Math Functions Quotes

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing—one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me.

Struggle is not optional—it's neurologically required: in order to get your skill circuit to fire optimally, you must by definition fire the circuit suboptimally; you must make mistakes and pay attention to those mistakes; you must slowly teach your circuit. You must also keep firing that circuit—i.e., practicing—in order to keep myelin functioning properly. After all, myelin is living tissue.

Think of those fingers as abilities. A creative person may write, paint, sculpt, or think up math formulae; he or she might dance or sing or play a musical instrument. Those are the fingers, but creativity is the hand that gives them life. & just as all hands are basically the same - form follows function - all creative people are the same once you get down to the place where the fingers join.

But she loved the simplicity of math, a number growing or shrinking depending on which function you performed. No surprises, just one logical step leading to another.

Asian children can perform basic functions, such as addition, far more easily. Ask an English-speaking seven-year-old to add thirty-seven plus twenty-two in her head, and she has to convert the words to numbers (37 + 22). Only then can she do the math: 2 plus 7 is 9 and 30 and 20 is 50, which makes 59. Ask an Asian child to add three-tens-seven and two-tens-two, and then the necessary equation is right there, embedded in the sentence. No number translation is necessary: It’s five-tens-nine.

Tengo's lectures took on uncommon warmth, and the students found themselves swept up in his eloquence. He taught them how to practically and effectively solve mathematical problems while simultaneously presenting a spectacular display of the romance concealed in the questions it posed. Tengo saw admiration in the eyes of several of his female students, and he realized that he was seducing these seventeen- or eighteen-year-olds through mathematics. His eloquence was a kind of intellectual foreplay. Mathematical functions stroked their backs; theorems sent warm breath into their ears.

Excel suffers from an image problem. Most people assume that spreadsheet programs such as Excel are intended for accountants, analysts, financiers, scientists, mathematicians, and other geeky types. Creating a spreadsheet, sorting data, using functions, and making charts seems daunting, and best left to the nerds.

A creative person may write, paint, sculpt, or think up math formulae; he or she might dance or sing or play a musical instrument. Those are the fingers, but creativity is the hand that gives them life. And just as all hands are basically the same—form follows function—all creative people are the same once you get down to the place where the fingers join.

We’ve now established three things. First, we don’t need willpower when we don’t desire to do something, and it isn’t a thing some of us have in excess and some of us don’t have at all. It’s a cognitive function, like deciding what to eat or solving a math equation or remembering your dad’s birthday. Willpower is also a limited resource; we have more of it at the beginning of the day and lose it throughout the day as we use it to write emails or not eat cookies. When you automate some decisions or processes (through forming habits), you free up more brain power. Second, for us to make and change a habit, we need a cue, a routine, and a reward, and enough repetition must occur for the process to move from something we have to think about consciously (“I need to brush my teeth,” “I don’t want to drink wine”) to something we do naturally, automatically. Third, throughout the day, we must manage our energy so that we don’t blow out and end up in the place of no return—a hyperaroused state where the only thing that can bring us down is a glass (or a bottle) of wine. Maybe

This world is of a single piece; yet, we invent nets to trap it for our inspection. Then we mistake our nets for the reality of the piece. In these nets we catch the fishes of the intellect but the sea of wholeness forever eludes our grasp. So, we forget our original intent and then mistake the nets for the sea. Three of these nets we have named Nature, Mathematics, and Art. We conclude they are different because we call them by different names. Thus, they are apt to remain forever separated with nothing bonding them together. It is not the nets that are at fault but rather our misunderstanding of their function as nets. They do catch the fishes but never the sea, and it is the sea that we ultimately desire.

she loved the simplicity of math, a number growing or shrinking depending on which function you performed. No surprises, just one logical step leading to another.

two of the most important function classes: exponential and trigonometric functions.

Quinn until he messaged me his nightly text, which had turned somewhat math-mushy recently: If I were a function, you would be my asymptote. I always tend toward you. He

The math is unavoidable: it is an inevitable property of long-lasting exponential growth that it ends up in a singularity, a point in time when a function reaches an infinite value, making anything instantly possible.

The reality is that most of us grow up strapped in an educational system that favors obedience over independent thinking. We’re rewarded for trusting authority, and punished for challenging it. We focus on memorizing the stuff other people came up with—formulas in math, grammar rules in English, theories in physics, cell functions in biology—rather than grasping the logic behind our most important breakthroughs and tracing the footsteps of their discovery. We answer test questions with what we think our teacher wants to hear. We chase grades instead of knowledge. And worst of all, we leave the classroom woefully unequipped with the thinking skills that matter most: how to balance open-mindedness with skepticism, how to identify bias, and how to challenge assumptions—including our own—in a way that’s truly objective.

I had to do a whole entire two page--both front side and back--of exponets. finding the product and finding the quotient. My brain has never stopped functioning that quick in the span of one second. Didn't even need to pick up the pencil, the paper(s) itself sparked fear into my soul

it is better to be loved than admired. It is better to be truly known and seen and taken care of by a small tribe than adored by strangers who think they know you in a meaningful way. We know that's true. But many of us, functionally, have gotten that math wrong in one season or another.

... I left Caen, where I was living, to go on a geological excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for convenience sake, I verified the result at my leisure.

At one point, the math nerd in me could not help but calculate, literally on the back of an envelope on an airplane, the fantastic improbability that a single functional protein was ever created by accident in the entire history of the universe. I was thunderstruck—it was an “Aha” moment. I remember staring at the calculations in disbelief—couldn’t others do the math, and see what seemed obvious? It was a “no-brainer.” At that moment, I knew modern science supported belief in God.

As was foreshadowed in Paragraphs 1 and 4, Cantor, and Dedekind's near-simultaneous appearance in math is more or less the Newton + Leibniz thing all over again, a sure sign that the Time Was Right for (Infinity)-type sets. Just as striking is the Escherian way the two men's work dovetails. Cantor is able to define and ground the concepts of 'infinite set' and 'transfinite number,' and to establish rigorous techniques for combining and comparing different types of (Infinity)s, which is just where Dedekind's def. of irrationals needs shoring up. Pro quo, the schnitt technique demonstrates that actually-infinite sets can have real utility in analysis. That, in other words, as sensuously and cognitively abstract as they must remain, (Infinity)s can nevertheless function in math as practical abstractions rather than as just weird paradoxical flights of fancy.

It happens.” He turned again to the garden. “To the brain, I mean. It’s the drink, naturally. And the hepatic function. But there is clearly something else occurring, too. Certainly in other men I’ve seen it. There is something in certain abilities that is never far from—far from—” He looked out at the lake. “I cannot really know.” “No, please go on.” “Far from terror, perhaps. It is not such a rare phenomenon, you see. I used to encounter it around the maths division when I was at university, and I have seen it here, even, in my little country practice. It seems to be quite primal. At its crudest, it is a bona fide paranoia. Plenty in the field are gone before the age of twenty. I’ve seen that, too. Perhaps it is a harbinger. I believe it to be physiological.” He looked down. “I sometimes imagine it as God’s revenge.” “Against mathematicians?” “One must bear in mind that they might be considered spies.” He was smiling now. “By the Deity, you mean?” “Indeed. Your dad’s cantankerous nature, by the way—you know that this is his liver, too, don’t you? And of course the drink plays a part in it—but it is also the man himself. The emotions are ablaze in him.” He set down the bucket. “For people like you and me—well, we are shielded by all our damping circuitry. We maintain a cushion against the world, if you will. A comfort against the ravage. But I believe it is not so for him.” He regarded me. “Think of what life must be like for a mind like your father’s. I mean, human existence is bounded by tragedy, is it not? And shot through with it, as well. I was born in Lahore, so I know this in a particular way. But your father, too—he knows it just as particularly, in his own way. I have learned to keep such thoughts somewhat at bay. And so have you. But for him, there is no ignoring it. There is no joy in God’s creation. No pleasure in sunlight or water. No pleasure in a good meal. There is no pleasure in the company of friends. There is nothing. Nothing that might assuage the maw. He

A few books that I've read.... Pascal, an Introduction to the Art and Science of Programming by Walter Savitch Programming algorithms Introduction to Algorithms, 3rd Edition (The MIT Press) Data Structures and Algorithms in Java Author: Michael T. Goodrich - Roberto Tamassia - Michael H. Goldwasser The Algorithm Design Manual Author: Steven S Skiena Algorithm Design Author: Jon Kleinberg - Éva Tardos Algorithms + Data Structures = Programs Book by Niklaus Wirth Discrete Math Discrete Mathematics and Its Applications Author: Kenneth H Rosen Computer Org Structured Computer Organization Andrew S. Tanenbaum Introduction to Assembly Language Programming: From 8086 to Pentium Processors (Undergraduate Texts in Computer Science) Author: Sivarama P. Dandamudi Distributed Systems Distributed Systems: Concepts and Design Author: George Coulouris - Jean Dollimore - Tim Kindberg - Gordon Blair Distributed Systems: An Algorithmic Approach, Second Edition (Chapman & Hall/CRC Computer and Information Science Series) Author: Sukumar Ghosh Mathematical Reasoning Mathematical Reasoning: Writing and Proof Version 2.1 Author: Ted Sundstrom An Introduction to Mathematical Reasoning: Numbers, Sets and Functions Author: Peter J. Eccles Differential Equations Differential Equations (with DE Tools Printed Access Card) Author: Paul Blanchard - Robert L. Devaney - Glen R. Hall Calculus Calculus: Early Transcendentals Author: James Stewart And more....

SOCIAL AND EMOTIONAL FUNCTIONING Another coexisting regulatory problem may be how the child feels about himself and relates to other people. • Poor adaptability: The child may resist meeting new people, trying new games or toys or tasting different foods. He may have difficulty making transitions from one situation to another. The child may seem stubborn and uncooperative when it is time to leave the house, come for dinner, get into or out of the bathtub, or change from a reading to a math activity. Minor changes in routine will readily upset this child who does not “go with the flow.” • Attachment problem: The child may have separation anxiety and be clingy and fearful when apart from one or two “significant olders.” Or, she may physically avoid her parents, teachers, and others in her circle. • Frustration: Struggling to accomplish tasks that peers do easily, the child may give up quickly. He may be a perfectionist and become upset when art projects, dramatic play, or homework assignments are not going as well as he expects. • Difficulty with friendships: The child may be hard to get along with and have problems making and keeping friends. Insisting on dictating all the rules and being the winner, the best, or the first, he may be a poor game-player. He may need to control his surrounding territory, be in the “driver’s seat,” and have trouble sharing toys. • Poor communication: The child may have difficulty verbally in the way she articulates her speech, “gets the words out,” and writes. She may have difficulty expressing her thoughts, feelings, and needs, not only through words but also nonverbally through gestures, body language, and facial expressions. • Other emotional problems: He may be inflexible, irrational, and overly sensitive to change, stress, and hurt feelings. Demanding and needy, he may seek attention in negative ways. He may be angry or panicky for no obvious reason. He may be unhappy, believing and saying that he is dumb, crazy, no good, a loser, and a failure. Low self-esteem is one of the most telling symptoms of Sensory Processing Disorder. • Academic problems: The child may have difficulty learning new skills and concepts. Although bright, the child may be perceived as an underachiever.

Math is not a constructor, or even a function. It’s an object. As you know, Math is a built-in object that you can use to do things like get the value of pi (with Math.PI) or generate a random number (with Math.random). Think of Math as just like an object literal that has a bunch of useful properties and methods in it, built-in for you to use whenever you write JavaScript code. It just happens to have a capital first letter to let you know that it’s built-in to JavaScript.

Linda Gottfredson, “Clinton’s New Form of Race-Norming.” Wall Street Journal, June 3, 1993. The national Assessment of Educational progress. . . has documented large gaps on specific skills and knowledge among high-school students. Throughout the 1980s, black 17-year-olds (excluding dropouts) had proficiency levels in math, reading, science and other subjects that were more comparable to white 13-year-olds than white 17-year-olds. A 1987 NAEP report found similarly large gaps in the functional literacy of young adults age 21 to 25. The average black college graduate could comprehend and use everyday reading materials, such as news articles, menus, forms, labels, street maps and bus schedules, only about as well as the average white high-school graduate with no college. In turn, black high school graduates function, on the average, only about as well as whites with no more than eight years of schooling. The pervasiveness of such huge gaps in current skills and knowledge explains why employment tests typically have disparate impact, especially in mid-to-high level jobs.

Though energy fields are invisible, they shape matter. Albert Einstein said that, “The field is the sole governing agency of the particle.” Many studies show that human beings are influenced by the energy fields of others. In a series of 148 1-minute trials involving 25 people, trained volunteers going into heart coherence were able to induce coherence in test subjects at a distance. They didn’t have to touch their targets to produce the effect. Their energy fields were sufficient. When you are in a heart coherent state, your heart radiates a coherent electromagnetic signal into the environment around you. This field is detectable by a magnetometer several meters away. When other people enter that coherent energy field, their heart coherence increases too, producing a group field effect. Not only are we affected by the fields of other people; we’re affected by the energies of the planet and solar system. A remarkable series of experiments, conducted by a research team led by Rollin McCraty, director of research at the HeartMath Institute, has linked individual human energy to solar cycles. McCraty and his colleagues track solar activity using large magnetometers placed at strategic locations on the earth’s surface. Solar flares affect the electromagnetic fields of the planet. The researchers compare the ebbs and flows of solar energy with the heart coherence readings of trained volunteers. They have found that when people are in heart coherence, their electromagnetic patterns track those of the solar system. 8.15. The heart coherence rhythms of a volunteer compared to solar activity over the course of a month. A later study of 16 participants over 5 months found a similar effect. McCraty writes: “A growing body of evidence suggests that an energetic field is formed among individuals in groups through which communication among all the group members occurs simultaneously. In other words, there is an actual ‘group field’ that connects all the members” together. The results of this research confirm a hypothesis McCraty and I discussed at a conference when I was writing Mind to Matter: Not only are these heart-coherent people in sync with large-scale global cycles, they’re also in sync with each other. McCraty continues, “We’re all like little cells in the bigger Earth brain—sharing information at a subtle, unseen level that exists between all living systems, not just humans, but animals, trees, and so on.” When we use selective attention to tune ourselves to positive coherent energy, we participate in the group energy field of other human beings doing the same. We may also resonate in phase with coherent planetary and universal fields. 8.16. The brain functions as receiver of information from the field. The Brain’s Ability to Detect Fields The idea of invisible energy fields has always been difficult for many scientists to swallow. Around 1900, when Dutch physician Willem Einthoven proposed that the human heart had an energy field, he was ridiculed. He built progressively more sensitive galvanometers to detect it, and he was eventually successful.

FIELD EFFECTS Emotional contagion is just one explanation for the growth of meditation. Another is field effects. Everything begins as energy, then works its way into matter. Though energy fields are invisible, they shape matter. Albert Einstein said that, “The field is the sole governing agency of the particle.” Many studies show that human beings are influenced by the energy fields of others. In a series of 148 1-minute trials involving 25 people, trained volunteers going into heart coherence were able to induce coherence in test subjects at a distance. They didn’t have to touch their targets to produce the effect. Their energy fields were sufficient. When you are in a heart coherent state, your heart radiates a coherent electromagnetic signal into the environment around you. This field is detectable by a magnetometer several meters away. When other people enter that coherent energy field, their heart coherence increases too, producing a group field effect. Not only are we affected by the fields of other people; we’re affected by the energies of the planet and solar system. A remarkable series of experiments, conducted by a research team led by Rollin McCraty, director of research at the HeartMath Institute, has linked individual human energy to solar cycles. McCraty and his colleagues track solar activity using large magnetometers placed at strategic locations on the earth’s surface. Solar flares affect the electromagnetic fields of the planet. The researchers compare the ebbs and flows of solar energy with the heart coherence readings of trained volunteers. They have found that when people are in heart coherence, their electromagnetic patterns track those of the solar system. 8.15. The heart coherence rhythms of a volunteer compared to solar activity over the course of a month. A later study of 16 participants over 5 months found a similar effect. McCraty writes: “A growing body of evidence suggests that an energetic field is formed among individuals in groups through which communication among all the group members occurs simultaneously. In other words, there is an actual ‘group field’ that connects all the members” together. The results of this research confirm a hypothesis McCraty and I discussed at a conference when I was writing Mind to Matter: Not only are these heart-coherent people in sync with large-scale global cycles, they’re also in sync with each other. McCraty continues, “We’re all like little cells in the bigger Earth brain—sharing information at a subtle, unseen level that exists between all living systems, not just humans, but animals, trees, and so on.” When we use selective attention to tune ourselves to positive coherent energy, we participate in the group energy field of other human beings doing the same. We may also resonate in phase with coherent planetary and universal fields. 8.16. The brain functions as receiver of information from the field. The Brain’s Ability to Detect Fields The idea of invisible energy fields has always been difficult for many scientists to swallow. Around 1900, when Dutch physician Willem Einthoven proposed that the human heart had an energy field, he was ridiculed. He built progressively more sensitive galvanometers to detect it, and he was eventually successful.

We intuitively know that the heart is the center of love and empathy, and studies are showing this to be true. In fact, empathy manifests in the electromagnetic field (EMF), which is generated by the heart in amounts greater than anywhere else in the body. The heart’s EMF emits fifty thousand femtoteslas (a measure of EMF), in contrast to the ten generated by the brain.37 Other research shows that when separated from the magnetic field, the heart’s electrical field is sixty times greater in amplitude than the brain’s field.38 Through this field, a person’s nervous system tunes in to and responds to the magnetic fields produced by the hearts of other people.39 The heart’s field is therefore one of the means by which a practitioner affects patients. This effect leads to the question, What do you want to share? To generate positive outcomes for a patient, a practitioner must hold positive feelings in his or her own heart. Not only does good will profit the client, but it also benefits the practitioner as a person. A set of studies by researcher Dr. Rollin McCraty of the HeartMath Institute in California, and described in his e-book, The Energetic Heart, helps explain the importance of positive energy.40 For decades, scientists have known that information is encoded in the nervous system in the time intervals between activities or in the pattern of electrical activity. Recent studies also reveal that information is captured in hormone pulses. Moreover, there is a hormone pulse that coincides with heart rhythms, which means that information is also shared in the interbeat intervals of the pressure and electromagnetic waves produced by the heart. Negative emotions such as anger, frustration, or anxiety disturb the heart rhythm. Positive emotions such as appreciation, love, or compassion produce coherent or functional patterns. Feelings, distributed throughout the body, produce chemical changes within the entire system. Do you want to be a healthy person? Be sincerely positive as often as you can. You thus “increase the probability of maintaining coherence and reducing stress, even during challenging situations.”41 What you as a practitioner believe will be shared—everywhere and with everyone you meet.

My junior year, I tested into the honors Advanced Math class—a hybrid of trigonometry, advanced algebra, and pre-calculus. The class’s instructor, Ron Selby, enjoyed legendary status among the students for his brilliance and high demands. In twenty years, he had never missed a day of school. According to Middletown High School legend, a student called in a bomb threat during one of Selby’s exams, hiding the explosive device in a bag in his locker. With the entire school evacuated outside, Selby marched into the school, retrieved the contents of the kid’s locker, marched outside, and threw those contents into a trash can. “I’ve had that kid in class; he’s not smart enough to make a functioning bomb,” Selby told the police officers gathered at the school. “Now let my students go back to class to finish their exams.

water waves are actually trochoidal in shape,

what practical uses are there for functions? We can use them to establish a connection between the value of one physical quantity and another. For example, through experiments or theoretical considerations we can determine a function f(p) that links the air pressure p to the air density D. It would allow us to insert any value for the air pressure p and calculate the corresponding value for the air density D, which can be quite useful.

roots of functions

Exponential functions appear in many real-world situations: population growth, technological growth, product value over time, compounded interest, radioactive decay, statistical analysis, ...

the curve will never touch the x-axis (in other words: it has no roots),

Just like how Python comes with several modules like random, math, or time that provide additional functions for your programs, the Pygame framework includes several modules with functions for drawing graphics, playing sounds, handling mouse input, and other things.

... 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.

test3 defines the sin function as a keyword argument, with its default value being a reference to the sin function within the math module. While we still do need to find a reference to this function within the module, this is only necessary when the test3 function is first defined. After this, the reference to the sin function is stored within the function definition as a local variable in the form of a default keyword argument. As mentioned previously, local variables do not need a dictionary lookup to be found; they are stored in a very slim array that has very fast lookup times. Because of this, finding the function is quite fast! While these effects are an interesting result of the way namespaces in Python are managed, test3 is definitely not “Pythonic.” Luckily, these extra dictionary lookups only start to degrade performance when they are called a lot (i.e., in the innermost block of a very fast loop, such as in the Julia set example). With this in mind, a more readable solution would be to set a local variable with the global reference before the loop is started. We’ll still have to do the global lookup once whenever the function is called, but all the calls to that function in the loop will be made faster. This speaks to the fact that even minute slowdowns in code can be amplified if that code is being run millions of times. Even though a dictionary lookup may only take several hundred nanoseconds, if we are looping millions of times over this lookup it can quickly add up. In fact, looking at Example 4-10 we see a 9.4% speedup simply by making the sin function local to the tight loop that calls it.

Calculate the factorial perl -MMath::BigInt -le 'print Math::BigInt->new(5)->bfac()' This one-liner uses the bfac() function from the Math::BigInt module in the Perl core. (In other words, you don’t need to install it.) The Math::BigInt->new(5) construction creates a new Math::BigInt object with a value of 5, after which the bfac() method is called on the newly created object to calculate the factorial of 5.

Take a look at the following list of numbers: 4, 8, 5, 3, 9, 7, 6. Read them out loud. Now look away and spend twenty seconds memorizing that sequence before saying them out loud again. If you speak English, you have about a 50 percent chance of remembering that sequence perfectly. If you're Chinese, though, you're almost certain to get it right every time. Why is that? Because as human beings we store digits in a memory loop that runs for about two seconds. We most easily memorize whatever we can say or read within that two-second span. And Chinese speakers get that list of numbers—4, 8, 5, 3, 9, 7, 6—right almost every time because, unlike English, their language allows them to fit all those seven numbers into two seconds. That example comes from Stanislas Dehaene's book The Number Sense. As Dehaene explains: Chinese number words are remarkably brief. Most of them can be uttered in less than one-quarter of a second (for instance, 4 is "si" and 7 "qi"). Their English equivalents—"four," "seven"—are longer: pronouncing them takes about one-third of a second. The memory gap between English and Chinese apparently is entirely due to this difference in length. In languages as diverse as Welsh, Arabic, Chinese, English and Hebrew, there is a reproducible correlation between the time required to pronounce numbers in a given language and the memory span of its speakers. In this domain, the prize for efficacy goes to the Cantonese dialect of Chinese, whose brevity grants residents of Hong Kong a rocketing memory span of about 10 digits. It turns out that there is also a big difference in how number-naming systems in Western and Asian languages are constructed. In English, we say fourteen, sixteen, seventeen, eighteen, and nineteen, so one might expect that we would also say oneteen, twoteen, threeteen, and five- teen. But we don't. We use a different form: eleven, twelve, thirteen, and fifteen. Similarly, we have forty and sixty, which sound like the words they are related to (four and six). But we also say fifty and thirty and twenty, which sort of sound like five and three and two, but not really. And, for that matter, for numbers above twenty, we put the "decade" first and the unit number second (twentyone, twenty-two), whereas for the teens, we do it the other way around (fourteen, seventeen, eighteen). The number system in English is highly irregular. Not so in China, Japan, and Korea. They have a logical counting system. Eleven is ten-one. Twelve is ten-two. Twenty-four is two- tens-four and so on. That difference means that Asian children learn to count much faster than American children. Four-year-old Chinese children can count, on average, to forty. American children at that age can count only to fifteen, and most don't reach forty until they're five. By the age of five, in other words, American children are already a year behind their Asian counterparts in the most fundamental of math skills. The regularity of their number system also means that Asian children can perform basic functions, such as addition, far more easily. Ask an English-speaking seven-yearold to add thirty-seven plus twenty-two in her head, and she has to convert the words to numbers (37+22). Only then can she do the math: 2 plus 7 is 9 and 30 and 20 is 50, which makes 59. Ask an Asian child to add three-tensseven and two-tens-two, and then the necessary equation is right there, embedded in the sentence. No number translation is necessary: It's five-tens-nine. "The Asian system is transparent," says Karen Fuson, a Northwestern University psychologist who has closely studied Asian-Western differences. "I think that it makes the whole attitude toward math different. Instead of being a rote learning thing, there's a pattern I can figure out. There is an expectation that I can do this. There is an expectation that it's sensible. For fractions, we say three-fifths. The Chinese is literally 'out of five parts, take three.' That's telling you conceptually

I am not concerned about proving anything to any other discipline outside the scope of math itself where my function lies nowadays; and when it comes to my own statements that I have built on top of numbers, they are restricted to my own context, hence, the quoting cladding.

For example, if healthy 30-year-olds are sleep deprived for six days (averaging, in this study, about four hours of sleep per night), parts of their body chemistry soon revert to that of a 60-year-old. And if they are allowed to recover, it will take them almost a week to get back to their 30-year-old systems. Taken together, these studies show that sleep loss cripples thinking in just about every way you can measure thinking. Sleep loss hurts attention, executive function, working memory, mood, quantitative skills, logical reasoning ability, general math knowledge. Eventually, sleep loss affects manual dexterity, including fine motor control, and even gross motor movements, such as the ability to walk on a treadmill.

All of this told of harm done, of a drug that made a child depressed, lonely, and filled with a sense of inadequacy, and when researchers looked at whether Ritalin at least helped hyperactive children fare well academically, to get good grades and thus succeed as students, they found that it wasn’t so. Being able to focus intently on a math test, it turned out, didn’t translate into long-term academic achievement. This drug, Sroufe explained in 1973, enhances performance on “repetitive, routinized tasks that require sustained attention,” but “reasoning, problem solving and learning do not seem to be [positively] affected.”26 Five years later, Herbert Rie was much more negative. He reported that Ritalin did not produce any benefit on the students’ “vocabulary, reading, spelling, or math,” and hindered their ability to solve problems. “The reactions of the children strongly suggest a reduction in commitment of the sort that would seem critical for learning.”27 That same year, Russell Barkley at the Medical College of Wisconsin reviewed the relevant scientific literature and concluded “the major effect of stimulants appears to be an improvement in classroom manageability rather than academic performance.”28 Next it was James Swanson’s turn to weigh in. The fact that the drugs often left children “isolated, withdrawn and overfocused” could “impair rather than improve learning,” he said.29 Carol Whalen, a psychologist from the University of California at Irvine, noted in 1997 that “especially worrisome has been the suggestion that the unsalutary effects [of Ritalin] occur in the realm of complex, high-order cognitive functions such as flexible problem-solving or divergent thinking.”30 Finally, in 2002, Canadian investigators conducted a meta-analysis of the literature, reviewing fourteen studies involving 1,379 youths that had lasted at least three months, and they determined that there was “little evidence for improved academic performance.”31

So, which is it? Does matter create mind? Or does mind create matter? Scientific materialists have never made any progress whatsoever in explaining how matter produces mind. Mind can easily explain the production of matter – via autonomous Fourier immaterial frequency functions finding collective expression in a Fourier material spacetime world. It’s the collective rather than individual nature of this mental activity that makes it seem “physical”, and which gives rise to the delusion of the existence of matter. It’s all in the math.

It can be useful for a function to accept any number of arguments. For example, Math.max computes the maximum of all the arguments it is given. To write such a function, you put three dots before the function’s last parameter, like this: function max(... numbers)

As we’ve seen, Math is a grab bag of number-related utility functions, such as Math.max (maximum), Math.min (minimum), and Math.sqrt (square root).

the mathematically curious reader, please search for a video on “greens function PDE” and you’ll see a sample of the level of difficulty in electrical engineering math at the graduate level.668

Research on the impact of mindfulness on children is still in the early stages, but studies have shown that in the school years these practices can lower levels of stress, aggression, and social anxiety, improve executive functions such as inhibition and working memory, and contribute to stronger performance in math.

Because I am a math guy, I like to visualize lifespan and healthspan in terms of a mathematical function, as in figure 2 on the following page—one of many graphs that I draw for my patients. The horizontal or x-axis of the graph represents your lifespan, how long you will live, while the vertical or y-axis represents a kind of sum total of your physical and cognitive function, the two age-dependent dimensions of healthspan. (Obviously, healthspan is not really quantifiable, but bear with my oversimplification.)

I never realised power of x for a function f(x) when I was in school.

I never realised power of x of function f(x) when I was in school.

Nevertheless, here’s the point for even a nontechnical reader: the Bitcoin blockchain gives a history that’s hard to falsify. Unless there’s an advance in quantum computing, a breakthrough in pure math, a heretofore unseen bug in the code, or a highly expensive 51% attack that probably only China could muster, it is essentially infeasible to rewrite the history of the Bitcoin blockchain — or anything written to it. And even if such an event does happen, it wouldn’t be an instantaneous burning of Bitcoin’s Library of Alexandria. The hash function could be replaced with a quantum-safe version, or another chain robust to said attack could take Bitcoin’s place, and back up the ledger of all historical Bitcoin transactions to a new protocol.

At the moment of measuring one electron, a collapse of the wave function of both electrons occurs regardless of the distance between them.

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.

It might surprise you to know that, for the most part, finance involves addition and subtraction. When finance people get really fancy, they multiply and divide. We never have to take the second derivative of a function or determine the area under a curve (sorry, engineers). So have no fear: the math is easy. And calculators are cheap. You don't need to be a rocket scientist to be financially intelligent.

if you give the learner enough of the appropriate data, it can approximate any function arbitrarily closely—which is math-speak for learning anything.

techniques that had been developed by the Institute of HeartMath. These positive emotion-focused techniques help individuals learn to self-generate and sustain a beneficial functional mode known as psychophysiological coherence, characterized by increased emotional stability and by increased synchronization and harmony in the functioning of physiological systems.

This is the essence of codebreaking, finding patterns, and because it’s such a basic human function, codebreakers have always emerged from unexpected places. They pop up from strange corners. Codebreakers tend to be oddballs, outsiders. The most important trait is not pure math skill but a deeper ability to pay attention.