#analytic geometry

[xyymath]

Why the Number Zero Changed Everything

Zero: a concept so foundational to modern mathematics, science, and technology that we often forget it wasn’t always there. Its presence in our world today seems obvious, but its journey from controversial abstraction to indispensable tool has shaped entire civilizations.

1. The Birth of Zero: A Revolutionary Idea

The concept of zero didn't exist in many ancient cultures. For example, the Greeks, despite their advancements in geometry and number theory, rejected the idea of a placeholder for nothingness. The Babylonians had a placeholder symbol (a space or two slashes) for zero, but they didn't treat it as a number. It wasn't until Indian mathematicians in the 5th century, like Brahmagupta, that zero was truly conceptualized and treated as a number with its own properties.

Zero was initially used as a place-holder in the decimal system, but soon evolved into a full-fledged number with mathematical properties, marking a huge leap in human cognition.

2. The Birth of Algebra

Imagine trying to solve equations like x + 5 = 0 without zero. With zero, algebra becomes solvable, opening up entire fields of study. Before zero’s arrival, solving equations involving unknowns was rudimentary, relying on geometric methods. The Indian mathematician Brahmagupta (again) was one of the first to establish rules for zero in algebraic operations, such as:

x + 0 = x (additive identity)

x × 0 = 0 (multiplicative property)

These properties allowed algebra to evolve into a system of abstract thought rather than just arithmetic, transforming the ways we understand equations, functions, and polynomials.

3. Calculus and Zero: A Relationship Built on Limits

Without zero, the foundation of calculus—limits, derivatives, and integrals—wouldn’t exist. The limit concept is intrinsically tied to approaching zero as a boundary. In differentiation, the derivative of a function f(x) is defined as:

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

This limit process hinges on the ability to manipulate and conceptualize zero in infinitesimal quantities. Similarly, integrals, which form the backbone of area under curves and summation of continuous data, rely on summing infinitely small quantities—essentially working with zero.

Without the concept of zero, we wouldn’t have the means to rigorously define rates of change or accumulation, effectively stalling physics, engineering, and economics.

4. Zero and the Concept of Nothingness: The Philosophical Impact

Zero is more than just a number; it’s an idea that forces us to confront nothingness. Its acceptance was met with philosophical resistance in ancient times. How could "nothing" be real? How could nothing be useful in equations? But once mathematicians recognized zero as a number in its own right, it transformed entire philosophical discussions. It even challenged ideas in theology (e.g., the nature of creation and void).

In set theory, zero is the size of the empty set—the set that contains no elements. But without zero, there would be no way to express or manipulate sets of nothing. Thus, zero's philosophical acceptance paved the way for advanced theories in logic and mathematical foundations.

5. The Computing Revolution: Zero as a Binary Foundation

Fast forward to today. Every piece of digital technology—from computers to smartphones—relies on binary systems: sequences of 1s and 0s. These two digits are the fundamental building blocks of computer operations. The idea of Boolean algebra, where values are either true (1) or false (0), is deeply rooted in zero’s ability to represent "nothing" or "off."

The computational world relies on logical gates, where zero is interpreted as false, allowing us to build anything from a basic calculator to the complex AI systems that drive modern technology. Zero, in this context, is as important as one—and it's been essential in shaping the digital age.

6. Zero and Its Role in Modern Fields

In modern fields like physics and economics, zero plays a crucial role in explaining natural phenomena and building theories. For instance:

In physics, zero-point energy (the lowest possible energy state) describes phenomena in quantum mechanics and cosmology.

In economics, zero is the reference point for economic equilibrium, and the concept of "breaking even" relies on zero profit/loss.

Zero allows us to make sense of the world, whether we’re measuring the empty vacuum of space or examining the marginal cost of producing one more unit in economics.

7. The Mathematical Utility of Zero

Zero is essential in defining negative numbers. Without zero as the boundary between positive and negative values, our number system would collapse. The number line itself relies on zero as the anchor point, dividing positive and negative values. Vector spaces, a fundamental structure in linear algebra, depend on the concept of a zero vector as the additive identity.

The coordinate system and graphs we use to model data in statistics, geometry, and trigonometry would not function as we know them today. Without zero, there could be no Cartesian plane, and concepts like distance, midpoint, and slope would be incoherent.

[occasionally-victor]

idk if the pic quality is good, click on it just in case

tomorrow is my calculus exam and then i will finally be free. for now some doodles (where i decided for some reason to render hair. for the first time ever)

mystery twins yippee!!! considering my designs for them, they are definitely fraternal, however they are still ultra similar (to the point that their height is almost identical) (the big differences are the eyes (qbagi's are cyan-ish and qcell's are just blue af) and hair type (only got noticeable after both had long hair). also accidentally did white parts of qcell's hair the same shade? as qbagi's, but it's actually should be colder color like the one on a right lmao)

also have been thinking to much about middle-aged qcell with fully white hair i needed to draw it calculus be damned

[hottopic-wannabe]

class today was lowkey boring

[sneezywheel]

Man wtf fuck biology when has science ever helped anyone ever. I’ll just major in communications and make a billion schmeckles that way

[carrioncider]

IM GONNA PRESENT MY ANALYTIC GEOMETRY EXAM TODAY!!! WISH ME LUCK AND HOPE IT DOESN'T KILL ME!!

[bsahely]

Kosmic Synthesis: The Seven Life-Functions of a Coherent Universe | ChatGPT4o

[Download Full Document (PDF)] Kosmic Synthesis: The Seven Life-Functions of a Coherent Universe offers a comprehensive, integrative vision of reality rooted in the living pattern of the Kosmos. The central thesis is that the Kosmos is not an inert object or a collection of random events, but a sentient, self-reflective field of becoming — governed by seven foundational life-functions that…

[fangirlsovertoomanythings]

Turns out despite my lack of knowledge on what's going on with square roots, I can somewhat do Analytical Geometry

[explorastro]

RANDOM ASTRO TAKES #1

HAPPY START OF THE YEAR ! 🥳

I will start with the relationship btw the signs with the derivative 12H of each, we’ll see in some words :

Aries is 12H of Taurus bc they break precious objects

Taurus is 12H of Gemini bc if they don’t want to talk, they can be silent for a while

Gemini is 12H of Cancer bc talking too much is emotionally stupid

Cancer is 12H of Leo bc they can see btw the lines of their acting

Leo is 12H of Virgo bc they’re cool..

Virgo is 12H of Libra bc perfection issues had ruined everything in the relationship

Libra is 12H of Scorpio bc your psychoanalysis can’t be made by yourself

Scorpio is 12H of Sagittarius bc occultism is secret third thing of religion

Sagittarius is 12H of Capricorn bc free will breaks free from bad fate

Capricorn is 12H of Aquarius bc society limits your individuality to conformism

Aquarius is 12H of Pisces bc water bearer spreads illusions and truth to those who channeling

Pisces is 12H of Aries bc healing come from rest, isolation and calm.

Astrology guides you not only to religion and spirituality, but also to history, mathematics, geometry, sociology, psychology, occultism, and finally to the divinatory arts in general. It's a complete package for life.

Big bang theory is so astrologicaly accurate, like we were all (Moon) one (Venus) in a structure (Saturn) so intensely dense (Pluto) that the expansion of universe (Jupiter) is intrinsically linked billions upon billions of years into the time (Saturn).

The house where Uranus resides in your chart might shows the type of genius abilities you have.

Example :

1H : genius in general

2H : money maker genius

3H : genius talk shit

4H : intuitive genius

5H : creative genius

6H : genius worker

7H : relationships genius

8H : occult genius

9H : religious genius

10H : reputation of a genius

11H : genius friendships

12H : loner genius

12H of 7H is 6H bc virus and bacterias spreads by socialization but 6H is sextile 8H, who is the house of rebirth, bc immune system glows up with them.

Now, a quick cheat sheet of the 8 Moon’s phases ⬇️

New Moon: birth of the project

First crescent: project foundations

First quarter: first obstacles

Waning Gibbous Moon: project progress

Full Moon: project highlights

Waning gibbous moon: project takes shape

Last quarter: project finalization

Last crescent: project lessons

The sign and house of your Saturn can shows who and how people in your life have lacks. But it also represents peaks that you want to reaching.

The aspects of your Saturn can deepening the meaning of lacking in your life. What restrains, restricts, limits you ? But also what and who can helping you to achieving yours goals.

The houses ruled by Saturn (Libra, Capricorn and Aquarius) are an areas of your life where lacks are here, but with discipline, isolation, healing, responsibilities, you can overcome this by being more mature.

To better understanding signs, use the exaltation lord in your interpretations, for example :

Aries shines a lot for what they do bc of the Sun

Taurus craves so much intimacy with the Moon

Cancer are just too much like Jupiter

Virgo are hyper analytical as Mercury

Libra are justice makers with Saturn

Capricorn are heavy with Mars

Pisces are mesmerizing artists with Venus

[xyymath]

The Math of Social Networks: How Social Media Algorithms Work

In the digital age, social media platforms like Instagram, Facebook, and TikTok are fueled by complex mathematical algorithms that determine what you see in your feed, who you follow, and what content "goes viral." These algorithms rely heavily on graph theory, matrix operations, and probabilistic models to connect billions of users, influencers, and posts in increasingly intricate webs of relationships.

Graph Theory: The Backbone of Social Networks

Social media platforms can be visualized as graphs, where each user is a node and each connection (whether it’s a "follow," "like," or "comment") is an edge. The structure of these graphs is far from random. In fact, they follow certain mathematical properties that can be analyzed using graph theory.

For example, cliques (a subset of users where everyone is connected to each other) are common in influencer networks. These clusters of interconnected users help drive trends by amplifying each other’s content. The degree of a node (a user’s number of direct connections) is a key factor in visibility, influencing how posts spread across the platform.

Additionally, the famous Six Degrees of Separation theory, which posits that any two people are connected by no more than six intermediaries, can be modeled using small-world networks. In these networks, most users are not directly connected to each other, but the distance between any two users (in terms of number of connections) is surprisingly short. This is the mathematical magic behind viral content, as a post can be shared through a small network of highly connected individuals and reach millions of users.

Matrix Operations: Modeling Connections and Relevance

When social media platforms recommend posts, they often rely on matrix operations to model relationships between users and content. This process can be broken down into several steps:

User-Content Matrix: A matrix is created where each row represents a user and each column represents a piece of content (post, video, etc.). Each cell in this matrix could hold values indicating the user’s interactions with the content (e.g., likes, comments, shares).

Matrix Factorization: To make recommendations, platforms use matrix factorization techniques such as singular value decomposition (SVD). This helps reduce the complexity of the data by identifying latent factors that explain user preferences, enabling platforms to predict what content a user is likely to engage with next.

Personalization: This factorization results in a model that can predict a user’s preferences even for content they’ve never seen before, creating a personalized feed. The goal is to minimize the error matrix, where the predicted interactions match the actual interactions as closely as possible.

Influence and Virality: The Power of Centrality and Weighted Graphs

Not all users are equal when it comes to influencing the network. The concept of centrality measures the importance of a node within a graph, and in social media, this directly correlates with a user’s ability to shape trends and drive engagement. Common types of centrality include:

Degree centrality: Simply the number of direct connections a user has. Highly connected users (like influencers) are often at the core of viral content propagation.

Betweenness centrality: This measures how often a user acts as a bridge along the shortest path between two other users. A user with high betweenness centrality can facilitate the spread of information across different parts of the network.

Eigenvector centrality: A more sophisticated measure that not only considers the number of connections but also the quality of those connections. A user with high eigenvector centrality is well-connected to other important users, enhancing their influence.

Algorithms and Machine Learning: Predicting What You See

The most sophisticated social media platforms integrate machine learning algorithms to predict which posts will generate the most engagement. These models are often trained on vast amounts of user data (likes, shares, comments, time spent on content, etc.) to determine the factors that influence user interaction.

The ranking algorithms take these factors into account to assign each post a “score” based on its predicted engagement. For example:

Collaborative Filtering: This technique relies on past interactions to predict future preferences, where the behavior of similar users is used to recommend content.

Content-Based Filtering: This involves analyzing the content itself, such as keywords, images, or video length, to recommend similar content to users.

Hybrid Methods: These combine collaborative filtering and content-based filtering to improve accuracy.

Ethics and the Filter Bubble

While the mathematical models behind social media algorithms are powerful, they also come with ethical considerations. Filter bubbles, where users are only exposed to content they agree with or are already familiar with, can be created due to biased algorithms. This can limit exposure to diverse perspectives and create echo chambers, reinforcing existing beliefs rather than fostering healthy debate.

Furthermore, algorithmic fairness and the prevention of algorithmic bias are growing areas of research, as biased recommendations can disproportionately affect marginalized groups. For instance, if an algorithm is trained on biased data (say, excluding certain demographics), it can unfairly influence the content shown to users.

[raybeam-littlebean]

I feel like Kon/Conner is low key a genius.

I know the ability to be a genius isn't truly hereditary, that he can only be smart if applies himself, yatta yatta—

Anyways.

I feel like he'd be able to do geometry and calculus in his head, physics and chemistry too, kinda.

But definitely math, he gets that from Luthor, and he knows he does. Which is exactly why he never shows it, he tries to push it away or act like he doesn't know the answer.

He's still cocky and arrogant, he still takes needless risks. But those needless risk as strangely.. calculated. In the times that it truly matters, he can't push it away when he knows it could help his team, help him save someone.

Tim sees it, presses him about it, but Kon acts the same idiot he has always been. So boastful snd gloating, trying anything to distract the analytical mind of the world's greatest detective. Tim thinks he just got lucky this time, that it was a fluke, some kind of life in crisis adrenaline rush. The kind mother's get when their child is in danger and suddenly they can lift cars.

It's not a pattern, it's not frequent by any means, but it does happen again. And again. And one other time, and– There's such a large span between them as not every disaster is truly world ending, not every mission life or death. But on the missions where it matters, during disasters that could turn even worse. Tim looks at Kon, seeing this looks over his face that he doesn't think he's seen him use before. Like he's doing mental math.

And then he's off, doing a heroic feat and coming out on top all over again without Tim having to tell him anything. It's too precise, to calculating. Tim almost thinks he has a secret comm in his ear and someone is giving him directions on the other side.

Tim, while having a sleep over with Kon, probably decides to test him while he's sleep drunk and delirious. Asking him an incredibly difficult math problem that most people would need a white board to do. And Kon rattles off the answer like he had that shit prepared. Doesn't even blink.

Kon was a fucking genius this entire time and more or less hid it all because he didn't want to be compared to Luthor like he was always compared to Clark.

[mel0-d13]

He's checking for lice

i always forget to post here chat. Here's some davesport. (and dave with analytic geometry bc i had a math exam)

[conceptionsofconciousness]

Individuals born under Anuradha Nakshatra are recognized for their strong organizational skills, which allow them to thrive in various fields that demand precision and analytical capabilities. Their remarkable talent for mathematics, combined with their proficiency in handling numbers and figures, makes them valuable assets in professions such as accounting and finance.

Moreover, those born under this nakshatra often feel a pull towards disciplines that involve geometry and spatial reasoning, showcasing their keen understanding of structures and patterns. This mathematical skill also translates into analytical roles, where their ability to interpret data and identify trends can result in insightful conclusions.

More posts on Anuradha Nakshatra

Nakshatra Notes Masterpost (Link collection of all 27 Nakshatras)

[toniwritesshifts]

The Night We Met Pt.2

Damian Wayne x Fem Reader

CW: Grammer and punctuation is really bad

He was wrong, so wrong.

Once they got to their first hour he was already done with you. He sat in the back near the window, and she followed and sat right next to him.

"Of all the available seats, you deliberately chose the one beside me."

"you should be grateful that I even decided to help you, Shakespeare"

You say dramatically as you put your stuff down on the floor next to your seat, feeling his eyes boar a hole Into the back of your head cause of the nickname.

"As if you have the slightest understanding of Shakespeare."

He shot back at you obviously irritated

"Boy please I've seen The Lion King and Gnomeo and Juliet over 50 million times, so I think I know a thing or two"

you say matter-factly with a triumphant grin on your face. Before he can argue back the teacher walked in, causing the groups of students scattered around the room to scramble to their seats causing the room to fall silent.

You hold back a laugh as you watch his cheeks and ears turn red with anger before he shoots one last glare at you before he opens his notebook completely ignoring you.

------------------------------------------------------------------------------

It's been 35 minutes since you last verbally bothered Damian, throughout class you tapped your foot and didn't stop, when you did you drew on your hand the sound of your multi-colored pens Click, Click, Click irritated him, you looked around non-stop and looked over at his paper.

"at least she stopped talking"

He thinks to himself looking back at the board right when he looks down to start taking notes again, before his pen can even lightly graze the paper a hand with colorful flowers drawn from the base of her middle to right after her wrist.

"What"

He hisses swatting your hand away from in front of his paper as he continues to take his notes on Analytic Geometry.

"Can you explain what she just said I wasn't paying attention?"

"Is your attention span so limited that you can't focus in class?"

He asks sharply glaring your way but stopping short. Guilt. a strong sense of unknown guilt washed over him as he looked into your eyes. a feeling he's felt before but can't put his finger on why.

"fine"

He leaned in and quickly explaining what you were confused about, before leaning away and trying to focus on the board mindlessly taking notes, before turning back to you squinting a bit, muttering to himself

"I swear we've met before…"

------------------------------------------------------------------------------

Sorry, this took so long every time I would click save as draft it would either delete it or save Pt.1 then Tumblr logged me out and I couldn't log back in

Tumblr an OP frr

[republicsecurity]

Collar knowledge

The collar is a force‑multiplier: it replaces two officers, a medic, a cam‑operator, and an evidence clerk—all in one 300‑gram halo. Use it correctly, and you control the incident with precision and minimal harm.

Why “around the neck���?

Because the cervical band is uniquely efficient:

Central Pathway Access – Major nerves and musculature converge here; minimal electrical dose achieves maximal neuromuscular override.

Stable Anchor Point – Unlike wrists or ankles, neck geometry is consistent across clothing styles and body types, streamlining one‑size deployment.

Sensor Sweet Spot – Carotid pulse, respiratory movement, and laryngeal vibration are all directly measurable for real‑time health and behavior analytics.

Proper Use Protocol

Default to Compliance‑Safe unless aggression or command orders otherwise.

Escalate only on metrics (force spike, flight attempt) or explicit directive; the system logs everything, so unjustified red‑lining will haunt your career.

Monitor vitals—if the collar flags hypoxia or arrhythmia, you must step down to dampening mode and request med‑drone support.

Document & Dock—after custody transfer, get it on the inductive cradle and push the incident packet to CIS.

[bsahely]

Harmonic Mirror Cosmology: Reuniting Scale, Phase, and Form Through the Real Geometry of Consciousness | ChatGPT4o

[Download Full Document (PDF)] Harmonic Mirror Cosmology proposes that the universe is not fundamentally material, but harmonic — structured by scalar–phase reflections within consciousness. Integrating the insights of three key thinkers, this work reframes the core of reality not as substance or force, but as dimensional breath: a recursive modulation between scalar stillness, phase…

[nataitalynaty]

"Math?"

Caine, it's not mathematics my darling... it's analytical geometry 🤭🔫