Exploring Mathematical Concepts

Multiplication doesn’t always begin with numbers. Sometimes, it begins with lines. Parallel lines for one number. Crossing lines for another. Count the intersections—and the answer appears. No memorization. No pressure. Just patterns revealing themselves. And that’s where the real insight lies. 💡 Learning becomes powerful the moment it becomes visible. For years, math has been taught as something to remember. Tables to memorize. Steps to follow. Answers to reproduce. But true understanding has never been about recall. It has always been about representation. Change the way a concept is presented… and suddenly, what once felt abstract becomes intuitive. What’s fascinating is that this idea has existed across cultures and centuries. From Vedic Mathematics to lattice multiplication—the principle remains the same: Make thinking visible, and learning follows naturally. This approach works even with larger numbers. More lines, more intersections, clearer patterns. For visual learners, it can transform multiplication from a task into an experience. And perhaps, the lesson goes beyond mathematics. Many challenges feel difficult not because they are inherently complex, but because they are presented in a way that hides their simplicity. Sometimes, the breakthrough isn’t about working harder. It’s about seeing differently. Because when perspective changes… understanding follows. 😎

Students of AI: Stop Getting Lost in Equations. I've been there—drowning in formulas, missing the deeper intuition. Then, I discovered a better way: focusing on the story behind the math. Let's walk through this with one of the hardest concepts I learned last year- The Natural Gradient. Typically textbooks give you the formula. Then they might walk you through the symbols and get deep into the derivations. Unless you love this stuff, there's a good chance that this is where you mentally check out and start scrolling through your phone (for me it was the football highlights on IG). Here's how I would suggest learning it instead. First we ask what’s Really Happening- 1. Standard gradient descent blindly follows steepest slopes. 2. The Natural Gradient introduces the Fisher Information Matrix (FIM), incorporating the geometry of your solution space. 3. This is interesting. Why Does the Fisher Information Matrix Matter? Looking more into this we see- High Fisher Value = Big fluctuations, tread lightly. Low Fisher Value = Smooth terrain, move boldly. Once we understand the function, the two bog blocks of math functions make a lot more sense- 1. Squared log-likelihood emphasizes magnitude regardless of direction—capturing change clearly. 2. Expectation (E[…]) filters out noise, spotlighting reliable, stable patterns. When I stopped fixating on derivations and asked “WHY this approach?”, the pieces fell into place. Don’t just memorize math. Understand the narrative behind it. In fact I'd go far enough and say that unless you're a researcher (or you LOVE) math, you're much better off ignoring all the math details and simply understanding the story. The math intuition can be built over time, as you come across the ideas repeatedly.

Are you ready to learn faster? People see things often as too complex. Sometimes the easier way is the best way. The Feynman Technique is a powerful learning method developed by physicist Richard Feynman. It's a simple yet effective way to deepen your understanding of complex concepts. Here's how it works: First, choose a concept you want to learn. Then, explain it as simply as possible, as if you were teaching it to a child. This process helps you identify gaps in your understanding. Next, review your explanation and simplify it further. Break down complex ideas into smaller, more manageable parts. Then, teach the concept again, refining your explanation as needed. Repeat this process until you can explain the concept confidently and clearly. The Feynman Technique helps solidify your understanding by forcing you to confront any areas of confusion and fill in the gaps in your knowledge. #personaldevelopment #humanresources #growth #people

Calculus in Sanskrit is primarily referred to as calana-kalana (चलन-कलन), representing the "motion-calculation" of change. While ancient, foundational concepts were developed further by the Kerala school of astronomy and mathematics (notably Madhava), formal modern terminology includes sūkṣmamāna - gaṇitam (differential) and samāhāra - gaṇitam (integral). Calculus is the mathematical study of continuous change, serving as the "Mathematical Backbone" for modeling dynamic systems where quantities vary, such as motion, acceleration, and fluid flow. It expanded upon ancient methods of exhaustion and infinite series to provide a formal language for calculating : - instantaneous rates of change (differential calculus) and  - total accumulation (integral calculus). Nature of Calculus Calculus breaks down complex, changing, or curved objects into infinitely small pieces to calculate exact values. Its core nature involves two inverse branches: #Differential Calculus: Studies instantaneous rates of change, such as slopes of curves, velocity, and acceleration. #Integral Calculus: Deals with the accumulation of quantities, calculating total areas under curves, volumes, and total distance traveled. #Fundamental Theorem of #Calculus: Proves that differentiation and integration are inverse operations, connecting the two branches. Role in Mathematics and Science Mathematics: It is the core of "mathematical analysis," separating higher mathematics from static classical mathematics. Physical Sciences: It is considered the "basic instrument of physical science," used to model almost all dynamic phenomena, including electricity, magnetism, and fluid mechanics. Biology: Calculus in biology enables the modeling and analysis of dynamic, changing systems such as population growth, disease spread, and enzyme kinetics. Application Examples: Engineering/Architecture: Designing curved structures like bridges, high-rise buildings, and vehicles. Economics: Modeling marginal costs, profit maximization, and financial trends. Medicine: Analyzing drug interaction in the body or blood flow. Technology: GPS navigation, space travel, and computer algorithms.

🤯 𝗦𝘁𝘂𝗰𝗸 𝗶𝗻 𝘁𝗵𝗲 ‘𝗪𝗵𝘆’ 𝗼𝗳 𝗠𝗮𝘁𝗵𝘀? 𝗟𝗲𝘁’𝘀 𝗖𝗵𝗮𝗻𝗴𝗲 𝗧𝗵𝗮𝘁! 𝗪𝗵𝗲𝗿𝗲 𝗺𝗮𝗻𝘆 𝘀𝘁𝘂𝗱𝗲𝗻𝘁𝘀 𝗮𝗿𝗲: Sitting with a complex maths problem, staring at the paper, feeling frustrated. They’ve memorized formulas, but when it comes to breaking down a tricky question – they freeze. ❄️ 𝗧𝗵𝗲𝗶𝗿 𝗱𝗿𝗲𝗮𝗺 𝗼𝘂𝘁𝗰𝗼𝗺𝗲? They want to approach any problem with confidence. To see complexity and think, “I’ve got this.” To understand every step—not just apply it. 𝗪𝗵𝘆 𝗗𝗼 𝗠𝗼𝘀𝘁 𝗦𝘁𝘂𝗱𝗲𝗻𝘁𝘀 𝗦𝘁𝗿𝘂𝗴𝗴𝗹𝗲? Most students fail because they focus on what to do, not why they’re doing it. They memorize steps but miss the logic. So, when they face an unfamiliar problem, they get lost.  • They don’t question the question.  • They skip the ‘why.’ 𝗧𝗵𝗲 𝗦𝗼𝗹𝘂𝘁𝗶𝗼𝗻: 𝗕𝗿𝗲𝗮𝗸 𝗗𝗼𝘄𝗻 𝗣𝗿𝗼𝗯𝗹𝗲𝗺𝘀 𝘄𝗶𝘁𝗵 ‘𝗪𝗵𝘆’ 𝗧𝗵𝗶𝗻𝗸𝗶𝗻𝗴 🧐 Teaching students to ask “why” at each step transforms their understanding. Here’s how it works: 1️⃣ 𝗦𝘁𝗮𝗿𝘁 𝘄𝗶𝘁𝗵 𝘁𝗵𝗲 𝗣𝗿𝗼𝗯𝗹𝗲𝗺, 𝗡𝗼𝘁 𝘁𝗵𝗲 𝗦𝗼𝗹𝘂𝘁𝗶𝗼𝗻: Instead of rushing to plug in formulas, encourage them to ask: • “What is this problem actually asking?” • “What information do I have?” • “What do I need to find?” 🧠 𝗧𝗶𝗽: Break the question into smaller parts. Each part should be a mini-problem to solve. 2️⃣ 𝗤𝘂𝗲𝘀𝘁𝗶𝗼𝗻 𝗘𝗮𝗰𝗵 𝗦𝘁𝗲𝗽: When they apply a formula or make a calculation, they should ask: • “Why am I doing this step?” • “How does this help me get closer to the solution?” 🔍 𝗘𝘅𝗮𝗺𝗽𝗹𝗲: Solving an equation? • Why do we move variables to one side? • Why do we simplify terms first? 3️⃣ 𝗥𝗲𝗳𝗹𝗲𝗰𝘁 𝗔𝗳𝘁𝗲𝗿 𝗦𝗼𝗹𝘃𝗶𝗻𝗴: Once they reach an answer, teach them to look back and ask: • “Did every step make sense?” • “Why did this method work?” • “Could I explain this to someone else?” This reflection cements learning 𝗧𝗵𝗲 𝗧𝗿𝗮𝗻𝘀𝗳𝗼𝗿𝗺𝗮𝘁𝗶𝗼𝗻 🌟 When students adopt ‘why thinking,’ they gain confidence, reduce mistakes, and develop deeper mastery of concepts. This approach shifts them from memorizing to truly understanding, paving the way to top grades. Let’s help them question the question. The answers will follow! 👍 Like | 💬 Comment | 🔁 Repost | 👤 Follow me, Faisal Naqvi #MathsMastery #GrowthMindset #QEDTuitions #CriticalThinking #ProblemSolving #WhyThinking #ConfidentLearners

Applied Mathematics in Trading Most traders think math ends at moving averages and Fibonacci retracements. That’s surface-level. Real trading edge begins when you treat the market as a system of measurable relationships: • Price is a function of liquidity, volatility, and time. • Every candle is a data point — variance, distribution, skew. • Structure isn’t “random”; it’s a probabilistic map where liquidity pools act as attractors. Institutions don’t gamble. They model. They run probability distributions on order flow, volatility clustering, and expected ranges. That’s why they win over the long run. Applied mathematics isn’t about predicting the future it’s about narrowing the uncertainty so that risk becomes calculable. That’s the only way to stop trading stories and start trading numbers.

Some more personal advice for those learning Mathematics: (1) Not all concepts are equal, so allocate your time accordingly. When learning new material, spend considerable time on the building blocks of the topic—see how to express them in different ways, look at many examples, and visualise them. For Riemannian geometry, you must know what the Riemannian metric tensor is; spend ample time trying to understand it. It is a worthy investment, given every geometric quantity derives from it (geodesics, angles, volumes, connections, curvature tensors, etc.)—these will be far more intuitive and easier to remember when you understand their foundation. Similarly, in analysis and group theory, if you don't fully understand injections and surjections, none of the morphisms (and there are many of them) will be clear or memorable. There should be an asymmetry in the time you spend on different concepts. (2) Step back and (subliminally) ponder. As one delves deeper into abstract mathematics, theorems and techniques become difficult to understand and apply, and forcing clarity in one sitting seldom works. At first passing, go through concepts without working through every detail, then step away. Let your mind coalesce them in the background—some theorems take weeks or months to process, and that's fine. If you continue this over a prolonged period, especially with a visual understanding of the foundations (from point one), you will be able to connect ideas from different areas of mathematics and (possibly) find creative ways to apply them. Just those two pieces of advice for now. The photo is of my desk at Oxford Mathematics (I ran out of space for my other books so I have now begun using my friends' shelves).

DIFFERENTIAL EQUATIONS: Imagine math is a toolkit for understanding the world. While arithmetic helps you count apples and algebra clears up an unknown in a still photo, differential equations are like a video camera: they serve to capture movement and change. Here's the gist of this concept, without a single variable in between. Concept: The relationship between "now" and "after" The central idea of a differential equation is very intuitive: the way something changes depends on how that thing is at the moment. Population Example: The more people in a city, the more babies are born. That is, the speed at which the population grows depends on the current size of the population. Example of coffee: A very hot cup of coffee cools down very quickly at first. As it gets closer to room temperature, it gets cooler slower. The speed of temperature change depends on the temperature the coffee has right now. How do they work in practice? To solve a differential equation, we do not look for a number (like “5” or “10”), but we look for a complete story or a trajectory. If you give a mathematician the "rule of change" (the equation) and the "starting point" (the initial conditions), he can rebuild everything that will happen in the future. It's as if, knowing only the direction of the wind and the strength of an engine every second, you could draw the complete map of an airplane's route. Why are they so important? Almost everything around us is dynamic. Differential equations are the invisible engine behind: Music: They describe how a guitar string vibrates or how air moves to bring sound to your ears. Medicine: They are used to calculate how long it takes for a drug to be distributed through the blood and when it will start to lose effect. Engineering: They allow us to know whether a bridge will withstand the sway of the wind or if a skyscraper will withstand an earthquake. Weather: Meteorologists use them to simulate how air masses will move and predict if it will rain tomorrow. The chaos and complexity Sometimes, these rules of change are so sensitive that a tiny change at the beginning alters the whole end result. This is what we know as the Butterfly Effect. Although differential equations give us the power to "predict the future," in complex systems (like the atmosphere or economy) they teach us that there are limits to our ability to predict, reminding us that nature always has a touch of unpredictability.

Statistics is a crucial tool for interpreting data and making evidence-based decisions, even for those without formal statistical training. Understanding basic statistical concepts allows professionals to analyze information critically and apply quantitative reasoning in their work. This document provides a practical introduction to statistics for non-statisticians, explaining key concepts in an accessible way without requiring advanced mathematical knowledge. The guide covers data types, descriptive statistics, probability fundamentals, and basic inferential techniques, helping users understand how to summarize, interpret, and draw conclusions from data. It introduces sampling methods, hypothesis testing, confidence intervals, and regression analysis, emphasizing practical applications over complex formulas. Special focus is placed on common statistical misconceptions and errors, ensuring users develop a solid foundation for accurate data interpretation. For professionals in business, social sciences, healthcare, and public policy, this document is a valuable resource for applying statistical thinking in real-world scenarios. It highlights how to assess data credibility, avoid misinterpretations, and make informed decisions. Whether analyzing survey results, tracking performance, or evaluating project outcomes, these insights help non-statisticians use statistics effectively for decision-making and problem-solving.

How school formulas helped me in calculating workout LGD.   When I explain mathematics to children, I often hear the question: "How can this formula be useful to me in life?"   This question prompted me to the post talking about a school formula that helped me solve a practical problem in credit risk.   A few years ago I was developing a LGD model in accordance with IRB approach for a mortgage portfolio. I needed to calculate workout LGD. To do this, I needed to discount cash flows from the observation date to the default date. Cash flows were collected on the dates when defaults were observed in the portfolio. However, when the collateral is placed on the Bank's balance sheet (when it cannot be sold at the legal stage), default stops to be observed in the portfolio.   The problem was that cash flows (costs of maintaining the collateral and its sale) were not recorded in the portfolio (Figure 1️⃣). If to be more precise the difficulty was in the constant costs that first had to be discounted and then summed up.   One solution was to artificially extend the default history from the last date of being in the portfolio to the date of sale of the collateral and analytically record the expenses for each new date. But it complicated the data set and, as a consequence, its explanation to auditors and the regulator.   School knowledge of mathematics came to the rescue. A number of discount factors are Geometric progression (Formula 1️⃣). Accordingly, the sum of a Geometric progression is Formula 2️⃣. Substituting the discount factors into the formula for the sum of a geometric progression, we obtain Formula 3️⃣. Next we simplify and have the final formula 4️⃣.   As a result, knowing the last month of default, the interest rate and average constant costs, it is possible to calculate the total constant costs without using loops or complicating the data set.   Now I have an argument for children that it was not in vain that I studied mathematics at school.