Applied Math Quotes

One might suppose that reality must be held to at all costs. However, though that may be the moral thing to do, it is not necessarily the most useful thing to do. The Greeks themselves chose the ideal over the real in their geometry and demonstrated very well that far more could be achieved by consideration of abstract line and form than by a study of the real lines and forms of the world; the greater understanding achieved through abstraction could be applied most usefully to the very reality that was ignored in the process of gaining knowledge.

Brilliant Muslim scholars applied Qur’anic insights to spark the medieval Islamic Golden Age filled with a mind-boggling outpouring of creativity in science, math, medicine, fashion, philosophy, economics, mental health therapy, architecture, art, and beyond.

It was as though applied mathematics was my spouse, and pure mathematics was my secret lover.

Poincaré [was] the last man to take practically all mathematics, pure and applied, as his province. ... Few mathematicians have had the breadth of philosophic vision that Poincaré had, and none in his superior in the gift of clear exposition.

The trick to overwriting a habit is to look for the pressure point—your reaction to a cue. The only place you need to apply willpower is to change your reaction to the cue.

I've written about 2,000 short stories; I've only published 300 and I feel I'm still learning. Any man who keeps working is not a failure. He may not be a great writer, but if he applies the old fashioned virtues of hard, constant labor, he'll eventually make some kind of career for himself as a writer. Ray Bradbury, 1967 interview (Doing the Math - that means for every story he sold, he wrote six "un-publishable" ones. Keep typing!)

It gave Jane a wicked sense of satisfaction that he’d noticed that aspect of her sister’s personality, but she tried not to sound too arrogant. “Savannah doesn’t worry about homework. Apparently they don’t care about your GPA when you apply for beauty school.” “Beauty school, huh? I would have thought she’d already graduated valedictorian from there.” Jane blinked at him in frustration. Fairy’s side note: Adults are constantly telling teenagers that it’s what’s on the inside that matters. It’s always painful to find out that adults have lied to you. Hunter shrugged. “I guess I shouldn’t have assumed you’d be like Savannah where math is concerned.” Meaning: After all, you aren’t pretty like she is.

Well, regular math, or applied math, is what I suppose you could call practical math," he said. "It's used to solve problems, to provide solutions, whether it's in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn't exist to provide immediate, or necessarily obvious, practical applications. It's purely an expression of form, if you will - the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.

If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes.

You can tell if a discipline is BS if the degree depends severely on the prestige of the school granting it. I remember when I applied to MBA programs being told that anything outside the top ten or twenty would be a waste of time. On the other hand a degree in mathematics is much less dependent on the school (conditional on being above a certain level, so the heuristic would apply to the difference between top ten and top two thousand schools). The same applies to research papers. In math and physics, a result posted on the repository site arXiv (with a minimum hurdle) is fine. In low-quality fields like academic finance (where papers are usually some form of complicated storytelling), the “prestige” of the journal is the sole criterion.

I do not think the division of the subject into two parts - into applied mathematics and experimental physics a good one, for natural philosophy without experiment is merely mathematical exercise, while experiment without mathematics will neither sufficiently discipline the mind or sufficiently extend our knowledge in a subject like physics.

Then the applied mathematician comes along and ruins everything by making topology useful.

If you’d like to see how to apply these ideas directly to memorizing formulas, try out the SkillsToolbox .com website for a list of easy-to-remember visuals for mathematical symbols.7

For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to convey just what the subject is about. Instead, the focus is on learning and applying various procedures to solve math problems. That's a bit like explaining soccer by saying it is executing a series of maneuvers to get the ball into the goal. Both accurately describe various key features, but they miss the \what?" and the \why?" of the big picture.

There is such a thing as nonnerdy applied mathematics: find a problem first, and figure out the math that works for it (just as one acquires language), rather than study in a vacuum through theorems and artificial examples, then change reality to make it look like these examples.

The harder you push your brain to come up with something creative, the less creative your ideas will be. So far, I have not found a single situation where this does not apply. Ultimately, this means that relaxation is an important part of hard work— and good work, for that matter.

As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer...

A synthesis—an abstraction, chunk, or gist idea—is a neural pattern. Good chunks form neural patterns that resonate, not only within the subject we’re working in, but with other subjects and areas of our lives. The abstraction helps you transfer ideas from one area to another. That’s why great art, poetry, music, and literature can be so compelling. When we grasp the chunk, it takes on a new life in our own minds—we form ideas that enhance and enlighten the neural patterns we already possess, allowing us to more readily see and develop other related patterns. Once we have created a chunk as a neural pattern, we can more easily pass that chunked pattern to others, as Cajal and other great artists, poets, scientists, and writers have done for millennia, Once other people grasp that chunk, not only can they use it, but also they can more easily create similar chunks that apply to other areas in their lives—an important part of the creative process.

HISTORY, IN CORK’S OPINION, was a useless discipline, an assemblage of accounts and memories, often flawed, that in the end did the world no service. Math and science could be applied in concrete ways. Literature, if it didn’t enlighten, at least entertained. But history? History was simply a study in futility. Because people never learned.

The easiest explanations are often the right ones," his math professor, Dr. Li, always said, and maybe the sam principle applied here. Except he knew it didn't. Math was one thing. Nothing else was that reductive.

This is apparently a little promotional ¶ where we’re supposed to explain “how and why we came to” the subject of our GD series book (the stuff in quotations is the editor’s words). The overall idea is to humanize the series and make the books and their subjects seem warmer and more accessible. So that people will be more apt to buy the books. I’m pretty sure this is how it works. The obvious objection to such promotional ¶s is that, if the books are any good at all, then the writers’ interest and investment in their subjects will be so resoundingly obvious in the texts themselves that these little pseudo-intimate Why I Cared Enough About Transfinite Math and Where It Came From to Spend a Year Writing a Book About It blurblets are unnecessary; whereas, if the books aren’t any good, it’s hard to see how my telling somebody that as a child I used to cook up what amounted to simplistic versions of Zeno’s Dichotomy and ruminate on them until I literally made myself sick, or that I once almost flunked a basic calc course and have seethed with dislike for conventional higher-math education ever since, or that the ontology and grammar of abstractions have always struck me as one of the most breathtaking problems in human consciousness—how any such stuff will help. The logic of this objection seems airtight to me. In fact, the only way the objection doesn’t apply is if these ¶s are really nothing more than disguised ad copy, in which case I don’t see why anyone reading them should even necessarily believe that the books’ authors actually wrote them—I mean, maybe somebody in the ad-copy department wrote them and all we did was sort of sign off on them. There’d be a kind of twisted integrity about that, though—at least no one would be pretending to pretend.

Pure math,” he replied. “How is that different from”—she laughed—“regular math?” Gillian asked. “Well, regular math, or applied math, is what I suppose you could call practical math,” he said. “It’s used to solve problems, to provide solutions, whether it’s in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn’t exist to provide immediate, or necessarily obvious, practical applications. It’s purely an expression of form, if you will—the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.

A synthesis—an abstraction, chunk, or gist idea—is a neural pattern. Good chunks form neural patterns that resonate, not only within the subject we’re working in, but with other subjects and areas of our lives. The abstraction helps you transfer ideas from one area to another. That’s why great art, poetry, music, and literature can be so compelling. When we grasp the chunk, it takes on a new life in our own minds—we form ideas that enhance and enlighten the neural patterns we already possess, allowing us to more readily see and develop other related patterns. Once we have created a chunk as a neural pattern, we can more easily pass that chunked pattern to others, as Cajal and other great artists, poets, scientists, and writers have done for millennia, Once other people grasp that chunk, not only can they use it, but also they can more easily create similar chunks that apply to other areas in their lives—an important part of the creative process.

HISTORY, IN CORK’S OPINION, was a useless discipline, an assemblage of accounts and memories, often flawed, that in the end did the world no service. Math and science could be applied in concrete ways. Literature, if it didn’t enlighten, at least entertained. But history? History was simply a study in futility. Because people never learned. Century after century, they committed the same atrocities against one another or against the earth, and the only thing that changed was the magnitude of the slaughter.

Note that there’s no option to answer “all of the above.” Prospective workers must pick one option, without a clue as to how the program will interpret it. And some of the analysis will draw unflattering conclusions. If you go to a kindergarten class in much of the country, for example, you’ll often hear teachers emphasize to the children that they’re unique. It’s an attempt to boost their self-esteem and, of course, it’s true. Yet twelve years later, when that student chooses “unique” on a personality test while applying for a minimum-wage job, the program might read the answer as a red flag: Who wants a workforce peopled with narcissists?

counterfactual emotions,” or the feelings that spurred people’s minds to spin alternative realities in order to avoid the pain of the emotion. Regret was the most obvious counterfactual emotion, but frustration and envy shared regret’s essential trait. “The emotions of unrealized possibility,” Danny called them, in a letter to Amos. These emotions could be described using simple math. Their intensity, Danny wrote, was a product of two variables: “the desirability of the alternative” and “the possibility of the alternative.” Experiences that led to regret and frustration were not always easy to undo. Frustrated people needed to undo some feature of their environment, while regretful people needed to undo their own actions. “The basic rules of undoing, however, apply alike to frustration and regret,” he wrote. “They require a more or less plausible path leading to the alternative state.” Envy was different. Envy did not require a person to exert the slightest effort to imagine a path to the alternative state. “The availability of the alternative appears to be controlled by a relation of similarity between oneself and the target of envy. To experience envy, it is sufficient to have a vivid image of oneself in another person’s shoes; it is not necessary to have a plausible scenario of how one came to occupy those shoes.” Envy, in some strange way, required no imagination. Danny spent the

A Puritan twist in our nature makes us think that anything good for us must be twice as good if it's hard to swallow. Learning Greek and Latin used to play the role of character builder, since they were considered to be as exhausting and unrewarding as digging a trench in the morning and filling it up in the afternoon. It was what made a man, or a woman -- or more likely a robot -- of you. Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult. What a perverse fate for one of our kind's greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you're an adult you'll never have to listen to music again. And this is mathematics we're talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down into our deepest intuitions and outward toward the nature of the universe -- mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together -- beyond, around, and within us. It doesn't just help us to balance our checkbooks; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our **architectural instinct** -- as deep in us as any of our urges.

Mathematical theories have sometimes been used to predict phenomena that were not confirmed until years later. For example, Maxwell's equations, named after physicist James Clerk Maxwell, predicted radio waves. Einstein's field equations suggested that gravity would bend light and that the universe is expanding. Physicist Paul Dirac once noted that the abstract mathematics we study now gives us a glimpse of physics in the future. In fact, his equations predicted the existence of antimatter, which was subsequently discovered. Similarly, mathematician Nikolai Lobachevsky said that "there is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.

What you learn after a long time in math-and I think the lesson applies much more broadly-is that there's always somebody ahead of you, whether they're right there in class with you or not. People just starting out look to people with good theorems, people with some good theorems look to people with lots of good theorems, people with lots of good theorems look to people with Fields Medals, people with Fields Medals look to the "inner circle" Medalists, and those people can always look toward the dead. Nobody ever looks in the mirror and says, "Let's face it, I'm smarter than Gauss." And yet, in the last hundred years, the joined effort of all these dummies-compared-to-Gauss has produced the greatest flowering of mathematical knowledge the world has ever seen.

I work in theoretical computer science: a field that doesn’t itself win Fields Medals (at least not yet), but that has occasions to use parts of math that have won Fields Medals. Of course, the stuff we use cutting-edge math for might itself be dismissed as “ivory tower self-indulgence.” Except then the cryptographers building the successors to Bitcoin, or the big-data or machine-learning people, turn out to want the stuff we were talking about at conferences 15 years ago—and we discover to our surprise that, just as the mathematicians gave us a higher platform to stand on, so we seem to have built a higher platform for the practitioners. The long road from Hilbert to Gödel to Turing and von Neumann to Eckert and Mauchly to Gates and Jobs is still open for traffic today. Yes, there’s plenty of math that strikes even me as boutique scholasticism: a way to signal the brilliance of the people doing it, by solving problems that require years just to understand their statements, and whose “motivations” are about 5,000 steps removed from anything Caplan or Bostrom would recognize as motivation. But where I part ways is that there’s also math that looked to me like boutique scholasticism, until Greg Kuperberg or Ketan Mulmuley or someone else finally managed to explain it to me, and I said: “ah, so that’s why Mumford or Connes or Witten cared so much about this. It seems … almost like an ordinary applied engineering question, albeit one from the year 2130 or something, being impatiently studied by people a few moves ahead of everyone else in humanity’s chess game against reality. It will be pretty sweet once the rest of the world catches up to this.

Which is actually good because we’re doing an AP Euro study group this week at the library—I mean good that it got canceled, not good that someone died—so I was wondering too if maybe I can use the car, so you won’t have to come pick me up super late every night?” Alma had been a wildly clingy kid, but now she is a mostly autonomous and wholly inscrutable seventeen-year-old; she is mean and gorgeous and breathtakingly good at math; she has inside jokes with her friends about inexplicable things like Gary Shandling and avocado toast, paints microscopic cherries on her fingernails and endeavors highly involved baking ventures, filling their fridge with oblong bagels and six-layer cakes. “I’m asking now because last time you told me I didn’t give you enough notice,” she says. She has recently begun speaking conversationally to Julia and Mark again after nearly two years of brooding silence, and now it’s near impossible to get her to stop. She regales them with breathless incomprehensible stories at the dinner table; she delivers lengthy recaps of midseason episodes of television shows they have never seen; she mounts elaborate and convincing defenses of things she wants them to give her, or give her permission to do. Conversing with her is a mechanical act requiring the constant ability to shift gears, to backpedal or follow inane segues or catapult from the real world to a fictional one without stopping to refuel. There’s not a snowball’s chance in hell that she won’t be accepted next month to several of the seventeen exalted and appallingly expensive colleges to which she has applied, and because Julia would like the remainder of her tenure at home to elapse free of trauma, she responds to her daughter as she did when she was a napping baby, tiptoeing around her to avoid awakening unrest. The power dynamic in their household is not unlike that of a years-long hostage crisis.

Andromeda said, “I see that you understand the paradox involved. These are axiomatic beliefs. If life is finite, there can be no math, no logic, nothing which says using the Eschaton Engine to obliterate the majority of the universe in self-preservation is wrong. No game theory applies, because there is no retaliation, no tit for tat. No punishment. But if life is infinite, then an infinite game theory applies, and no act where the ends justifies the means is allowed, because there is no Concubine Vector, no eternal imbalance, no chance of any act escaping unpunished.

Fiqh is man-made!" By that reasoning and applying your standards, so are logic, math, natural sciences, and virtually every single body of knowledge and its fruits. If being man made is sufficient reason to reject fiqh, it's more than sufficient reason to reject those, too. And before arguing that science is special, know that the epistemology and philosophy of science are also man-made. UPDATE. Pay particular attention to where it says the "body of knowledge." And if you want to eliminate that qualifier, read up a bit on antirealism.

Data Science takes the guesswork/emotions out of answering business questions by applying logic and mathematics to find better solutions.

So the way to think about the problem is this. You and I and computers and bacteria and viruses and everything else material are made of molecules and atoms, which are themselves composed of particles like electrons and quarks. Schrodinger's equation works for electrons and quarks, and all evidence points to its working for things made of these constituents, regardless of the number of particles involved. This means that Schrodinger's equation should continue to apply during a measurement. After all, a measurement is just one collection of particles (the person, the equipment, the computer...) coming into contact with another (the particle or particles being measured). But if that's the case, if Schrodinger's math refuses to bow down, then Bohr is in trouble. Schrodinger's equation doesn't allow waves to collapse. An essential element of the Copenhagen approach would therefore be undermined. So the third question is this: If the reasoning just recounted is right and probability waves don't collapse, how do we pass from the range of possible outcomes that exist before a measurement to the single outcome the measurement reveals? Or to put it in more general terms, what happens to a probability wave during a measurement that allows a familiar, definite, unique reality to take hold? Everett pursued this question in his Princeton doctoral dissertation and came to an unforeseen conclusion.

I believe maths should be applied to the arts. So I claim that it was in Britain that the famous equation E = mc2 was proved: E = exposure (of nether parts) m = much c = chuckling chuckling squared = helpless laughter

Of course, topologists don’t care about any of this “applied math” nonsense. They’re just trying to find all the shapes.

Contrary to what many well-intentioned people believe, the fact that we have multiple social media platforms today has little effect on spreading genuinely diverse narratives and perspectives. Social media is not only increasingly in the hands of a few billionaires strongly connected to the ruling class (e.g., Meta acquiring some of the most popular and active platforms), but also the fact that social media platforms operate based on carefully designed and manipulated algorithms to promote the viewpoints of the ruling class in what Cathy O’Neil has called ‘weapons of math destruction’, and what Safiya Umoja Noble insightfully calls ‘algorithms of oppression’, which apply not only to racial matters, but extend to every other matter that is potentially at odds with the desires of the ruling class.

Fibonacci’s new numbering system became a hit with the merchant class and for centuries was the preeminent source for mathematical knowledge in Europe. But something equally important also happened around this time: Europeans learned of double-entry bookkeeping, picking it up from the Arabians, who’d been using it since the seventh century. Merchants in Florence and other Italian cities began applying these new accounting measures to their daily businesses. Where Fibonacci gave them new measurement methods for business, double-entry accounting gave them a way to record it all. Then came a seminal moment: in 1494, two years after Christopher Columbus first set foot in the Americas, a Franciscan friar named Luca Pacioli wrote the first comprehensive manual for using this accounting system. Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalita, written in Italian rather than Latin so as to be more accessible to the public, would become the first popular work on math and accounting. Its section on accounting was so well received that the publisher eventually published it as its own volume. Pacioli offered access to the precision of mathematics. “Without double entry, businessmen would not sleep easily at night,” Pacioli wrote, mixing in the practical with the technical—Pacioli’s Summa would become a kind of self-help book for the merchant class.

The most common anxiety I hear about learning to program is that people think it requires a lot of math. Actually, most programming doesn’t require math beyond basic arithmetic. In fact, being good at programming isn’t that different from being good at solving Sudoku puzzles. To solve a Sudoku puzzle, the numbers 1 through 9 must be filled in for each row, each column, and each 3×3 interior square of the full 9×9 board. You find a solution by applying deduction and logic from the starting numbers. For

Big data is not about trying to “teach” a computer to “think” like humans. Instead, it’s about applying math to huge quantities of data in order to infer probabilities: the

... the development of mathematics, for the sciences and for everybody else, does not often come from pure math. It came from the physicists, engineers, and applied mathematicians. The physicists were on to many ideas which couldn’t be proved, but which they knew to be right, long before the pure mathematicians sanctified it with their seal of approval. Fourier series, Laplace transforms, and delta functions are a few examples where waiting for a rigorous proof of procedure would have stifled progress for a hundred years. The quest for rigor too often meant rigor mortis. The physicists used delta functions early on, but this wasn’t really part of mathematics until the theory of distributions was invoked to make it all rigorous and pure. That was a century later! Scientists and engineers don’t wait for that: they develop what they need when they need it. Of necessity, they invent all sorts of approximate, ad hoc methods: perturbation theory, singular perturbation theory, renormalization, numerical calculations and methods, Fourier analysis, etc. The mathematics that went into this all came from the applied side, from the scientists who wanted to understand physical phenomena. [...] So much of mathematics originates from applications and scientific phenomena. But we have nature as the final arbiter. Does a result agree with experiment? If it doesn’t agree with experiment, something is wrong.

One might think that math aptitude would apply across the code-writing world, but this was not so. The logical operations of a computer, while abstract, were abstract in a familiar way. When computers added two numbers together, for instance, they did so in much the same way as people do: One plus one equals two. A computer performed this addition faster than a person, but the route to the answer was the same. This was not so with graphics. When an ordinary person (as opposed to a civil engineer) wanted to draw a circle, he did not apply a special algorithm to his choice of a diameter. He just drew a circle with his hand. Having no hands, computers drew circles—and all the other shapes—by applying mathematical formulas. The better the formulas, the more accurate and versatile the shapes.

At its core, big data is about predictions. Though it is described as part of the branch of computer science called artificial intelligence, and more specifically, an area called machine learning, this characterization is misleading. Big data is not about trying to “teach” a computer to “think” like humans. Instead, it’s about applying math to huge quantities of data in order to infer probabilities:

Applied war is where the money's at. I don't mean that figuratively. Defense is 140% of the UVE budget. Offense is twice that. This is possible because there is no budget for math education.

felicity. It’s simple math, though. Add up the pleasurable aspects of your life, then subtract the unpleasant ones. The result is your overall happiness. The same calculations, Bentham believed, could apply to an entire nation. Every action a government took, every law it passed, should be viewed through the “greatest happiness” prism.

String theory is potentially the next and final step in this progression. In a single framework, it handles the domains claimed by relativity and the quantum. Moreover, and this is worth sitting up straight to hear, string theory does so in a manner that fully embraces all the discoveries that preceded it. A theory based on vibrating filaments might not seem to have much in common with general relativity's curved spacetime picture of gravity. Nevertheless, apply string theory's mathematics to a situation where gravity matters but quantum mechanics doesn't (to a massive object, like the sun, whose size is large) and out pop Einstein's equations. Vibrating filaments and point particles are also quite different. But apply string theory's mathematics to a situation where quantum mechanics matters but gravity doesn't (to small collections of strings that are not vibrating quickly, moving fast, or stretched long; they have low energy-equivalently, low mass- so gravity plays virtually no role) and the math of string theory morphs into the math of quantum field theory.

A photograph, which used to be a pattern of pigment on a sheet of chemically coated paper, is not a string of numbers, each one representing the brightness and color of a pixel. An image captured on a 4-megapixel camera is a list of 4 million numbers-no small commitment of memory for the device shooting the picture. But these numbers are highly correlated with each other. If one pixel is bright green, the next one over likely to be as well. The actual information contained in the image is much less than 4 million numbers' worth-and it's precisely this fact that makes it possible to have compression, the critical mathematical technology that allows images, videos, music, and text to be stored in much smaller spaces than you'd think. The presence of correlation makes compression possible; actually doing it involves much more modern ideas, like the theory of wavelets developed in the 1970s and 80s by Jean Morlet, Stephane Mallat, Yves Meyer, Ingrid Daubechies, and others; and the rapidly developing area of compressed sensing, which started with a 2005 paper by Emmanuel Candes, Justin Romberg, and Terry Tao, and has quickly become its own active subfield of applied math.

Circuit design is one of the great applied-maths problems.

This quest to take a problem and see what happens in different situations is called generalizing, and it is this force that drives mathematics forward. Mathematicians constantly want to find solutions and patterns which apply to as many situations as possible, i.e. are as general as possible. A maths puzzle is not complete when you merely find an answer, a maths puzzle is complete when you've then tried to generalize it to other situations as well-and minds including Leonard Euler and Lord Kelvin have excelled in mathematics by displaying just this kind of curiosity. Because mathematicians like the puzzles which work on the pure number rather than the symbolic digit and the system we happen to be writing our numbers down in, there is a sense that, when a puzzle works only in one given base, there is something rather, well, 'secind class' about it. Mathematicians do not like things which work only in base-10; it is only because we have ten fingers that we find that system interesting at all. Mathematics is the search for universal, not base-specific, truth.

I now follow the rule of thumb that is basically: The harder you push your brain to come up with something creative, the less creative your ideas will be. So far, I have not found a single situation where this does not apply. Ultimately, this means that relaxation is an important part of hard work—and good work, for that matter.

Cellular biologist Glen Rein, Ph.D., conceived of a series of experiments to test healers’ ability to affect biological systems. [...] In Dr. Rein’s experiment, he first studied a group of ten individuals who were well practiced in using techniques that Heart-Math teaches to build heart-focused coherence. They applied the techniques to produce strong, elevated feelings such as love and appreciation, then for two minutes, they held vials containing DNA samples suspended in deionized water. When those samples were analyzed, no statistically significant changes had occurred. A second group of trained participants did the same thing, but instead of just creating positive emotions (a feeling) of love and appreciation, they simultaneously held an intention (a thought) to either wind or unwind the strands of DNA. This group produced statistically significant changes in the conformation (shape) of the DNA samples. In some cases the DNA was wound or unwound as much as 25 percent! A third group of trained subjects held a clear intent to change the DNA, but they were instructed not to enter into a positive emotional state. In other words, they were only using thought (intention) to affect matter. The result? No changes to the DNA samples. [...] Only when subjects held both heightened emotions and clear objectives in alignment were they able to produce the intended effect. An intentional thought needs an energizer, a catalyst—and that energy is an elevated emotion.

This caveat applies to teachers, too! In one study, we taught students a math lesson spiced up with some math history, namely, stories about great mathematicians. For half of the students, we talked about the mathematicians as geniuses who easily came up with their math discoveries. This alone propelled students into a fixed mindset. It sent the message: There are some people who are born smart in math and everything is easy for them. Then there are the rest of you. For the other half of the students, we talked about the mathematicians as people who became passionate about math and ended up making great discoveries. This brought students into a growth mindset. The message was: Skills and achievement come through commitment and effort. It’s amazing how kids sniff out these messages from our innocent remarks.

Maya’s face as though wondering what to tell her. ‘It’s just I know they weren’t always happy, and I did once wonder if they’d have stayed together… There was something my husband, George, said when you were first in my maths class. As you know, he taught the other year one class at your primary school and mentioned how once he’d had to break up an argument between your parents when they were waiting to pick you up from school. It must have been pretty heated for him to remember it after all that time – he wasn’t one to gossip. Apparently, Mrs Lyons wouldn’t let you out of your classroom until George had managed to calm them down.’ Maya feels her stomach clench. ‘All couples argue.’ ‘I know.’ Mrs Ellis pats her hand. ‘And that’s why you mustn’t worry about it. It was a long time ago, anyway.’ The bus is stopping. Bending to her bag, Mrs Ellis moves it so that it’s not in the way of the people getting on. ‘But if you ever feel you want to spread your wings, you mustn’t feel your dad would be on his own. He’s a grown man, and you can’t make him your responsibility. I’m sure he has friends, neighbours, even work colleagues who would keep an eye on him. Doesn’t he have his own private practice in Lyme Regis?’ ‘Yes, but it’s not the same. He needs me.’ Maya’s voice slips away, so it’s barely a whisper. ‘Yes, he needs me. It’s why I couldn’t go to university.’ She doesn’t want to talk about that time for, although her dad had been encouraging when she’d first told him she was applying, a week after the forms were filled in, a cloud had settled over him. One that was darker than previous ones. Maya had tempted him with his favourite food, enticed him out for healing walks along the clifftop, but nothing she’d done could lift it. Eventually, telling herself it was because of what she’d done, she’d deleted her application from the computer. When her dad had found out and asked why she’d done it, she’d told him it was because she couldn’t face more studying. Would rather earn a living. Whether he’d believed her or not, she couldn’t say. What she did know was that he’d never tried to change her mind. ‘Do you like your job, Maya?’ Maya lowers her eyes and studies her hands. It’s something she hasn’t given much thought to. Her job is just something she does to get through

That’s the reason we like music. Music is far more complex than [the ratios of] Pythagoras. The reason doesn’t have to do with mathematics, it has to do with biology.”7 I might temper this a little bit by saying that the harmonics our palates and vocal cords create might come into prominence because, like Archimedes’s vibrating string, any sound-producing object tends to privilege that hierarchy of pitches. That math applies to our bodies and vocal cords as well as strings, though Purves would seem to have a point when he says we have tuned our mental radios to the pitches and overtones that we produce in both speech and music.

To apply first principles thinking to the field of value investing, consider several fundamental truths. Understand and practice the following if you want to become a good investor: 1. Look at stocks as part ownership of a business. 2. Look at Mr. Market—volatile stock price fluctuations—as your friend rather than your enemy. View risk as the possibility of permanent loss of purchasing power, and uncertainty as the unpredictability regarding the degree of variability in the possible range of outcomes. 3. Remember the three most important words in investing: “margin of safety.” 4. Evaluate any news item or event only in terms of its impact on (a) future interest rates and (b) the intrinsic value of the business, which is the discounted value of the cash that can be taken out during its remaining life, adjusted for the uncertainty around receiving those cash flows. 5. Think in terms of opportunity costs when evaluating new ideas and keep a very high hurdle rate for incoming investments. Be unreasonable. When you look at a business and get a strong desire from within saying, “I wish I owned this business,” that is the kind of business in which you should be investing. A great investment idea doesn’t need hours to analyze. More often than not, it is love at first sight. 6. Think probabilistically rather than deterministically, because the future is never certain and it is really a set of branching probability streams. At the same time, avoid the risk of ruin, when making decisions, by focusing on consequences rather than just on raw probabilities in isolation. Some risks are just not worth taking, whatever the potential upside may be. 7. Never underestimate the power of incentives in any given situation. 8. When making decisions, involve both the left side of your brain (logic, analysis, and math) and the right side (intuition, creativity, and emotions). 9. Engage in visual thinking, which helps us to better understand complex information, organize our thoughts, and improve our ability to think and communicate. 10. Invert, always invert. You can avoid a lot of pain by visualizing your life after you have lost a lot of money trading or speculating using derivatives or leverage. If the visuals unnerve you, don’t do anything that could get you remotely close to reaching such a situation. 11. Vicariously learn from others throughout life. Embrace everlasting humility to succeed in this endeavor. 12. Embrace the power of long-term compounding. All the great things in life come from compound interest.

If you were to note every manmade item within your field of vision, chances are that nearly every last gadget and trinket was invented by a white man. According to Charles Murray’s book Human Accomplishment, whites have historically dominated the fields of physics, math, chemistry, medicine, biology, and technology. What’s grossly ironic is the specter of people using white computers hooked up to white electricity sent across white power grids to criticize the very white people who made their whining possible. Even worse is the ubiquity of white people pejoratively using the term “white people” as if it somehow doesn’t apply to them. That right there is a collective mental illness for the ages.

Unit circle is one of the important math concepts that every student must learn and understand. There are numerous concepts related to Trigonometry and geometry that needs to understand basics before solving the problems. Unit circle is known as the foundation of projectile motion, sine, cosine, tangents, degrees and radians. If you are learning the concept of geometry and trigonometry then you must have a unit circle chart as reference sheet. Most of the school teachers us this sheet while teaching the concepts of applied mathematics. This basic circle will be helpful throughout your life. It is necessary to learn this Blank Unit Circle Printable by heart and to practice it regularly for a solid foundation.

The social psychologist Claude Steele demonstrated the power of what he calls “stereotype threat” in the U.S. context: Women do better on math tests when they are explicitly told that the stereotype that women are worse in math does not apply to this particular test; African Americans do worse on tests if they have to start by indicating their race on the cover sheet.33 Following Steele’s work, two researchers from the World Bank had lower-caste children in the Indian state of Uttar Pradesh compete against high-caste children in solving mazes.34 They found that the low-caste children compete well against the high-caste children as long as caste is not salient, but once low-caste children are reminded that they are low castes competing with high-caste children (by the simple contrivance of asking them their full names before the game starts), they do much worse. The authors argue that this may be driven in part by a fear of not being evaluated fairly by the obviously elite organizers of the game, but it could just as well be the internalization of the stereotype. A child who expects to find school difficult will probably blame herself and not her teachers when she can’t understand what is being taught, and may end up deciding she’s not cut out for school—“stupid,” like most of her ilk—and give up on education altogether, daydreaming in class or, like Shantarama’s children, just refusing to go

Euler's general equation stands out because it forged a fundamental link between different areas of math, and because of its versatility in applied mathematics. After Euler's time it came to be regarded as a cornerstone in "complex analysis," a fertile branch of mathematics concerned with functions whose variables stand for complex numbers.

Repetition until Your Learning Becomes Unconscious (Outsourced to Environment) While I implemented what I learned, my teacher would watch me from a distance. He let me struggle as I tried to remember what he had just shown me. The first time, applying what he taught took a lot of time and effort. So we did it again, and again, and again. Over time, I became competent and thus confident. Learning something new is all about memory and how you use it. At first, your prefrontal cortex—which stores your working (or short-term) memory—is really busy figuring out how the task is done. But once you’re proficient, the prefrontal cortex gets a break. In fact, it’s freed up by as much as 90 percent. Once this happens, you can perform that skill automatically, leaving your conscious mind to focus on other things. This level of performance is called automaticity, and reaching it depends on what psychologists call overlearning or overtraining. The process of getting a skill to automaticity involves four steps, or stages: Repeated learning of a small set of information. If you’re playing basketball, for instance, that might mean shooting the same shot over and over. The key here is to go beyond the initial point of mastery. Make your training progressively more difficult. You want to make the task harder and harder until it’s too hard. Then you bring the difficulty back down slightly, in order to stay near the upper limit of your current ability. Add time constraints. For example, some math teachers ask students to work on difficult problems with increasingly shortened timelines. Adding the component of time challenges you in two ways. First, it forces you to work quickly, and second, it saps a portion of your working memory by forcing it to remain conscious of the ticking clock. Practice with increasing memory load—that is, trying to do a mental task with other things on your mind. Put simply, it’s purposefully adding distractions to your training regimen.

Skin in the game can make boring things less boring. When you have skin in the game, dull things like checking the safety of the aircraft because you may be forced to be a passenger in it cease to be boring. If you are an investor in a company, doing ultra-boring things like reading the footnotes of a financial statement (where the real information is to be found) becomes, well, almost not boring. But there is an even more vital dimension. Many addicts who normally have a dull intellect and the mental nimbleness of a cauliflower—or a foreign policy expert—are capable of the most ingenious tricks to procure their drugs. When they undergo rehab, they are often told that should they spend half the mental energy trying to make money as they did procuring drugs, they are guaranteed to become millionaires. But, to no avail. Without the addiction, their miraculous powers go away. It was like a magical potion that gave remarkable powers to those seeking it, but not those drinking it. A confession. When I don’t have skin in the game, I am usually dumb. My knowledge of technical matters, such as risk and probability, did not initially come from books. It did not come from lofty philosophizing and scientific hunger. It did not even come from curiosity. It came from the thrills and hormonal flush one gets while taking risks in the markets. I never thought mathematics was something interesting to me until, when I was at Wharton, a friend told me about the financial options I described earlier (and their generalization, complex derivatives). I immediately decided to make a career in them. It was a combination of financial trading and complicated probability. The field was new and uncharted. I knew in my guts there were mistakes in the theories that used the conventional bell curve and ignored the impact of the tails (extreme events). I knew in my guts that academics had not the slightest clue about the risks. So, to find errors in the estimation of these probabilistic securities, I had to study probability, which mysteriously and instantly became fun, even gripping. When there was risk on the line, suddenly a second brain in me manifested itself, and the probabilities of intricate sequences became suddenly effortless to analyze and map. When there is fire, you will run faster than in any competition. When you ski downhill some movements become effortless. Then I became dumb again when there was no real action. Furthermore, as traders the mathematics we used fit our problem like a glove, unlike academics with a theory looking for some application—in some cases we had to invent models out of thin air and could not afford the wrong equations. Applying math to practical problems was another business altogether; it meant a deep understanding of the problem before writing the equations.

Applying math to practical problems was another business altogether; it meant a deep understanding of the problem before writing the equations. But if you muster the strength to weight-lift a car to save a child, above your current abilities, the strength gained will stay after things calm down. So, unlike the drug addict who loses his resourcefulness, what you learn from the intensity and the focus you had when under the influence of risk stays with you. You may lose the sharpness, but nobody can take away what you’ve learned. This is the principal reason I am now fighting the conventional educational system, made by dweebs for dweebs. Many kids would learn to love mathematics if they had some investment in it, and, more crucially, they would build an instinct to spot its misapplications.

College classes teaching math are nice clean places, therefore math itself can't apply to life situations that aren't nice and clean

Pure mathematics is useless to ordinary people, but useful to those who wish to understand Absolute Truth, God or Infinity. Applied maths is useful in our daily life.

I’ve wondered if I could take my computer programming skills and apply them to learning math.