#introduction to trigonometry

[upennmanuscripts]

LJS 476 is a partial copy of an introduction to astronomy. Three books, addressing geometry and trigonometry; the structure and position of the Earth; and the movement of the sun and moon and equinoxes and eclipses. Written in Spain or Italy, ca. 1460

🔗:

[idontknowreallywhy]

A little lunchtime rewatch of one of my top favourite episodes (for so many reasons) - Runaway 🚅

And a cute little canon moment supporting my adhd flyboy theory…

I mean, it wasn’t that complicated…

(Apologies for poor quality video but it makes me giggle so had to capture it)

Episode also notable for

- the introduction of the concept of Celebrity Alligator Wrestling

- “Sorry, forgot to use my trying to make a joke voice”

- ALL the Brains screaming and reciting trigonometry

- that shot with TB1 and tiny hologram TB1

- “Scott can I…?” “No.”

- “UGH, TASTES LIKE A FOOT”

- “But don’t use C247 model… it’ll blow up in your face. *MAX beep* or the C247A.”

“With no extra charge for sudden dismemberment”

[asimplearchivist]

𝑪𝑯. 𝑰𝑽 — 𝑭𝑼𝑳𝑳 𝑶𝑭 𝑳𝑶𝑽𝑬.

𝐂𝐇. 𝐈𝐕 𝐨𝐟 𝐓𝐇𝐎𝐔𝐆𝐇𝐓 𝐈 𝐃𝐑𝐄𝐀𝐌𝐄𝐃 𝐇𝐄𝐑.

[𝓪𝓼𝓲𝓶𝓹𝓵𝓮𝓪𝓻𝓬𝓱𝓲𝓿𝓲𝓼𝓽'𝓼 𝓶𝓪𝓼𝓽𝓮𝓻𝓵𝓲𝓼𝓽] [ 𝐒𝐏𝐈𝐃𝐄𝐑𝐕𝐄𝐑𝐒𝐄 𝐌𝐀𝐒𝐓𝐄𝐑𝐋𝐈𝐒𝐓 ] AO3 | SPOTIFY | PINTEREST summary 🕷️ ⤏ spider-woman of earth 928c has a painfully short adjustment period to her new predicament. she isn’t the only one. pairing 🕷️ miguel o’hara/spider!reader word count 🕷️ 4.0k a/n 🕷️ [gif credit] ⤏ this chapter was originally planned to be entirely passive, just to catch up on reader’s feelings through the last couple of chapters. however, jess (or should I say miguel) had other plans. they took it from me and I had to run with it. next chapter we’ll see a little more progress in reader and miguel’s dynamic, hopefully. 🕷️ MASTERPOST 🕷️ 🕷️ PREVIOUS CHAPTER ⤎ 🕷️ ⤏ NEXT CHAPTER 🕷️

The infamous, so-called Spider-Society wasn’t nearly as…prestigious as you’d initially anticipated, after all the buildup that the alternate Miguel had laid down convincing you to join. Meeting so many other Spider-People was certainly as elating as it was stressful (honestly, you never thought that you would see as many of the same person at one time, multiverse theory be damned, much less the stark differences between the lot of them—it was a good thing that you weren’t allergic to rocks, because there sure was a lot of Peter). The majority of them seemed to come and go as they pleased, only turning up to HQ to report in occasionally or to share information about any new anomalies that may have cropped up in their dimension.

Oh, yeah, and the whole anomaly thing…combined with the canon events theory…there was a reason you’d focused on robotics and nanotech rather than physics during university. Being told that because of your involvement in the arachno-humanoid poly-multiverse (and Christ, Miguel needed to get a new name for that, it was just too damn big of a mouthful) you were basically guaranteed to undergo negative experiences that would drive most people into depression or madness didn’t…sit well. Although it was a slight comfort to know that virtually every other Spider-Person had gone through something similar to you in regards to grief, it still didn’t sit quite right with you—but perhaps that was the hero complex in you speaking. If I had just tried harder, been faster, put my life further out on the line…

Nevertheless, it did help you to relax slightly. Peter B. was infectious with his easy affection and lighthearted demeanor, cracking jokes and knowing just what to say to help you circumnavigate your awkward interactions with the others. His wife was about five months along, expecting their first child—a little girl they’d already decided to name Mayday after his deceased aunt. He introduced you to Jessica Drew, who herself was due to have a baby in seven months—she was cool and calm but had a lot of heart and humor, and you appreciated her honestly as much as her sincerity.

Some of the others were…quite a handful. You weren’t quite sure what to think of Hobie Brown, besides the fact that he was hilarious. You had the distinct impression that he was always watching you, though, dark eyes as scrutinous as your trigonometry teacher back in high school. You didn’t really want to think that he was waiting for you to mess up or something, but…he always seemed to know something more than you at all times and it made you just a tad uncomfortable. The quiet, observant sort like him often were the greatest troublemakers—his standing track record in his universe, which he claimed with no small amount of pride, notwithstanding. You liked him, you really did—he’d been nothing but kind to you during your brief introduction—but you couldn’t help but feel like he had something up his sleeve.

The rest were just as, if not more, colorful characters. The mind-boggling quantity of alternate Spider-People overwhelmed you after Peter B.’s tour, so after he herded you back to the cafeteria and helped you to unload your cart, he showed you the basics on opening the portals.

“Try not to use them in crowded areas—you don’t want a bunch of junk getting in the way. Or, you know, splicing into your body like a redshirt being beamed up. Type in the universe number here, enter the coordinates—looks like Miguel already saved your previous location as the default, so you can teleport directly from there to HQ whenever you need to—then tap this, and…voila! There’s a communications tab here, so you can contact LYLA or call any one of us if you ever need backup, or to report an anomaly.”

You nodded along, squinting in hopes that you could commit the motions to memory rather than have to ask multiple times how the damn thing worked—you were a roboticist, yes, and you could probably whip up a circuit out of cardboard, gum, and a lemon—but even this level of technology was beyond your paygrade. Miguel’s work was truly phenomenal, you had to give him that. (And…well. You didn’t want to wind up like a redshirt, either.)

You imparted him with another half-dozen cupcakes to take home to the missus as thanks for taking his time with you, dragged your cart through the portal back into the kitchen of your bakery, and as the maelstrom of mindfuck physics disintegrated into thin air within the blink of an eye as though it had never been there to start with…you sank against the countertop and buried your face in your hands.

This Miguel was different from your late husband, in countless ways that you’d been doing your best not to take notice of too closely. (Would it be considered rude to compare them? They were inherently the same person, just…under different circumstances.) But, at the same time, they behaved so similarly: that same low, even, soft-spoken cadence; those half-lidded eyes feigning disinterest but to hide the exact opposite after years of having any enthusiasm beat out of him as a child; the tension he always carried in his shoulders to hold them back, his chin angled up, his hands always ready at his sides, always ready to move at a moment’s notice (and those shoulders…you hadn’t been joking, asking Peter B. about the man’s dieting habits—he easily dwarfed your husband twice over, and you couldn’t help but wonder if it was intentional or whether it was due to whatever sort of spider abilities he’d inherited).

Even still, that was only the start of the differences. This Miguel was snippier, shorter-tempered, and mumbled as though he had a mouthful of something packed under his bottom lip all the time. He had a clipped walk like a panther, a long, stalking stride that made far too little noise for a man of his bulk. He kept his hair slicked back rather than allowing his curls freedom. His mildly unsettling carmine eyes were tired, and you were half-convinced that if he were to sit too long he’d likely pass out based on the way he hyper-focused on everything around him as though to compensate. His exhaustion was obvious, but maybe that only had to do with your familiarity of the canvas rather than the difference in hues and brushstrokes.

They said your name the same way, however. You weren’t sure how to feel about that.

By the time you pulled yourself together (because you had wept enough the night before and didn’t want a repeat incident—you’d probably tripled your water bill hunkering down in the shower to muffle your misery and you were not going to cry again, thank you very much) and made it back upstairs, the night had set in and plunged your apartment in total darkness. It was a bit odd, as in Earth 928B it had been about noon when you left, but you supposed that temporal unalignment was the least of your concerns.

Your cat, a silver tabby your husband had named Alba upon finding her abandoned as a kitten in the park, greeted you at the door letting you know exactly how late you were for serving her evening meal. You long-sufferingly doled out the kibble in the kitchen and watched her inhale it in less than a minute before coiling around your legs in a figure-eight once and retreating to her bed set in the windowsill overlooking the street below.

You shook your head with a sigh, trudging around to feed your other pet Horchata before retreating into your bedroom, stripping off your clothes as you went. You tossed them all into the overflowing hamper, stored the compact belt containing the technology housing your suit in your nightstand drawer, and slipped into the bathroom to wash away the flour and confectioner’s sugar. You carefully removed the not-watch (as Peter B. had been so kind to inform you that Miguel hated them to refer to it as a watch) and set it on the counter alongside the ring you wore on a delicate chain around your neck.

You wondered what this new Miguel saw in you, to want you on his top-secret strike team so badly. You weren’t exactly perky yourself. You’d laid out all your flaws for him, and yet he hadn’t even flinched. He’d only seemed agitated at your immediate refusal, if nothing else.

The next day passed in relative normalcy—you woke early to start work on your stock, had coffee with your aunt (who spent more time daydreaming about her doctor than actually sharing the report of her yearly physical, although everything seemed normal, fortunately), and opened up shop all before the sun rose. You dealt with all the catering and pickup orders as they came in and kept tabs on everything baking in the back while Maya dealt with the storefront. You dipped out at noon to get lunch for the both of you, listened to the police reports on the way, and breathed a sigh of relief that Nueva York managed to retain its peace for another day.

The afternoon crept by far more slowly, and by the time closing rolled around you got to share the news with your business partner.

“I’ve made a deal with a company to sell out our stock at the end of every day,” you told her while you swept and she dusted the countertops. It was ten ‘til but it helped to clean up before since she always went straight home once the sign was turned off. “It’s, uh…an all-day type-thing, and they needed extra stock in their cafeteria. I think it’ll help with keeping up.”

“Oh, that’s wonderful!” Maya exclaimed, smiling broadly with glittering eyes. “I always did hate seeing it go to waste since the donation centers never took all of it. Do you need any help packing it all up?”

“No, I’ve got it. It’ll only take a few minutes, and I can be the only one to deliver it. Kind of a hush-hush type thing.”

She quirked a brow, glancing over your shoulder as the bell over the door jangled. “So long as it’s not for the mafia.”

You rolled your eyes. “I’m not working for the mafia. I’m not pretty enough to work for the mafia.”

“You don’t have to be pretty to work for the mafia.”

You turned to find Jess standing there with a lazy smile, dressed in sleek civvies. You swallowed. “Hey. I thought I was dropping all this off?”

“I’m in a bit of a time crunch,” she said with a smile, but you didn’t miss the subtle urgency in her eyes. “I’ll help you pack it up and get it there.” She nodded to your aunt. “How’re you doing today?”

“I’m just fine,” Maya beamed, coming around the counter. “Thank you for your business. What’s your name?”

“Jessica Drew,” the other Spider-Woman introduced smoothly, extending a hand. Maya shook it. “Your niece told me good things about you.”

“Oh, as she should!” Maya laughed, eyes twinkling. “I’ve been taking care of her since she was knee-high to a grasshopper!”

“And I think it’s about time for you to head home to take your meds,” you interjected, face warming as you set the broom to the side and moved towards the back to get her things for her. “Want me to call you an Uber?”

Maya scoffed quietly, casting Jess a look that screamed, ‘can you believe the nerve?’ “I’ll walk. I didn’t get all my steps in this morning.”

“Be careful, then.” You dipped into the kitchen, grabbed her purse, and reemerged to find them chatting about children—your aunt had a sixth-sense when it came to babies, and evidently she’d picked up on Jess’ glow. “I’ll see you tomorrow for supper.”

“Don’t forget to bring the pickled beets,” she reminded you, slipping into her jacket despite the sun still shining bright upon the pavement outside.

“Believe me, I won’t. They’ll never get eaten if I don’t deliver them.”

“They’re good for you.”

“I’m sure they are. But my tongue says otherwise.”

You waved her out, locked the door behind her, and flicked off the neon sign. You turned back to Jess, brow furrowing. “What’s wrong?”

“Under normal circumstances, we’d introduce you to anomaly-catching a little more gradually,” she responded, face schooling into business-mode, “but Peter B.’s out and Miguel thought he could handle it on his own. Again. I need backup.”

You nodded, already moving to shed your apron and the unnecessary outer layers that would interfere with the UMF. You slipped back into the kitchen to toss them into the hamper and, this time, she followed. “Do I need to be concerned about his evident tendency to jump in head-first without looking?”

“He’s been like this for as long as I’ve known him.” Jess began to type into her (not) watch. “He’s one of the most headstrong people I’ve ever met.”

“You’ll meet few others that are more stubborn,” you muttered without really thinking. You tried to hide your wince by tapping your belt to activate your suit. “Where to?”

The portal blossomed open, and Jess gestured towards it. “After you.”

You swallowed, eyed it for just a moment, then moved through the vortex.

It spat you out on top of a brick office building in a version of New York only slightly less futuristic than your own. The sheer noise of screaming in the streets below, both from humans and vehicle brakes alike, precluded the rumble of an explosion that rocked the infrastructure beneath your feet. You whipped around to the cloud of smoke that arched up into the dusky sky, the lenses in your mask automatically adjusting to locate the heat signatures of two individuals caught in a grapple amongst the flames.

“I’ve got a visual,” you told Jess as she emerged next to you. “Over there.”

“Let’s go. Looks like it’s already gotten out of control.”

The pair of you leapt off the rooftop, swinging out into open air. You watched a rapidly moving shape bulleted from the plume off to the left, and you could hear maniacal laughter even with the wind battering your ears.

“Oh, great,” you muttered, “we love the crazy-ass grenade enthusiast.” You jerked your head towards the crumbling building—the other heat signature had yet to move. “Go check on Miguel, I’ll track our escapee.”

Jess looked like she wanted to argue, but she only released a sigh. When it came to split-second decisions like this, there was never enough time to deliberate. “Be careful. Don’t get in over your head.”

“You got it.” You diverted to swing wide around a warehouse, setting a waypoint on the villain-of-the-week rocketing through the buildings. He was a fast bastard, but fortunately he had yet to start dropping bombs on the bewildered population below—that’s when things with the Goblins always got tricky.

Soon enough, you managed to pick up speed to catch up, and with a well-timed turn and tight swing you lunged for the glorified hoverboard. You connected, latched on to the edge, and the added weight caused the entire platform to dip and sway dangerously.

“Another little spider!” crooned the Goblin, banking sharply to the right in an attempt to knock you off with a light pole. The reverberating collision with your hip hurt like a bitch. “How lucky am I to have as many new friends with which to play?”

“Save it, Osborn!” you growled, using the momentum of his next whirl to get a knee up over the edge. You grabbed his ankle and dislodged his foot from the hook, cooling a loop of webbing around it and securing it out of place to hinder his balance. “If you’ll just make this easier on every one of us, that would be appreci—”

You should have expected him to slam his heel into your elbow, but in hindsight you’d thought him not nimble enough to outspeed you.

“But that would take all the fun out of the game,” he crowed, stooping down and planting a fist into your temple. Your vision swam and your grip loosened. “And teaming up two against one is hardly fair to start with!”

You gritted your teeth and grabbed the nape of his neck, wrenching him down towards you and flipping the board in the process. You managed to yank his other foot out of the stirrup and then the both of you were freefalling. You snatched the board with a web before he could reach it, slinging it at the nearest building and embedding it into the side. 

Unfortunately, he decided to latch onto you, instead. With the talons built into his gloves, no less. Sharp pain pricked your thigh and side, respectively, and when you landed on a terrace all the wind was knocked from your lungs. The Goblin dug in deep, hefting you up and over his shoulder to throw you bodily over into the street below.

The next impact was softer than you expected. You blinked the tears from your eyes only to find that Miguel was the reason for it, the lenses of his mask narrowed dangerously at you.

“You shouldn’t have gone after him alone,” he growled, reaching out with his free hand to pierce the mortar with his claws and suspend you against the wall. You struggled to catch your breath, especially with his tight, unyielding grip around your waist. “That was stupid. This one could easily do a lot of damage, and—”

The squawk of the Goblin over your heads caused both of your gazes to snap upwards, watching through the bars as Jess kicked him down. Within seconds she had him webbed in place, face pressed against the iron, glaring down at you. His wrists were bound together over his head, and you caught the subtle movement of his fingertips against the device coiled around his wrist. The grind of stone, the hum of fission, and the flash of light all activated your senses.

“Mig—!” you started, head whipping to the other side. You grappled at his rigid form to wedge yourself into his side, blocking him from the glider that launched itself at him at full speed. You absorbed the blow, but the force of it knocked both of you from the wall and tumbled down to the street. Your vision blacked out briefly when you landed roughly on your side.

“Por los clavos de Cristo!” Miguel snarled, breathing harshly even as he scrambled up from the concrete. “Idiota!*”

“Imagine cussing someone out for keeping your ribs from getting caved in,” you wheezed, hearing the glider sputter. You twisted to squint up at Jess decommissioning the damned thing. “De nada, cabrón.**”

Miguel’s silhouette shaded your eyes as he kneeled next to you, lenses narrowed to slits, but the quiver in his hands as he reached out to you caught you by surprise. He carefully prodded your bruised side, retracting his touch instantly when you hissed.

“I’m fine,” you said, slowly sitting up with a low groan. “Just a scratch.”

“Hey!” Jess called, hauling the goblin over her shoulder. “Do you have the field?”

He didn’t move an itch, save to hover his hands over you as you clambered back to your feet. There was scarcely half a foot of space between you, his hunkered, hulking form blocking your view of much else beyond the glowing nanites highlighting his contours.

“Miguel!” Jess tried again, jumping nimbly down near you. “You all right?”

That finally seemed to break him out of whatever trance he’d been caught in, and you watched the tension return in full force to his body as he straightened to his full height once more. “Here.” He snatched the device from his hip and tossed it on the ground. She unceremoniously dumped the Goblin into the field, then moved over to pick up the glider. He didn’t look away from you, finally croaking, “That was reckless.”

“You don’t seem to have the Spidey-Sense,” you pointed out wryly, rubbing your ribs gingerly. “And it could’ve been worse.”

He brandished a pointed finger in your face, the hook of his claw gleaming with the UMF stretched over its surface. “Don’t,” he growled, “do it again. I mean it.”

You quirked a brow at him, despite knowing he wouldn’t be able to see it. You folded your arms over your chest so he’d get the message. “Don’t jump in by yourself again,” you returned evenly. “You’re not invincible either, Miguel.”

And he did look worse for wear. Now that your vision had cleared, you could see that he was favoring his left leg, the UMF along his shoulders was glittering as the nanobots attempted to stitch themselves back together, and the lens over his right eye was cracked.

“This isn’t about me,” he started curtly.

“It is, because you came here without backup.” You smacked his hand away and returned his previous gesture, fingertip centimeters away from his nose under the mask. “Don’t tell me you’re stupid enough to think that you could handle all this by yourself.”

His extended silence, grating and aggravated, was answer enough.

You rubbed your forehead and let out a heavy sigh. “Let’s just…finish this up, shall we? You need to get checked out.”

He tilted his head, and you had the distinct impression that he was scowling at you. “Since when did you become team leader?”

“Since when did you become so damned irresponsible?” you shot back.

Jess’ face slackened into shock, her eyes cutting away as she pursed her lips. She looked on the verge between laughter and wanting to leave immediately. “Opening a portal back to base,” she offered mildly, turning her back on the both of you for some illusion of privacy.

“Look,” you hissed, gesturing towards Jess and the Goblin, who was chuckling quietly to himself despite being caught in a rather humiliating predicament of being tied ass-over-head. “I understand that you’re the head honcho and that you seem to have your bluff in on everybody else in the Society. Most of them are terrified of you, from what I gathered yesterday. And I don’t know to what extent that you’re familiar with me, like you said, but I know you, too, Miguel. I know that you’d sooner eat drywall than admit you’re wrong, and that your head is harder than vibranium. I know that you always take on too damn much on your shoulders because you feel like no one else could or should have to handle it. But I swear to God, I am not going to let you run yourself into the ground. You’re just like…” You stopped abruptly, swallowed, then dropped your head to shake it in frustration. “...look. The whole reason you have these dozens of Spider-People recruited in the first place is to help you manage all this, not to pick you back up when you inevitably fall apart trying to carry it all by yourself. You’re not fucking Atlas. And if you’re going to be like this all the time, I’m not going to have any part of it. Comprendes?***”

He glowered at you for a long moment before his low, gritted tone reached you. “Comprendo.****”

“Great.” You thumped him in the sternum for good measure. “Now I’m sure there’s a long, drawn-out, laborious process of paperwork following this, so the sooner I can get it done and deliver the heaping pile of croissants I had leftover today, the better.”

You turned as the portal blossomed open once more, waving Jess off and slinging the Goblin over your shoulder, and marched right on through back to HQ, feeling Miguel’s eyes burning into your back the entire way.

[adarkromantic]

remote student!reader x stalker

-gender neutral reader

-cisgender male stalker

you live in the middle of the woods and take your uni lectures in the comfort of your own home. the woods rustle every so often but its just an animal. right..?

[cw: swearing, stalking, sex, dubious consent, choking, restraint, infantilisation, praise kink, dacryphilia- arousal when seeing someone cry]

"What do you mean 6.3 isn't the answer? I swear to-" You curse and bang your head against the algebra textbook that is currently the bane of your existence. It's too late to keep on doing this to yourself and your phone showing that it's nearly midnight is proof. Sighing, you roll your chair over to the window and press your forehead against the cool glass, listening to the wind whistling past your little home in the woods.

Pine trees line the land outside your front garden, dark green needles poke through the wooden fence that encircles your bungalow. You rub away condensation on the window with the ball of your palm, squinting to see the woods again; the trees are blurry towers of the Earth.

A shadow darts from one tree trunk to another. The hairs on the back of your neck stand up but you run a hand down your face. You let out a long breath and whisper calming words to yourself to warm the cold pit of dread in your stomach.

"Just a deer."

Several weeks later...

Your English coursework was draining the life out of you, so you close your laptop and give up on writing an introduction to your essay. A headache is brewing behind your eyes and you groan, spinning around in your desk chair to face the window that paints a clear picture of the woods.

The trees sway outside your woodland bungalow, a sea of green you can traverse with a good flashlight. An owl swoops down from the treetops, near crashing into your window, and causing you startle back falling to the floor with a clatter. On your ass, you laugh and haul yourself back up again, face flushed pink.

You look out the window but... you can't see your reflection or the trees anymore.

Eyes as green as the swaying tree-line stare back at you and two pupils dilate to blackholes, fixed on you. A hand clad in black leather slams against the window and you jump back, hitting your foot on the wheels of your desk chair after you grab your phone from the desk.

"Ow, shit-"

You curse and scramble on your hands and knees to your bedroom, slamming the door behind you. There is a crash and you know it's a window being smashed, followed by heavy thumping footsteps.

Your chest feels too tight for your lungs to move and your hands shake gripping your phone, cold and sweaty, as you press  'emergency' instead of tapping out your password. The wifi bar isn't there and neither is a byte of data. But you paid your data plan for this month.

"No, no, no!"

Your outside lights are still on and the lamp at your bedside too, so the power isn't out. It hits you and you curse again. The router is in your office. Your face scrunches at your own idiocy and you slump against your bedroom door, brows furrowed. 

Those heavy footsteps are getting closer.

You tremble and set your phone down gently, searching for anything close to enough to stave off the intruder but can only reach one of your trigonometry books. Your fingers barely brush the spine of the thick paperback, even as you try to shuffle closer.

The door suddenly rattles against your back, knocking your head back hard and forcing the air from your lungs. You wince but push back against the brute force, mouth clamped shut to silence any whimpers. A voice calls to you.

"I saw you looking at me a few weeks back, night owl. You should really go to bed earlier."

The voice is male, rich like the scent of wood that clings to the very air around you.

"Get the fuck out of my home! I've called the police and you will be put away for a long time, you sick creep." You shout, willing your voice not to break or crack.

His deep laugh echoes and reverberates through your bones, a chill down your spine.

"W-what's so funny?" Your voice wavers despite your attempts to keep it steady, knowing he is listening.

"I smashed your router. As well as the living room window. Sorry, night owl." He answers in a sarcastic drawl that makes your face heat. You know when you're being made fun of. "So the police aren't coming and I won't be put away."

The intruder laughs again and he rattles the doorknob this time, making you start with a jump. Goosebumps ripple over your clammy skin and the hair on the back of your neck stands up.

"Open the door, night owl." He croons from behind the door and kicks it. The hinges rattles from the force. Cold dread spreads through your chest and fear trickles down as the sweat on your crinkled brow.

"Are you armed?"

You gasp out, shoulders still tense as you continue to push back against the door.

He bangs what sounds like his fists against the door. "No. Bare handed. Just open the door." Another bang. "Let me in, night owl. I'd love to meet you face to face instead of watching you through your window."

Your stomach drops and bile rises in your throat. Bang.

"Open. The door." He pants, slamming a shoulder into the door and splintering the doorframe. The door sags against your back and you crawl out of the way before it can crush you. It slams into the floor with a thundering crash.

A figure looms in the doorway. The intruder.

"Oh, my night owl-" He grins, baring his palms to you; calloused and smooth. "don't fly away now." 

He's tall, at least 6'2 because he has to duck his head to pass through the door frame. His hair falls in brown waves across his forehead, cropped just above his thick eyebrows raised in surrender. Pink lips are stretched into a smirk and dimples mark his cheeks, highlighting his sharp jawline. The intruder is dressed in black tech wear, fitting for his crimes, and pulls off leather gloves as he enters your bedroom.

You scream and launch your trigonometry book at him. It smacks into his chest but bounces off, like a squeaky toy against a brick wall. He cocks his head to the side and grins, running his tongue over his top row of teeth wet with saliva.

"I'm not armed." He kicks the book to one side. "So there is no need for that. Not that this textbook would stop me, I was good at trig."

The intruder slinks towards you, teeth bared in a wide smile and those forest green eyes sparkling under thick black lashes. "My name's Finch, night owl. I already know yours but I want you to say it anyway because I love to hear your voice."

You splutter for a moment, dumbfounded before blurting out. "My name is Y/n. H-hello Finch."

Finch grins and reaches out a hand, brushing his knuckles against your cheek. "I've imagined what you look like when you cry for a month now. Seeing it in real life is so much nicer."

A tear rolls down your face as one of your eyes twitch, your teeth grinding and your jaw set. Finch wraps his other arms around your waist, pulling you flush against his chest and leans down. He sticks out his tongue and licks the tear from your cheek.

"Fuck, night owl. You look so good when you cry."

You tense as Finch presses his hard on into your thigh, biting your lip to stop any more tears from falling for fear of making him more excited.

"Don't do that." Finch whispers, running his thumb over your bottom lip before hooking it behind your teeth to stop you biting it again.

"I'm not going to hurt you, night owl."

His voice blows warm against your neck, lips almost touching the column of your throat.

You can't stop your shoulders shaking as you sob, squeezing your thighs together as a warmth builds between them but not from friction although you wish they were rubbing against something. "P-please. Leave me alone, Finch."

Finch pushes you down onto your bed, then gripping both your wrists in just one of his hands. "I'm afraid I can't do that."

He croons, pushing his hips against yours as you fight the urge to grind against him.

"I've been watching you for too long to scare you like this and just leave. I don't want to scare you, night owl, I want to make you happy. I like you, Hell, I might even love you. So I will make you happy..."

Finch lowers his voice to a deep growl that makes the fire in the pit of your stomach burn with the heat of a thousand suns.

"and you will take it, won't you?"

Animal instinct compels you to nod, say 'yes' and survive. Just animal instinct.

"Y-yes, Finch."

"Good." Finch pats your cheek with his spare hand before letting it drop to gently wrap around your throat. "So spread your legs, night owl."

Finch leans closer and runs his teeth over the column over your throat, nipping at the soft skin there. Your breaths come in short, harsh pants as your legs fall open for him and Finch murmurs.

"If I let go of your wrists, you won't fight me, will you?" He draws back and pouts mockingly, shaking his head before patting your cheek again. "Good pet."

You don't move your wrists, even as Finch releases them from his grip.

"What are you going to- to do to me, Finch?"

The words barely fall from your lips as Finch grins down at you and straddles your hips.

"Well, night owl, since you asked nicely..."

Finch undoes the buttons to his trousers and exposes his throbbing cock. The tip is pink and uncut, you want to wrap your mouth around it- You blink that thought away. Finch smirks as if he heard you and quickly tugs down your shorts.

"You are going to suck my cock, night owl and when I cum in your mouth I want you to take it like a good little pet, okay?"

Animal instinct. You nod and Finch kisses your forehead, patting you like a puppy waiting for a treat from its master. A good little pet. Ready. Finch pulls you up and pushes you down onto your knees on your bed, facing him as he stands beside you. He is tall enough that even when you are kneeling on the added height of your bed, the tip of his cock still brushes your nose.

You lick your lips and look up at Finch as he tangles his hand in your hair, yanking your head forwards so your mouth wraps around his cock. "Now suck, night owl."

He groans and tugs your hair at a slow pace at first, your head bobbing as your throat works to swallow his size. Finch has to be 6 inches at least. "Good fucking pet."

He coos and gently slaps your cheek so the tears forming in your eyes as you choke roll down your cheeks. It only makes him harder.  You gag and splutter as Finch slams his cock down your throat, groaning as his cum floods your mouth.

You have to take large gulps to not choke and even then, as Finch yanks your head back, cum from your mouth is connected to his cock in a long strand before he wipes it away whispering.

"Good job, pet. You did so well."

Finch kisses you and licks your lips, tasting his own cum from your spit. "Are you happy?"

You nod dumbly, like an animal. Your head feels like it's stuffed with cotton, floating almost.

"You are happy, aren't you? You love sucking my cock, pet. Mhmm." Finch pouts and wrap his hand around your throat again, grinding his cock between your legs. "Do you want me to fuck you now, pet? Make you feel all nice and happy."

Just nod. Do what he wants. Good little pet. You smile and lay on your back as Finch crawls on top of you, giving you no time to think before slamming his cock inside of you.

No lube needed when you're already soaking wet. He pumps in and out once, tightening his grip on your throat.

"Over this month I've been wondering if you are the type of pet who likes it slow or fast. Rough or gentle."

You moan and your head lolls back against the sheets, feeling too heavy to hold up anymore as your throat aches from taking Finch's cock. 

"I've decided I'm going to make that decision for you, pet, and it's going to be fast and rough."

Your fuzzy brain lights up at that and your eyes widen, lips stretching into a submissive smile.

Finch grins and nods. "Oh, you want that, pet?"

You nod with him and yelp as he suddenly thrusts into you, his balls slapping against your skin as he does it again and again. Your mattress starts to squeak and the bed frame moves with each brutal thrust from Finch.

"Good fucking pet, night owl." He groans and kneads your thighs with his calloused hands. Pressure builds between your legs, nerves imploding like stars as Finch rolls his hips.

"F-Finch, I can't- please, I-" You babble incoherently whilst skin claps against skin, eyelids fluttering as Finch edges you closer.

"You're gonna make me cum too quickly, pet." Finch grits out through a clenched jaw, his head leaning back and brown waves of hair fall across his knitted brow. "Fuck, pet. I'm gonna- cum." Whole galaxies seem to explode behind your eyes as your body writhes in orgasmic bliss.

Your body sags limp into your mattress and Finch rises to his knees on either side of you, cum still leaking from the tip of his cock, now softer.

"You okay, pet? You took me so well, didn't you?" He runs his thumb over your bottom lip, wearing a smile that exposes his dimples to you.

"But you need to go to bed, night owl."

Lifting you effortlessly, Finch tucks you into bed with his cum still sticky between your thighs.

He draws back and walks over your bedroom door, left broken on the floor.

"I'll see you tomorrow, night owl." Finch hums and leans against the doorframe before turning.

"But you might not see me."

[mocktortis]

So recently I've been seeing this bit of information around that "Lewis Carroll was a mathematician and he was inspired to write Alice in Wonderland because he was really frustrated by the contemporary math of his time." Specifically people kept bringing up imaginary numbers.

And that struck me as weird, because I have read a lot of writings on Alice, and I had never come across this before. I have the mega-version of the Annotated Alice, and multiple copies of Alice with introductions from pretty well-known Carrollian scholars. They all mentioned that Carroll's real identity was the mediocre mathematician Charles Lutwidge Dodgson, but never really tied that with Alice. Even The Annotated Alice, which was edited by the math guy Martin Gardner, only has math related footnotes when math (usually arithmetic) is specifically brought up by the text (such as when Alice is struggling to remember her times-tables). I should probably warn you now, there will be math in this post.

So... where does this claim come from?

I asked people for sources, and I got one response that was actually useful. They admitted they didn't have an academic source, but told me to try googling "Alice in wonderland math". It was the best lead I had (one person told me verbatim "Google is free". Classy), so I put it into Google and came up with a decent amount of results. The first article I found linked to another article from 2009 by a doctoral candidate at Oxford called Melanie Bayley. Unfortunately, the article is only available if you make an account with New Scientist, which I was not keen to do. I moved on, continued reading through poorly written articles and 10 year-old blogs, looking through their sources to see where they were getting their information... and every single article and blog post linked back to Bayley's article from 2009, or an op-ed she wrote for NY Times (also blocked behind a give-your-name-to-the-fae type deal). Fortunately, one of my family members actually has an NY Times account already, so I just asked if I could use their account to access this article.

Eureka. After reading through so many misinformed and poorly explained sources (one of which just copy-pasted Bayley's article into their blog), reading Bayley's actual article was like a miracle. It was so well-written, well-researched, and actually solidly convincing. I was nearly crying at just how beautiful the thing everyone else was ripping off really was. This encouraged me to make an account to read her New Scientist article.

Bayley references back to a paper published in 1984 by Helena Pycior, At the Intersection of Mathematics and Humor. Pycior is a Professor at University of Wisconsin Milwaukee, an MA in math, and a historian. Her paper is rigorously researched and does a fabulous job of explaining mathematical advancements (especially symbolical algebra) in the late-18th to mid-19th centuries, Lewis Carroll's own education in mathematics, and his more serious writings on math as Charles Dodgson. Pycior also highlights a line in the chapter Alice's Evidence, when Alice remarks, "I don't believe there's an atom of meaning in it, which is eerily similar to a line in Augustus De Morgan's Trigonometry and Double Algebra ("With one exception, no word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter,"), a math textbook Carroll definitely read. Pycior's paper is very technical, however, and might not be for everyone. But it is a great foundation for Bayley to base her claims.

While I would highly recommend reading Bayley's articles, I understand not everyone will be interested, so I will summarize:

Bayley's analysis mainly focuses on sections added by Lewis Carroll after 1864. Carroll's original manuscript, written for Alice Liddell and her sisters, is missing several scenes from the final published version, and there are some scenes in the manuscript that were extended in the final version. The scenes Bayley dissects are Advice from a Caterpillar, Alice's first meeting with the Duchess, Alice's conversation with the Cheshire Cat, and A Mad Tea Party.

Bayley first covers the chapter Advice From a Caterpillar. She connects it to De Morgan's Trigonometry and Double Algebra. Helena Pycior points out that Carroll was "clearly inspir[ed]" by De Morgan, and references De Morgan in a few of his academic works published under his real name. Bayley draws attention to De Morgan's use of the Arabic phrase: "al jebr e al mokabala" —the original Arabic name for algebra. Bayley explains that it means "restoration and reduction". I actually don't think I can put it much better than she did, so I will use an excerpt from her article:

"Restoration was what brought Alice to the mushroom: she was looking for something to eat or drink to “grow to my right size again”, and reduction was what actually happened when she ate some: she shrank so rapidly that her chin hit her foot."

A little later, Bayley adds:

"De Morgan... proposed... that symbolic algebra should be considered as a system of grammar. “Reduce” algebra from a universal arithmetic to a series of logical but purely symbolic operations, he said, and you will eventually be able to “restore” a more profound meaning to the system – though at this point he was unable to say exactly how."

Part of the Caterpillar's "advice" to Alice is "Keep your temper", after Alice complains that she keeps changing in size. Alice assumes he means not to get angry but, as Bayley explains "To intellectuals at the time, though, the word “temper” also retained its original sense of “the proportion in which qualities are mingled”" ie. tempered steel, tempered glass, tempered chocolate etc. Bayley proposes that the Caterpillar is using this meaning of the word temper— meaning his remark would be advising her to keep her proportions the same, even if she can't stay the same size. That remark becomes relevant after Alice tries changing her size with the two sides of the mushroom: when she tries just the small side, her torso shrinks and brings her face so close to her feet she can scarcely open her mouth; when she tries just the large size, her neck stretches to ridiculous lengths. Only when she tries a bit of both in a balance— tempering them— is she able to change size while keeping her proportions.

Alice next encounters the Duchess in her kitchen, and the Duchess' notably ugly baby. As the Duchess leaves for croquet she throws (literally throws) her baby at Alice, who catches the baby and takes it outside, reasoning to herself that the violent Duchess and her Cook would likely kill the baby if Alice were to leave it there. As Alice looks down at the baby, she realizes it is turning into a pig, and she releases the baby-turned-pig into the wood.

Bayley's interpretation of this scene is as a satire of projective geometry— and specifically the "principle of continuity", laid out by French mathematician Jean-Victor Poncelet. Poncelet's description of the principle (via Bayley) is “Let a figure be conceived to undergo a certain continuous variation, and let some general property concerning it be granted as true, so long as the variation is confined within certain limits; then the same property will belong to all the successive states of the figure.” In Bayley's NY Times article, she explains it more clearly as, "[the principle of continuity] involves the idea that one shape can bend and stretch into another provided it retains the same basic properties— a circle is the same as an ellipse or parabola."

Bayley suggests that Carroll's rebuttal to this is based off Poncelet's use of the word "figure". If the figure of a triangle can change its shape while remaining a triangle, then the figure of a person (or baby, in this case) can also change its shape. As Bayley puts it "What works for a triangle should also work for a baby."

Skipping ahead to the Mad Tea Party, Bayley proposes that the characters of the March Hare, the Mad Hatter, and the Dormouse, are paralleling the concept of quaternions, proposed by William Rowan Hamilton in 1843. Hamilton's Lectures on Quaternions was the first way of representing rotations in three dimensions with algebra, and it was well-known enough at the time that it is reasonable to assume Carroll had read it, or at least seen arguments relating to it.

Here is Bayley's explanation of Hamilton's quaternions:

"Just as complex numbers work with two terms, quaternions belong to a number system based on four terms. Hamilton spent years working with three terms – one for each dimension of space – but could only make them rotate in a plane. When he added the fourth, he got the three-dimensional rotation he was looking for, but he had trouble conceptualising what this extra term meant. Like most Victorians, he assumed this term had to mean something, so in the preface to his Lectures on Quaternions of 1853 he added a footnote: “It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time.”"

Breaking that down a little more (serious math alert): a complex number is a number with two terms, a real portion (represented by a), and an imaginary portion (represented by bi), and is written as a+bi. One of Hamilton's quaternions would be represented like this: a+bi+cj+dk. (I don't really know how they work either.)

In A Mad Tea Party, the Mad Hatter says, "It's always six o'clock now", trapping the party at perpetual teatime. The Hatter's explanation for this is that he quarreled with the personification of Time, and in retaliation, Time is keeping the clocks at six for the foreseeable future. Without Time, the Mad Hatter, the March Hare, and The Dormouse keep rotating around the table, as if in a two-dimensional plane. It is possible that this is Carroll's way of poking fun at the absurd idea that time would factor into an expression meant to determine the movement of objects in space.

And it is this final section of Bayley's article which gets misinterpreted into the claim, "Lewis Carroll was inspired to write Alice in Wonderland because he was frustrated by imaginary numbers."

I want to end with this: we have no proof Carroll intended any sort of mathematical allegory in Alice's Adventures in Wonderland. He seemingly did his best to keep his life as a mathematician and his life as a popular children's author separate from one another. Most of his surviving writing's on his inspirations for Alice make no mention of Math. That said, Melanie Bayley's article provides a truly fascinating interpretation of some of the most beloved episodes in Alice, and I wouldn't begrudge anyone who wants to believe it.

If you're interested in reading more, this is a free PDF of Melanie Bayley's NY Times op-ed. The first page is an email someone sent to friends that contained the article, but the full article is underneath.

Also, this article by Art Publika has a great overview of both of Melanie Bayley's articles, plus some extra background on Carroll, and so many pictures.

[oliviabutsmart]

Physics Friday #17 [Opinion]: The Great Tau vs Pi Debate

I'm really going for these hot takes now! Just look at me help tear the internet apart!

Education Level: Primary School (Y5/6)

Topic: Trigonometry (Mathematics)

Introduction: What is Tau? (And also pi)

Chances are, are that you already know what pi is ...

π = 3.14159265358979323 ....

I wrote that from memory, yes, I remember 18 digits of pi.

Anyways, we all know what pi is. It's the ratio between a circle's circumference and diameter, it's the ratio between a square and a circle inscribed in the square.

Image Credit: GeeksforGeeks

Pi is also an angle, well, every real number is technically an angle. But pi is a special angle.

It represents a 180˚ turn in radians.

For those who also don't know what a radian is, it's sort-of a special unit of angle measure. Much like how we measure length in meters or feet, we can measure angles in degrees, radians, or revolutions.

A 1 Radian sector of a circle of radius 1 will have an arc length of 1. This makes a 1 rad ≈ 57.2958˚.

Image Credit: Mometrix Test Preperation

You may be able to see why we would prefer radians to degrees. Radians often clean up our equations and even more, it allows us to express common angles in terms of pi.

Because a circle's circumference is equal to 2π times it's radius, a full revolution is equal to 2π units around the circle.

Thus we can express:

270˚ = 3π/2 rad 180˚ = π rad 90˚ = π/2 rad 60˚ = π/3 rad 45˚ = π/4 rad 30˚ = π/6 rad

Looks pretty nice? Well, there's actually another constant we can use to express angle measurements.

Tau, or τ, is equal to 2π. That means that τ = 2π, and a full revolution is equal to τ radians.

Thus we can express:

270˚ = 2τ/3 rad 180˚ = τ/2 rad 90˚ = τ/4 rad 60˚ = τ/6 rad 45˚ = π/8 rad 30˚ = π/12 rad

Tau is a relatively new symbol in the world of mathematics. And it's rather prolific online now. Pi is still used as the universal constant to represent radians and other circle-based coordinates.

While Pi is standard, there's been a growing movement to make Tau the new standard for angle measures. Let's look at the argument.

(Common) Arguments for and against switching

This section will only focus on the main arguments for and against using Tau as a common constant for angle measures. Below is a Numberphile video that goes into some detail over the main arguments for the pi vs tau debate.

Let's look at the pro's of switching to Tau:

Taking portions of a full revolution is significantly easier to grasp conceptually (Half a revolution is τ/2, a quarter revolution is τ/4)

It simplifies several equations in physics and mathematics by removing a factor of 2, e.g. Circumference = 2πr = τr

And here are the cons:

It's an unnecessary level of effort to change everyone over to a different constant when π is already doing a good enough job

It introduces an additional factor of 1/2 in several equations in physics and maths, e.g. Area = πr² = τr²/2

Alright, now it's time for the hot takes/opinions

Pi is better. That's it.

Okay okay, so there are obviously reasons.

Changing Standards

Firstly, I do subscribe to the idea of "if it ain't broke don't fix it", Pi is universally used. When I mean universal I do mean universal. Outside of the online maths-education-space world, tau is relatively unknown if at all.

If you want to switch over to tau, you will need to convince a supermajority of establishments, teachers, or professors globally - across multiple different disciplines like Economics, Engineering, Mathematics, Physics, and Computer Science.

Not just that, but you want to absolutely avoid the relevant xkcd:

Which will come inevitably when you make Tau popular enough that a non-negligible amount of people use it, but not popular enough that you have a majority of the population convinced.

You might as well try change the number base we use.

Changing Hardware/Software

Pi has become so entrenched in our information age modern society that you'll also need to now alter lots of computer software and hardware.

On MacOS (and Windows to), π is a default character on a standard English keyboard layout (using the option keys). Whereas other greek letters are relegated to your typing program of choice's maths function.

This makes writing τ a lot less convenient than π on a lot of computers. One can be written with Option+P, and the other needs to be copy-pasted into every text every time you want to use it.

Not just that, but we also need to consider that most applications that use calculation software (i.e. Microsoft Excel), uses π.

Do you know why windows is so backwards compatible? A lot of Excel's software is legacy, meaning that a lot of Excel software is old and at risk of breaking with new updates. Many, many, financial systems require π otherwise they too would break.

What about programming languages? Most modern languages include a mathematics module, and in order to fully switch to τ, every single last one of them needs to implement changes.

This becomes difficult, especially with older languages that don't get as much updates or developments, but are still used regularly in a lot of programs.

If you fully fully want to change to tau you'd have to go through every instance of pi and change it to tau/2 in order to not confuse future readers of the program.

It's in my opinion that doing all of this ... is not needed when you're getting very diminished returns.

Introducing More Fractions

The second point, that tau introduces extra fractions, is also something I agree with.

Fractions are innately more difficult for a layperson to grasp. Especially more difficult than multiples of a number.

Switching to τ means introducing an additional factor of 1/2 in every equation. This is okay for small fractions like 1/2 or 1/4. But angle measures like 30˚ end up having factors of 12.

Do you know what 1/12 is? Could you reasonably ask a layperson to write out 1/12 of the top of their head?

But this spreads much more widely. Every integral now has extra fractions. Fractions are the most common reason why you fuck up an integral. Because doing arithmetic with fractions is innately harder than arithmetic with integers.

The main argument here is that in order to trade the conceptual-ness of simplifying angle measurements slightly and conceptually, you end up making a mess of a lot of other aspects of trigonometry in terms of the arithmetic.

The Online World is not the Only World

The most annoying thing I find about the tau vs pi debate is in how people advocate for tau. It's a microcosm of a lot of online activism.

Just because your movement has a presence online, does not mean that it's popular elsewhere. All you do when you promote things online is confuse or offend people when reality slaps you in the face.

The tau vs pi debate is the most tame of these online vs real world disparities, but it's a good example in the light form.

I've often seen that people who use tau generally use it without clarifying what tau means. This means that someone who is not familiar with the existence of tau (of which there are many), will be confused when you start using e^iτ/2 to represent -1.

It also isn't helpful as it doesn't actually extend the movement's reach outside of a very minimal niche ...

There are two Numberphile videos on Tau, often credited as what really kick-started everything. Both videos have only 1 million views. About 1/400th the population of the US, and 1/8000th the population of the globe.

When you realise the scale of how small your movement is, it can really put in perspective what is required. And also why people may ask more things of you.

If you want to advocate for τ or anything else, there's a right way to do it, and a wrong way. And ignoring feedback or requests is more a sign of stubbornness or immaturity.

Conclusion

If I were to have it my way, I'd actually prefer making our angle measure constant smaller. Instead, let us have a symbol representing a right-angled turn.

This would be great, as people often deal with right angles a lot more often than 180˚ turns and 360˚ turns. It also handily removes an extra factor of 1/2 from all equations.

Of course, there are obvious disadvantages. But let's be honest. The first criticism I gave still absolutely applies.

Relevant xkcd, again.

Anyways, hope you enjoyed the post. Of course, this is an opinion post, meaning that I would very much like to hear your own thoughts on tau vs pi! While I sounded a bit agitated at the end there its more because of this:

Tumblr is being a bitch and is fucking up my computer's processing power for some reason. Like seriously the speed at which I type is making the website load poorly.

Outside of that, next week will be on Baryon Acoustic Oscillations. See you later!

[soul-has-left-the-chat]

Introduction

I see everyone else doing introductions so might as well do a quick one myself (don't be afraid to say hi, I don't bite ^^)

Name: Soul

Nicknames:

- Soup

- Suop

- Soulie

- Soil

- Soiler

- Soupie (Woopie)

- Newt

- And whatever else you wanna call me

Nationalities: 🇧🇩🇨🇦

Any pronouns :)

INFP-T, 9w1, Chaotic Neutral

Interests:

- Music (bands and stuff)

• STARSET!!!

• Sleep Token

• Bad Omens

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• Palaye Royale

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• CG5

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And so on

- Art

• OC content

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• Art collabs and trades

• Requests (depends)

- Writing

• Lore!!!

And anything else related to writing idk

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• Metal Gear Solid and Rising

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• Soulsborne franchise (Dark Souls, Bloodborne, Elden Ring, so on)

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• ULTRAKILL

• Ghost of Tsushima

• Sky: Children of the Light

• Geometry Dash

• Stardew Valley

• Minecraft

• Obey Me! (Nightbringer)

• Warframe!!!!

- Some animes and shows

• Jujutsu Kaisen

• Trigun (Stampede and Maximum)!!!

• Beastars

• To Your Eternity

• The Great Pretender

• Chainsaw Man

• Bungo Stray Dogs

• Lucifer

• BFDI, BFB, ONE, TPOT, LOTS

- Some book franchises and comics, also manga and manhwa

• Alex Rider!!!

• Villain to Kill

• Market of Monsters

• Arc of the Scythe

• The Hunger Games

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- Science and trigonometry

• Chemistry

• Biology (mostly genetics)

• Microbiology

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- WANNABE BIKER!!!

• dream bike Kawasaki Z400

I'm an introvert with pretty bad social anxiety and extremely asocial, but I love making new friends, so feel free to drop by :))

If you wanna find my art/writing/oc stuff, you can here:

#soupie woopie art

#soupie woopie writing

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For my gen posts:

#soupie woopie

Personal OC tags:

#oc tag: crossfire/hundred

#oc tag: sage the skykid

#oc tag: helios

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Other stuff:

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PLEASE READ!!!

Do you need to talk to me?

ABOUT MY ASK BOX

Edited blog 00:28 / 12:28am 10/9/24

[22ums115]

Algebra - Introduction and Basic Formula

INTRODUCTION AND BASIC FORMULAS IN ALGEBRA

Algebra is the part of mathematics that helps to represent the problems or situations in the form of mathematical expressions. Algebraic formula are used to simplify the algebraic statement. Algebraic formulas are useful for resolving algebraic, quadratic, polynomials, trigonometry, probability, and more.

Introduction of Algebra

Algebra is the branch of mathematics in which arithmetical operations and formal manipulations are applied to abstract symbols rather than specific numbers. The notion that there exists such a distinct subdiscipline of mathematics, as well as the term algebra to denote it, resulted from a slow historical development. This article presents that history, tracing the evolution over time of the concept of the equation, number systems, symbols for conveying and manipulating mathematical statements, and the modern abstract structural view of algebra. For information on specific branches of algebra, see elementary algebra, linear algebra and modern algebra.

Algebraic Formula

Algebraic formulas are the combination of numbers and letters to form an equation or formula. The algebraic formula is a short quick formula to solve complex algebraic calculations.

Algebraic properties

The properties of algebra enable us to solve mathematical equations. Notice that these properties hold for addition and multiplication. These properties include the associative property, commutative property, distributive property, identity property, inverse property, reflexive property, symmetric property, and transitive property.

Algebraic Formula

a2 – b2 = (a – b)(a + b)

(a + b)2 = a2 + 2ab + b2

a2 + b2 = (a + b)2 – 2ab

(a – b)2 = a2 – 2ab + b2

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

(a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca

(a + b)3 = a3 + 3a2b + 3ab2 + b3 

( a + b )3 = a3 + b3 + 3ab(a + b)

(a – b)3 = a3 – 3a2b + 3ab2 – b3 or a3 – b3 – 3ab(a – b)

a3 – b3 = (a – b)(a2 + ab + b2)

a3 + b3 = (a + b)(a2 – ab + b2)

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

(a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4

a4 – b4 = (a – b)(a + b)(a2 + b2)

a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)

If n is a natural number an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)

If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)

If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +an-3b2…- bn-2a + bn-1)

(a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….)

Laws of Exponents (am)(an) = am+n ; (ab)m = ambm ; (am)n = amn

Properties of Algebra :

Commutative property:

Addition : a + b = b + a

Changing the order of addons does not change the sum.

Multiplication : a x b = b x a

Changing the order of the factor does not change the product.

Associative Property:

Addition : (a + b)+ c = a + (b + c)

Changing the grouping of the addends does not change the sum.

Multiplication : (a x b) xc = a x (b x c)

Changing the grouping of the factors does not change the product.

Distributive properties: 

Addition : a × (b + c) = a × b + a × c

Multiplication :  (a + b) × c = a × c + b × c

The distributive property states that multiplying each element by a single term and then adding and subtracting the products is the same as multiplying each component by a single term and then adding and subtracting the products.

Rule of multiplication over subtraction: p (q-r) = p*q – p*r 

If p, q, and r, are all integers.

Left distributive law if p* (q-r) = (p * q) – (p*r)- and

Right distributive law if (p-q)*r = (p*r) – (q*r)-

What are the properties of Algebra?

Associative Property

Commutative Property

Distributive Property

Identity Property

Inverse Property

Reflexive Property

Symmetric Property

Transitive Property

Read more...

[cscclibrary]

We've recently acquired dozens of STEM titles, with an emphasis on study guides and computer programming. Here's a sample of our latest additions!

Mathematics

Testimonios: Stories of Latinx and Hispanic Mathematicians

Mathematics for ESL Learners

Painless Pre-Algebra

The Manga Guide to Linear Algebra

Painless Geometry

Trigonometry: A Complete Introduction

Must Know High School Trigonometry

Trigonometry: With Calculator-Based Solutions

Painless Calculus

Painless Statistics

General Science

Scientific Writing = Thinking in Words

Too Big for a Single Mind: How the Greatest Generation of Physicists Uncovered the Quantum World

Physics: An Illustrated Guide for All Ages

El Libro de la Fisica

Biology: An Illustrated Guide for All Ages

Painless Biology

Chemistry: An Illustrated Guide for All Ages

Painless Earth Science

Computer Science

The Pattern on the Stone: The Simple Ideas That Make Computers Work

Introducing Microsoft Access Using Macro Programming Techniques: An Introduction to Desktop Database Development by Example

AutoCAD for Dummies

Network Security: Private Communication in a Public World

Gray Hat Hacking: The Ethical Hacker's Handbook

Practical Linux DevOps: Building a Linux Lab for Modern Software Development

Java 17 for Absolute Beginners: Learn the Fundamentals of Java Programming

You can use the "What's New" section of our catalog to browse all of our latest items! Current Columbus State students and employees can check out items using a photo ID.

[inkbottlecap]

Inkbottlecap's introduction

Hello, I am Inkbottlecap, or Ink for short. This is my Dark Academia, study, math, and science blog. I'll talk about dark academia, studying, and different math and science studies, such as calculus, algebra 2, trigonometry, chemistry, biology, and pharmacology/other medical... Subgenres? Not sure what to call it. I'll call it that for now. I'm going into pharmacology, but I'm currently in high school.

Anyways, I hope I can meet some people on here. Seeya!

[blogcareerwill66]

Complete Guide to SSC CPO Syllabus 2025: Paper 1 & 2 Detailed Breakdown

Introduction

The Staff Selection Commission (SSC) conducts the Central Police Organization (CPO) examination to recruit candidates for posts like Sub-Inspector (SI) in Delhi Police, CAPFs, and ASI in CISF. Aspirants aiming to clear this prestigious exam must be well-acquainted with the detailed SSC CPO syllabus. This article provides a comprehensive overview of the SSC CPO Syllabus for Paper 1 and Paper 2, helping you prepare strategically and efficiently.

Understanding the SSC CPO Exam Structure

Before diving into the syllabus, let’s briefly understand the SSC CPO exam pattern. The recruitment process includes:

Paper 1 (Computer-Based Test)

Physical Standard Test (PST) / Physical Endurance Test (PET)

Paper 2 (Computer-Based Test)

Medical Examination

Both Paper 1 and Paper 2 are objective-type exams conducted online. Paper 1 tests candidates on four major subjects, while Paper 2 focuses solely on English language skills.

SSC CPO Syllabus: Paper 1 Breakdown

1. General Intelligence and Reasoning This section evaluates the candidate’s logical and analytical abilities. Topics include:

Analogies

Similarities and Differences

Spatial Visualization

Spatial Orientation

Problem-Solving and Analysis

Decision Making

Visual Memory

Discrimination

Observation

Relationship Concepts

Arithmetical Reasoning

Coding and Decoding

Syllogisms

Venn Diagrams

Statement-Conclusion

Blood Relations

2. General Knowledge and General Awareness This segment tests current events and general facts. Key topics include:

Current Affairs (National & International)

History of India

Indian Polity & Constitution

Geography

Economy

Scientific Research & Development

Environmental Issues

Awards and Honors

Books and Authors

Important Days

3. Quantitative Aptitude This section measures numerical and mathematical ability. Topics include:

Number Systems

Computation of Whole Numbers

Decimals and Fractions

Ratio and Proportion

Percentage

Profit and Loss

Simple and Compound Interest

Time and Distance

Time and Work

Averages

Mensuration

Data Interpretation

Algebra

Geometry and Trigonometry

4. English Comprehension This part tests understanding and usage of English. Key areas are:

Vocabulary

Grammar

Sentence Structure

Synonyms and Antonyms

Spotting Errors

Cloze Test

Comprehension Passages

Fill in the Blanks

Idioms and Phrases

SSC CPO Syllabus: Paper 2 Overview

Paper 2 is designed to test the candidate’s proficiency in English. The entire paper revolves around:

Reading Comprehension

Error Spotting

Sentence Improvement

Fill in the Blanks

Synonyms and Antonyms

One-word Substitution

Active and Passive Voice

Direct and Indirect Speech

Para Jumbles

Idioms and Phrases

Conclusion

The SSC CPO exam is competitive, and understanding the syllabus thoroughly is the first step toward success. Whether you're aiming for Delhi Police or CAPFs, having a clear grasp of the syllabus for both Paper 1 and Paper 2 ensures targeted preparation. Focus on each section systematically, take regular mock tests, and stay updated with current affairs to improve your chances. With dedication and strategic planning, cracking the SSC CPO exam is absolutely achievable.

[govindhtech]

New Python Package And Quantum Machine Learning Models

Combining machine learning and quantum computing,  quantum machine learning (QML) is an interdisciplinary field that is rapidly expanding. Studying how machine learning can be applied to quantum problems and how quantum systems might enhance machine learning is fascinating. Python is vital in this business because to its robust libraries and frameworks.

Introduction to Quantum Machine Learning and Python

Machine learning or quantum computing expert to learn QML. Quantum computing, which began in physics research, is now available to high school students as software. Math and linear algebra are the key requirements, along with basic Python. Trigonometry, vectors, matrices, polar and Cartesian coordinate systems, complex numbers, functions, gradients, eigenvalues, eigenvectors, and linear combinations are important math concepts. Although a basic understanding is sufficient, understanding qubit representation and manipulation requires these mathematical building blocks.

Python underpins numerous prominent classical and quantum machine learning tools and frameworks, including PyTorch, scikit-learn, and PennyLane. Free online courses or, if you've coded before, grammar videos, cheat sheets, and little projects are good ways to learn Python. QML benefits from NumPy, a popular Python scientific computing library.

After mastering these basics, you can study QML's three pillars: optimisation, machine learning, and quantum computing.

Optimisation is crucial and often involves minimising a “cost function” through progressive “cost landscape” modifications. Optimisation methods use gradient, which shows a function's steepest change, to find the lowest cost point.

Machine learning allows computers to recognise patterns in data and extrapolate them to new data without programming. This may involve training a model on a dataset, optimising a cost function, then testing it on a new dataset to ensure broad trends. The correct prediction rate or squared distance between model output and label, which is useful for gradient-based optimisation due to its continuity, can be used to measure classification progress.

Quantum computing QML tasks often use neural networks, a key machine learning concept. They are trained using backpropagation to estimate the gradient of the cost function with respect to the weights and have nodes and weighted edges that process data from inputs to outputs. Besides picture classification, machine learning tasks include regression, clustering, and reinforcement learning.

Physical quantum systems and their special characteristics are used in quantum computing to perform calculations. Quantum computers employ qubits, such as photons, superconducting qubits, or trapped ions, in contrast to classical computers. Qubits, which are complex-valued unit vectors or their linear combinations, are the building blocks of quantum information.

The idea of superposition, in which a qubit might be 0 or 1 like a spinning coin, is crucial. Entanglement and interference are also used in computation. Qubit gates, which are similar to classical logic gates, can superpose, entangle, and change measurement probabilities. These processes are usually depicted as a quantum circuit with gates and qubit wires. The final measurement compresses superpositions into classical states.

Quantum machine learning Python packages: PennyLane and Beyond

Combining these components makes Python packages crucial. PennyLane, a cross-platform Python quantum computer programming package with differentiability, is an example. This makes writing and running quantum computing algorithms easier and allows customers to use quantum computers from multiple manufacturers.

The following steps are typical for PennyLane QML program development:

Explain a device: State its quantum device type (e.g., ‘default.qubit’ simulator) and how many qubits (wires) it needs.

Define your quantum circuit (QNode): Write a Python function that performs the quantum circuit and returns a measurement using parameters.

Describe optional pre-/postprocessing: Hybrid models often use preprocessing or postprocessing methods like simple additions or complex neural networks.

Define cost function: Your QNode output and any traditional pre/postprocessing are used to minimise this Python function during training.

Execute optimisation: Choose an optimiser (PennyLane offers many).

Determine step size.

Quantum circuit parameters should be estimated beforehand. Repeat a set number of times to lower costs and adjust parameters.

Appreciate your results: Print or graph the optimisation result to see if the model found the data pattern.

Training a quantum circuit to replicate a sine function shows how to train a quantum model to recognise patterns.Outside PennyLane, specialised Python packages are being created. A new Python library that extends PennyLane's capabilities was designed to simplify Fourier model analysis and training for quantum machine learning models. This program, detailed in “QML Essentials A framework for working with Quantum Fourier Models” by Melvin Strobl, Maja Franz, Eileen Kuehn, Wolfgang Mauerer, and Achim Streit, provides strong analytical tools to understand QML model behaviour and maximise performance.

The main features

Main characteristics of this new package:

Noise addition: By merging different noise models, it can replicate genuine quantum hardware conditions, helping researchers test algorithm resilience and create noise-resistant circuits.

Circuit parameter initialisation methods: The package offers several approaches that can affect training and model quality.

Expression and entanglement calculations: These assess a model's learning and generalisation to new inputs. Expressibility is a circuit's ability to match any target function, while entanglement measures quantum interactions.

Fourier spectrum calculations: It uses two methods to calculate a quantum circuit's Fourier spectrum: an analytical trigonometric polynomial expansion method and the computationally efficient Fast Fourier Transform. This reveals the circuit's core dynamics and capabilities, revealing optimisation options.

Because the package is modular, the quantum machine learning community may simply add new features and encourage code reuse and collaboration. The development team values community feedback and strives towards improvement.

A new Python library, LazyQML, benchmarks and compares many QML models based on architectures and ansatzes from the literature. The conference paper LazyQML addresses the lack of a clear and systematic framework for comparing QML models due to the rapid expansion of quantum computing and the rapidly evolving QML frameworks like Qiskit and PennyLane.

In conclusion, Python libraries like PennyLane make QML accessible by defining quantum circuits, integrating them into machine learning algorithms, and optimising. Dedicated benchmarking packages like LazyQML and PennyLane's Fourier model extension improve the capacity to analyse, train, and compare complex QML models.

[manasadefenceacademy]

NDA Classes in India After 12th: Manasa Defence Academy Providing the Best NDA Training to Students

Introduction

So, you’ve just completed your 12th grade and are wondering what’s next? If you have a passion for serving your country and are looking for a career that’s both challenging and rewarding, then the National Defence Academy (NDA) might just be the perfect fit for you. In this article, we’ll explore the significance of NDA classes after 12th and how Manasa Defence Academy stands out as a premier training institution for aspiring defence personnel.

What is NDA?

The National Defence Academy (NDA) is a prestigious military academy in India that trains future leaders of the Indian Armed Forces. It’s not just a gateway to a career in defence; it’s a place where young minds are molded into disciplined, courageous, and skilled officers. The NDA plays a crucial role in shaping the future of the Indian military, and getting into it is no small feat.

Eligibility Criteria for NDA

Before you dive into NDA classes, it’s essential to know if you meet the eligibility criteria. Here’s what you need:

Educational Qualifications

You must have completed your 12th grade with Physics and Mathematics. This is crucial as these subjects form the foundation of the NDA syllabus.

Age Limit

Candidates must be between 16.5 to 19.5 years old. This age range ensures that you are at the right stage of life to take on the challenges of military training.

Nationality Requirements

You should be a citizen of India, or a subject of Bhutan or Nepal. Additionally, certain categories of refugees and migrants may also be eligible.

Why Choose NDA After 12th?

Choosing to pursue NDA after 12th opens up a world of opportunities. Here’s why it’s a great choice:

Career Opportunities in Defence

A career in the NDA leads to various roles in the Indian Army, Navy, and Air Force. You’ll have the chance to serve your country while enjoying a stable and respected career.

Personal Development and Discipline

The training at NDA instills discipline, leadership, and teamwork. These qualities are not just essential for a military career but are valuable in any field.

Serving the Nation

There’s something incredibly fulfilling about serving your country. The NDA offers you a chance to contribute to national security and pride.

NDA Exam Structure

Understanding the NDA exam structure is vital for your preparation. Here’s a breakdown:

Written Examination

The NDA exam consists of two papers: Mathematics and General Ability Test (GAT). Each paper is designed to test your knowledge and aptitude.

SSB Interview

After clearing the written exam, candidates undergo the Services Selection Board (SSB) interview, which assesses your personality, intelligence, and suitability for a career in the armed forces.

Medical Examination

Finally, a medical examination ensures that you meet the physical standards required for a career in defence.

NDA Syllabus Overview

The NDA syllabus is comprehensive and covers a range of subjects. Here’s a quick overview:

Subjects Covered in the Exam

Mathematics: Algebra, Trigonometry, Calculus, and Statistics.

General Ability Test: English, General Knowledge, and Science.

Importance of Each Subject

Each subject is crucial for your overall performance in the exam and your future role in the armed forces. A strong foundation in these subjects will set you up for success.

Benefits of Joining NDA Classes

Now, you might be wondering, “Why should I join NDA classes?” Here are some compelling reasons:

Structured Learning Environment

NDA classes provide a structured approach to learning, ensuring that you cover all necessary topics systematically.

Expert Guidance

With experienced instructors, you’ll receive valuable insights and tips that can make a significant difference in your preparation.

Peer Support and Motivation

Studying alongside like-minded peers can be incredibly motivating. You’ll share experiences, study together, and push each other to excel.

Manasa Defence Academy: An Overview

Introduction to Manasa Defence Academy

Manasa Defence Academy is one of the leading institutions in India for NDA training. With a focus on holistic development, the academy prepares students not just for exams but for a successful career in defence.

Mission and Vision of the Academy

The academy aims to nurture young talents and transform them into competent leaders of the armed forces. Their vision is to create a generation of disciplined and dedicated officers.

Why Manasa Defence Academy Stands Out

Experienced Faculty

The faculty at Manasa Defence Academy comprises retired defence personnel and experienced educators who understand the nuances of NDA training.

Comprehensive Training Programs

The academy offers a range of training programs tailored to meet the needs of aspiring NDA candidates, ensuring that every student receives personalized attention.

Success Stories of Alumni

Many alumni of Manasa Defence Academy have successfully cleared the NDA exam and are now serving in various capacities in the armed forces. Their success stories inspire current students and showcase the effectiveness of the training.

Training Programs Offered

Classroom Training

In-person classes provide an interactive learning experience, allowing students to engage directly with instructors and peers.

Mock Tests and Assessments

Regular mock tests help students gauge their preparation levels and identify areas for improvement.

Facilities at Manasa Defence Academy

Infrastructure and Resources

The academy boasts modern infrastructure, including well-equipped classrooms, laboratories, and study areas.

Library and Study Materials

A vast library filled with relevant books and resources ensures that students have access to the best study materials.

Sports and Physical Training

Physical fitness is crucial for NDA candidates. The academy provides ample opportunities for sports and physical training to keep students in top shape.

Student Testimonials

Success Stories from Former Students

Many students have shared their success stories, highlighting how Manasa Defence Academy played a pivotal role in their journey to the NDA.

Impact of Training on Their Careers

Alumni often mention the confidence and skills they gained during their training, which have been instrumental in their careers.

How to Enroll in NDA Classes at Manasa Defence Academy

Admission Process

Enrolling in NDA classes is straightforward. Interested candidates can visit the academy’s website or contact them directly for more information.

Important Dates and Deadlines

Keep an eye on important dates for admissions, as they can vary each year. It’s best to stay updated to ensure you don’t miss out.

Tips for Success in NDA Exam

Study Strategies

Develop a study plan that covers all subjects and stick to it. Consistency is key!

Time Management

Practice managing your time effectively during mock tests to ensure you can complete the exam within the allotted time.

Importance of Physical Fitness

Don’t neglect your physical training. A fit body contributes to a sharp mind, which is essential for success in the NDA.

Conclusion

In conclusion, pursuing NDA classes after 12th is a fantastic opportunity for those looking to serve their country while building a rewarding career. Manasa Defence Academy stands out as a premier institution that provides the best training and support for aspiring NDA candidates. So, if you’re ready to take the plunge, consider enrolling at Manasa Defence Academy and embark on your journey towards becoming a proud officer in the Indian Armed Forces!

Contact:

Manasa Defence Academ

New Gajuwaka Depot, Visakhapatnam.

Andhra Pradesh-530012.

Call: 7799799221, 7799799229

Gmail: manasadefenceacademy@gmail.com  

Join Manasa Defence Academy & Achieve Your Defence Career Dreams! 

[manasastuff-blog]

UPSC NDA II Syllabus Full Details

Introduction to UPSC NDA II Exam

The UPSC NDA (National Defence Academy) II exam is a significant stepping stone for students aspiring to join the Indian Armed Forces. Conducted by the Union Public Service Commission, this exam is not just a test of knowledge but also a gateway to a prestigious career in defense. Understanding the syllabus is crucial for effective preparation, and in this post, we will delve into the full details of the UPSC NDA II syllabus.

UPSC NDA II Syllabus Breakdown

The syllabus for UPSC NDA II is categorized into two primary sections: Mathematics and the General Ability Test (GAT). Each section is designed to assess different skills and knowledge areas essential for a career in the defense services.

Mathematics Syllabus

The Mathematics section of the NDA exam tests candidates on various mathematical concepts. Here’s a detailed breakdown:

Algebra: This includes concepts like sets, relations, and functions, as well as quadratic equations and linear inequalities.

Matrices and Determinants: Understanding operations on matrices, determinants, and their applications.

Trigonometry: Basic identities, trigonometric ratios, and properties of triangles.

Geometry: This covers the properties of lines, angles, triangles, and circles, along with coordinate geometry.

Statistics and Probability: Basic concepts of statistics, measures of central tendency, and probability theory.

Calculus: Fundamental concepts of limits, continuity, and differentiation.

General Ability Test (GAT) Syllabus

The syllabus for UPSC NDA II is categorized into two primary sections: Mathematics and the General Ability Test (GAT).

English

The syllabus for UPSC NDA II is categorized into two primary sections: Mathematics and the General Ability Test (GAT).

Vocabulary: Synonyms, antonyms, and word meanings.

Grammar: Comprehending components of speech, verb tenses, and the structure of sentences.Writing Skills: Essay writing and letter writing.

General Knowledge

Current Affairs: Awareness of national and international events.

History: Important historical events, freedom struggle, and significant personalities.

Geography: Physical and political geography of India and the world.

General Science: Fundamental principles in physics, chemistry, and biology.Defence Studies: Understanding the structure and functions of the Indian Armed Forces.

Manasa Defence Academy: Best NDA Coaching for Students

At Manasa Defence Academy, we understand the challenges students face while preparing for the NDA exam. Our coaching program is tailored to provide comprehensive support, ensuring that students grasp the syllabus thoroughly.

Why Choose Manasa Defence Academy?

Experienced Faculty: Our instructors have extensive experience in teaching and guiding students for competitive exams.

Structured Curriculum: We follow a well-organized curriculum that aligns with the NDA syllabus, ensuring no topic is left uncovered.

Mock Tests and Assessments: Regular mock tests help students evaluate their preparation and identify areas for improvement.

Personalized Attention: We believe in small batch sizes to provide individual attention to each student, enhancing their learning experience.

"At Manasa Defence Academy, we are committed to shaping the future of aspiring defense personnel."

Conclusion

Comprehending the UPSC NDA II syllabus is crucial for successful preparation. With appropriate guidance and resourcesstudents can navigate this challenging journey successfully. If you are looking for the best NDA coaching, consider joining Manasa Defence Academy, where we are dedicated to helping you achieve your dreams of serving in the Indian Armed Forces.

[inversetrigonometricfunctions]

00:00:00 Introduction for inverse trigonometry identities

00:03:34 Derivation of all inverse trigonometric identities

00:15:14 NCERT Solutions Exercise 2.2 Question 1

00:20:04 NCERT Solutions Exercise 2.2 Question 2

00:21:34 NCERT Solutions Exercise 2.2 Question 3

00:23:44 NCERT Solutions Exercise 2.2 Question 4

00:27:54 NCERT Solutions Exercise 2.2 Question 6

00:34:54 NCERT Solutions Exercise 2.2 Question 7

00:36:24 NCERT Solutions Exercise 2.2 Question 8

00:38:54 NCERT Solutions Exercise 2.2 Question 5

00:42:54 NCERT Solutions Exercise 2.2 Question 9

00:45:44 NCERT Solutions Exercise 2.2 Question 10

00:48:54 NCERT Solutions Exercise 2.2 Question 11

00:50:24 NCERT Solutions Exercise 2.2 Question 12

00:51:04 NCERT Solutions Exercise 2.2 Question 13

00:52:54 NCERT Solutions Exercise 2.2 Question 14

00:54:44 NCERT Solutions Exercise 2.2 Question 15

00:59:54 NCERT Solutions Exercise 2.2 Question 16

01:01:24 NCERT Solutions Exercise 2.2 Question 17

01:03:04 NCERT Solutions Exercise 2.2 Question 19

01:04:24 NCERT Solutions Exercise 2.2 Question 20

01:05:34 NCERT Solutions Exercise 2.2 Question 21

01:07:04 NCERT Solutions Exercise 2.2 Question 18

Detailed explanation of 'how to drive identities of Inverse Trigonometric Functions' and questions related to NCERT Solutions for Exercise 2.2 of Inverse Trigonometric Functions Class 12 Maths Chapter 2.

💎 Purchase the Class 12 Maths Membership for full access to video lectures, notes, and assignments: https://mathyug.com/class-12-maths

[sunenajain-blog]

IGCSE Math vs IB Math: What Changes and How to Adapt

For many students, the transition from IGCSE Mathematics to IB Mathematics (AA/AI, SL/HL) can feel like stepping into a new world of complexity, abstraction, and independence. While both programs aim to build strong mathematical foundations, the IB Math curriculum demands significantly more from students in terms of conceptual depth, real-world application, and critical thinking.

In this article, we’ll explore the key differences between IGCSE and IB Math, how to adapt successfully, and how platforms like IBGram can support that academic leap.

📊 Key Differences Between IGCSE Math and IB Math

CategoryIGCSE MathIB Math (AA/AI)Curriculum StyleProcedural, focused on problem-solvingConceptual, emphasizes real-life applicationLevel of RigorModerateHigh (especially at HL level)Assessment StyleShort questions, straightforward answersMulti-step, analytical problemsTechnology UseMinimalEssential (e.g., GDC calculators)Internal AssessmentsNoneIncludes IA (Internal Assessment essay)Focus AreasAlgebra, geometry, number theoryStatistics, calculus, modeling

🔁 What Changes: From IGCSE to IB Math

1. Complexity of Concepts

IGCSE introduces foundational topics with relatively straightforward methods.

IB Math delves deeper, especially in Calculus, Vectors, and Statistics, where derivations, justifications, and applications matter more.

2. Application Over Memorization

IB emphasizes understanding over rote learning. You’re expected to justify your reasoning, not just show correct answers.

3. Internal Assessment (IA)

A major shift is the introduction of the Math IA—a 12–20 page investigation where you explore a math concept of personal interest using real-world data or modeling.

4. Course Options: AA vs AI

Applications and Interpretation (AI) leans towards statistics, modeling, and real-world use.

Analysis and Approaches (AA) is more abstract and proof-based, ideal for students pursuing engineering, physics, or math-intensive fields.

5. Use of Technology

IB Math requires use of graphical display calculators (GDCs) and sometimes tools like Desmos, GeoGebra, or spreadsheets.

🧠 How to Adapt Successfully

✅ 1. Build Strong Algebraic Fluency

Algebra is the backbone of both IB AA and AI. Master equations, inequalities, and functions early.

✅ 2. Practice Real-World Problems

Start applying math to real-life situations—model the height of a basketball shot, or use statistics to analyze survey results.

✅ 3. Understand, Don’t Memorize

Ask why something works—not just how. This is especially crucial in IB assessments.

✅ 4. Get Comfortable with Your GDC

Know how to use functions like graph plotting, regression analysis, and probability simulations.

✅ 5. Start Prepping for the IA Early

Explore math topics that interest you and try simple investigations in your IGCSE years.

🤝 How IBGram Supports the Transition

IBGram is a trusted learning platform tailored to the unique needs of IB students, and it plays a vital role in helping students bridge the gap between IGCSE and IB Math.

🚀 What IBGram Offers:

Transition Modules: Designed specifically for students entering IB from IGCSE, covering advanced algebra, trigonometry, and precalculus.

IA Support: Step-by-step IA guidance, example investigations, and one-on-one mentorship.

Concept Videos: Bite-sized explainer videos focused on AA/AI topics.

Live Doubt Sessions: Weekly Q&A with experienced IB tutors.

GDC Tutorials: Learn your calculator through practical walkthroughs.

📈 Student Impact:

Students who use IBGram during their transition report:

Faster adaptation to IB-style questions

Higher engagement with the Math IA process

Improved scores in formative assessments and mock exams

📝 Final Tips for New IB Math Students

Don’t underestimate IB Math—it’s a major leap from IGCSE, even for high achievers.

Choose the right course (AA vs AI) based on your future aspirations.

Don’t wait until term 2 to seek help—start early with resources like IBGram to build confidence.

📌 Summary

Key AreaWhat to Focus OnCore SkillsAlgebra, functions, calculator usageMindsetConceptual learning > memorizationAssessment PreparationPractice IB-style questions earlySupport SystemUse IBGram’s structured resources