it had personally wronged her. She leaned forward and let her forehead thunk against the open textbook. If she pressed hard enough, maybe the knowledge would seep into her brain through osmosis. Or maybe she’d just knock herself unconscious and wake up after the exam was over. Either option sounded appealing.
A knock echoed from her bedroom door. “Nene-chan! Can I come in?”
Nene lifted her head just enough to sigh. Emu, she thought, already imagining the boundless energy on the other side of the door. “Yeah… come in.”
The door swung open immediately, and Emu practically bounced into the room, her eyes sparkling the moment she spotted Nene at her desk. “Nene-chan! You look super serious!”
“That’s because I’m fighting for my life,” Nene replied flatly, gesturing toward the textbook. “And math is winning.”
Emu leaned over the desk, peering at the page with intense focus. “…Wow! Those numbers look really angry!”
“They are,” Nene muttered. “I don’t understand any of this. I study, I read, I practice, and then it all disappears the second I look at the test.”
Emu straightened up, hands on her hips, and smiled brightly. “Then that means we need a different strategy!”
Nene eyed her suspiciously. “Your ‘strategies’ usually involve yelling.”
“And jumping!” Emu added cheerfully. “But they work sometimes!”
Despite herself, Nene felt a small smile tug at her lips. Maybe she wasn’t going to survive this math exam—but with Emu here, at least she wouldn’t suffer alone.
Emu tilted her head, watching Nene stare at the page like it might start attacking her again. “Nene-chan,” she said gently, “why are you so focused on math all of a sudden?”
Nene hesitated, fingers tightening around her pencil. “…Because I have to be.”
She sat up straighter and pushed the textbook a little closer to herself, even though she clearly didn’t want to look at it. “My grade’s on the edge. If I mess up the next test, it drops. And if it drops…” She trailed off, then shook her head. “I just want to make it work. I really do. I just—”
Nene let out a frustrated breath. “I have no idea how any of this works.”
Emu’s smile softened, the usual sparkle in her eyes turning thoughtful. “And the test is…?”
“In a week,” Nene said miserably. “Which isn’t a lot of time when your brain refuses to cooperate.”
Emu gasped. “A week?! That’s plenty of time!”
“That is not plenty of time,” Nene replied instantly.
But Emu was already pulling a chair closer. “Can I see the textbook?”
Nene slid it over without argument. Emu leaned in, eyes scanning the page. “…Oh! Ohhh! I remember this!”
Nene blinked. “You do?”
“Mhm!” Emu nodded enthusiastically. “We learned this before! Adding fractions with complex numbers—see? Like this one, with the imaginary numbers!” She pointed at the problem like it was an old friend.
Nene stared. “That does not help.”
Emu laughed. “Sorry! But that means I can help you!”
“You can?” Nene asked, hope creeping into her voice despite herself.
“Yep!” Emu said confidently. “I have my own little method for this. It makes way more sense in my head than the way the textbook explains it.”
Nene sighed, then nodded. “…At this point, I’ll take anything.”
Emu beamed. “Yay! Then let’s start from the beginning, Nene-chan! Math won’t know what hit it!”
Nene wasn’t sure she believed that—but for the first time all evening, the textbook didn’t look quite as terrifying.
Emu grabbed Nene’s pencil before she could protest and flipped to a clean page in her notebook. “Okay! So! Imagine complex numbers are people at a party!”
Nene blinked. “…A party.”
“Yes!” Emu nodded enthusiastically. “A very awkward party.”
She scribbled something down. “A complex number is made of two people: the constant, or real number, and the imaginary number. They’re different kinds of people, so they don’t really get each other. They don’t hate each other, but they’re like…” She waved her hand. “The people who only talk because they’re stuck next to each other.”
Nene stared at the page, then muttered, “That actually tracks.”
“So!” Emu continued, clearly encouraged. “When we have fractions with complex numbers on the bottom, it’s like those awkward people are blocking the exit. And math doesn’t like that.”
“Math doesn’t like anything,” Nene said.
Emu giggled and kept going. “But every complex number has a conjugate!” She drew another number beside it. “That’s its buddy! Same constant, but the imaginary part has the opposite sign.”
Nene leaned closer. “So… like a plus becomes a minus.”
“Exactly!” Emu beamed. “When we multiply the top and bottom by the conjugate, it’s like bringing everyone’s buddy to the party. Now the imaginary numbers aren’t alone anymore!”
She tapped the bottom of the fraction. “And when imaginary numbers meet their buddies down here, something funny happens. They cancel each other out!”
Nene frowned. “Because… i squared?”
“Mhm!” Emu said, pointing excitedly. “And since i² turns into a negative number, the imaginary parts stop being imaginary and turn into regular constants instead!”
Nene’s eyes widened slightly. “So the bottom becomes… normal.”
“Yep!” Emu said proudly. “No more awkward imaginary people blocking the exit!”
She then drew a line between the two fractions. “Now, the fractions are like two sides of the party. When we add them, the people who are alike have to stand together. Constants go with constants. Imaginary numbers go with imaginary numbers.”
Nene watched as Emu grouped the terms together. “…And since they’re back together—”
“They’re forced to be complex again!” Emu finished, throwing her hands up dramatically. “The party resets!”
There was a long pause.
Nene slowly sat back. “…I hate that this makes sense.”
Emu grinned. “But it does make sense!”
“…Yeah,” Nene admitted quietly, looking back at the page. “It kind of does.”
“Okay!” Emu said, clapping her hands once. “Let’s try one for real!”
She pointed to the problem in the textbook. Both fractions had the same denominator—three plus i.
Nene’s shoulders tensed automatically, but Emu was already smiling. “See? Same party! Same people blocking the exit!”
“…Right,” Nene said slowly. “So we bring their buddies.”
“Exactly!”
They multiplied both fractions by the conjugate, Emu explaining each step as they went. Nene followed along, repeating it in her head—same number on top and bottom, don’t forget the buddy, don’t split the group.
“When the imaginary numbers meet their buddies down here,” Emu said, tapping the denominator, “they stop being imaginary.”
Nene paused. “Because i squared turns into a negative.”
Emu’s eyes lit up. “Yes!!”
With the imaginary part gone from the bottom, everything suddenly felt… simpler. Both fractions now shared the same, normal number underneath, which meant they could finally be added together.
“So now,” Nene said, a little more confidently, “we combine the tops.”
She hesitated for only a second before realizing something. “They’re the same. Both tops end up being three minus i.”
“Which means?” Emu prompted, leaning in.
“They stick together,” Nene finished, writing it down. “So it’s just three times that… all over ten.”
She stared at the final answer, waiting for the familiar wave of doubt. It didn’t come.
“…I didn’t hate that,” Nene admitted.
Emu gasped, hands flying to her cheeks. “Nene-chan didn’t hate math?! This is a miracle!”
Nene huffed a quiet laugh, but her eyes stayed on the page. For the first time that night, the numbers didn’t feel like enemies.
They felt manageable. Enough, at least.
~
By the time they finished the last problem, the sky outside Nene’s window had shifted to a deeper shade, the light softer than it had been when they started.
Nene set her pencil down and stared at the page. Every problem she’d been stuck on earlier was crossed out, rewritten, solved. “…We did it,” she said quietly.
Emu leaned over her shoulder, eyes sparkling. “We did do it!”
Nene turned to her, a small but genuine smile on her face. “Emu. Seriously—thank you. I don’t think I would’ve gotten through any of this without you.”
Emu waved her hands immediately. “It was no problem at all!” she said brightly. “Math is way more fun when you do it together. And if there’s anything else you want to review—any topic at all—I’m always free to help!”
Nene nodded, warmth settling in her chest.
A second passed. Then Emu slumped dramatically onto the desk. “Wow… I’m suddenly very thirsty.”
Nene blinked, then laughed softly. “Now that you mention it… yeah.” She stood up and stretched, rolling her shoulders. “I’ll grab us some water. You stay here.”
“Okay!” Emu said cheerfully, already making herself comfortable.
As Nene headed out of the room, she glanced back once—at the finished pages, at Emu humming to herself—and felt something she hadn’t when the night started.
Relief settled in her chest, along with a quiet confidence she hadn’t thought she’d have for the test.
She finally felt like she could do this. She finally felt as if she could take on the world and no one could stop her.
"It's just a math test, Nene," a voice in her head said.
"Shut up," another responded.