there is so much existing content out there, takes time to digest, I don't have much for my own opinions - I spend time and money learning what other people share. I read in iBooks, save to the app Bear, keep as a journal for revisiting. Posting some cool shit here from Steve Strogatz's book "The Joy of X" - beauty of math, from a visual perspective, less memorizing formulas. The notes are sporadic, my highlights, not intended as a book summary. Maybe just enough to make you want to play with #'s visually, or that math was taught differently.
NOTES FROM The Joy of x by Steven Strogatz
“Relationships,” generalizes from working with numbers to working with relationships between numbers. These are the ideas at the heart of algebra. What makes them so crucial is that they provide the first tools for describing how one thing affects another, through cause and effect, supply and demand, dose and response, and so on—the kinds of relationships that make the world complicated and rich
Part 3, “Shapes,” turns from numbers and symbols to shapes and space—the province of geometry and trigonometry. Along with characterizing all things visual, these subjects raise math to new levels of rigor through logic and proof.
In part 4, “Change,” we come to calculus, the most penetrating and fruitful branch of math. Calculus made it possible to predict the motions of the planets, the rhythm of the tides, and virtually every other form of continuous change in the universe and ourselves. A supporting theme in this part is the role of infinity. The domestication of infinity was the breakthrough that made calculus work. By harnessing the awesome power of the infinite, calculus could finally solve many long-standing problems that had defied the ancients, and that ultimately led to the scientific revolution and the modern world.
any journey needs to begin at the beginning, so let’s start with the simple, magical act of counting
This dual aspect of numbers—as part heaven, part earth —is perhaps their most paradoxical feature, and the feature that makes them so useful. It is what the physicist Eugene Wigner had in mind when he wrote of “the unreasonable effectiveness of mathematics in the natural sciences
Just as numbers are a shortcut for counting by ones, addition is a shortcut for counting by any amount. This is how mathematics grows. The right abstraction leads to new insight, and new power.
Yet despite this infinite vista, there are always constraints on our creativity. We can decide what we mean by things like 6 and +, but once we do, the results of expressions like 6 + 6 are beyond our control. Logic leaves us no choice. In that sense, math always involves both invention and discovery: we invent the concepts but discover their consequences. As we’ll see in the coming chapters, in mathematics our freedom lies in the questions we ask—and in how we pursue them—but not in the answers awaiting us.
Subtraction forces us to expand our conception of what numbers are. Negative numbers are a lot more abstract than positive numbers—you can’t see negative 4 cookies and you certainly can’t eat them—but you can think
Perhaps the most unsettling thing is that a negative times a negative is a positive. So let me try to explain the thinking behind that.
ut it becomes more intuitive if you conceive of multiplication visually. Think of 7 × 3 as the number of dots in a rectangular array with seven rows and three columns
Maybe we’re wired to doubt the commutative law because in daily life, it usually matters what you do first. You can’t have your cake and eat it too. And when taking off your shoes and socks, you’ve got to get the sequencing right.
Gell-Mann had probably been sensitized to the importance of non-commutativity. As a quantum physicist he would have been acutely aware that at the deepest level, nature disobeys the commutative law. And it’s a good thing, too. For the failure of commutativity is what makes the world the way it is. It’s why matter is solid, and why atoms don’t implode.
Specifically, early in the development of quantum mechanics, Werner Heisenberg and Paul Dirac had discovered that nature follows a curious kind of logic in which p × q ≠ q × p, where p and q represent the momentum and position of a quantum particle. Without that breakdown of the commutative law, there would be no Heisenberg uncertainty principle, atoms would collapse, and nothing would exist.
Like Christy’s father and the Verizon service reps, I couldn’t accept something that had just been proven to me. I saw it but refused to believe it. (This might remind you of some people you know.)
Given how contrived this decimal is, you might suppose irrationality is rare. On the contrary, it is typical. In a certain sense that can be made precise,
Once you accept these astonishing facts, everything turns topsy-turvy. Whole numbers and fractions, so beloved and familiar, now appear scarce and exotic. And that innocuous number line pinned to the molding of your grade-school classroom? No one ever told you, but it’s chaos up there.
Finding a good way to represent numbers has always been a challenge. Since the dawn of civilization, people have tried various systems for writing numbers and reckoning with them, whether for trading, measuring land, or keeping track of the herd. What nearly all these systems have in common is that our biology is deeply embedded in them. Through the vagaries of evolution, we happen to have five fingers on each of two hands
The Babylonians were not nearly as attached to their fingers. Their numeral system was based on 60—a clear sign of their impeccable taste, for 60 is an exceptionally pleasant number. Its beauty is intrinsic and has nothing to do with human appendages.
eometry is good for the mind; it trains you to think clearly and logically. It’s not the study of triangles, circles, and parallel lines per se that matters. What’s important is the axiomatic method, the process of building a rigorous argument, step by step, until a desired conclusion has been established
Jefferson wrote, “We hold these truths to be self-evident,” he was mimicking the style of the Elements. Euclid had begun with the definitions, postulates, and self-evident truths of geometry—the axioms—and from them erected an edifice of propositions and demonstrations, one truth linked to the next by unassailable logic. Jefferson organized the Declaration in the same way so that its radical conclusion, that the colonies had the right to govern themselves, would seem as inevitable as a fact of geometry
at intellectual legacy seems far-fetched, keep in mind that Jefferson revered Euclid. A few years after he finished his second term as president and stepped out of public life, he wrote to his old friend John Adams on January 12, 1812, about the pleasures of leaving politics behind: “I have given up newspapers in exchange for Tacitus and Thucydides, for Newton and Euclid; and I find myself much the happier.”
arabolas and ellipses: Why is it that they, and only they, have such fantastic powers of focusing? What’s the secret they share? They’re both cross-sections of the surface of a cone.
nce you begin to appreciate the focusing abilities of parabolas and ellipses, you can’t help but wonder if something deeper is at work here. Are these curves related in some more fundamental way
The reason for calling F the focus becomes clear if we think of the parabola as a curved mirror
So that’s the secret: a parabola is an ellipse in disguise, in a certain limiting sense. No wonder it shares the ellipse’s marvelous focusing ability. It’s been passed down through the bloodline.
’s one additional sibling: if the cone is sliced very steeply, on a bias greater than its own slope, the resulting incision forms a curve called a hyperbola. Unlike all the others, it comes in two pieces, not one
When you speak and I hear, both our bodies are using sine waves—yours in the vibrations of your vocal cords to produce the sounds, and mine in the swaying of the hair cells in my ears to receive them. If we open our hearts to these sine waves and tune in to their silent thrumming, they have the power to move us. There’s something almost spiritual about them. When a guitar string is plucked or when children jiggle a jump rope, the shape that appears is a sine wave
Sine waves are the atoms of structure. They’re nature’s building blocks. Without them there’d be nothing, giving new meaning to the phrase “sine qua non.” In fact, the words are literally true. Quantum mechanics describes real atoms, and hence all of matter, as packets of sine waves. Even at the cosmological scale, sine waves form the seeds of all that exists
Yet trigonometry, belying its much too modest name, now goes far beyond the measurement of triangles. By quantifying circles as well, it has paved the way for the analysis of anything that repeats, from ocean waves to brain waves. It’s the key to the mathematics of cycles.
This shape is a sine wave. It arises whenever one tracks the horizontal or vertical excursions of something—or someone—moving in a circle
More precisely, sin a is defined as the girl’s altitude measured from the center of the wheel when she’s located at the angle a.
d then define “sin a” as the girl’s height above or below the center of the circle. The corresponding graph of sin a, as a keeps increasing (or even goes negative, if the wheel reverses direction), is what we mean by a sine wave. It repeats itself every time a changes by 360 degrees, corresponding to a full revolution.
The ripples on a pond, the ridges of sand dunes, the stripes of a zebra—all are manifestations of nature’s most basic mechanism of pattern formation: the emergence of sinusoidal structure from a background of bland uniformity.
The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces
As we take more and more slices, something marvelous happens: the scalloped shape approaches a rectangle. The arcs become flatter and the sides become almost vertical
But now the problem is easy. The area of a rectangle equals its width times its height, so multiplying πr times r yields an area of πr² for the rectangle
lenses. The eerie point is that light behaves as if it were considering all possible paths and then taking the best one. Nature—cue the theme from The Twilight Zone—somehow knows calculus
Calculus is the mathematics of change.
ing through. All the rest, as Rabbi Hillel said of the golden rule, is just commentary. Those two ideas are the derivative and the integral. Each dominates its own half of the subject, named, respectively, differential and integral calculus
Roughly speaking, the derivative tells you how fast something is changing; the integral tells you how much it’s accumulating
Change is most sluggish at the extremes precisely because the derivative is zero there. Things stand still, momentarily
e, the lasting legacy of integral calculus is a Veg-O-Matic view of the universe. Newton and his successors discovered that nature itself unfolds in slices
Together with the governing laws, the conditions in each slice of time or space determine what will happen in adjacent slices.
Close behind is i, the it-number of algebra, the imaginary number so radical it changed what it meant to be a number
Say hello to e. Nicknamed for its breakout role in exponential growth, e is now the Zelig of advanced mathematics. It pops up everywhere, peeking out from the corners of the stage, teasing us by its presence in incongruous places. For example, along with the insights it offers about chain reactions and population booms, e
To reach this conclusion, I’ve assumed that Romeo’s behavior can be modeled by the differential equation which describes how his love (represented by R) changes in the next instant (represented by dt). According to this equation, the amount of change (dR) is just a multiple (a) of Juliet’s current love (J) for him. This reflects what we already know—that Romeo’s love goes up when Juliet loves him—but it assumes something much stronger. It says that Romeo’s love increases in direct linear proportion to how much Juliet loves him. This assumption of linearity is not emotionally realistic, but it makes the equations much easier to solve.
Juliet’s behavior, by contrast, can be modeled by the equation The negative sign in front of the constant b reflects her tendency to cool off when Romeo is hot for her. The only remaining thing we need to know is how the lovers felt about each other initially (R and J at time t = 0). Then everything about their affair is predetermined. We can use a computer to inch R and J forward, instant by instant, changing their values as prescribed by the differential equations. Actually, with the help of the fundamental theorem of calculus, we can do much better than that. Because the model is so simple, we don’t have to trudge forward one moment at a time. Calculus yields a pair of comprehensive formulas that tell us how much Romeo and Juliet will love (or hate) each other at any future time.
While the laws of the heart may elude us forever, the laws of inanimate things are now well understood. They take the form of differential equations, which describe how interlinked variables change from moment to moment, depending on their current values.
As for what such equations have to do with romance—well, at the very least they might shed a little light on why, in the words of another poet, “the course of true love never did run smooth.
To illustrate the approach, suppose Romeo is in love with Juliet but that, in our version of the story, Juliet is a fickle lover. The more Romeo loves her, the more she wants to run away and hide. But when he takes the hint and backs off, she begins to find him strangely attractive. He, however, tends to mirror her: he warms up when she loves him and cools down when she hates him.
What happens to our star-crossed lovers? How does their love ebb and flow over time? That’s where calculus comes in. By writing equations that summarize how Romeo and Juliet respond to each other’s affections and then solving those equations with calculus, we can predict the course of their affair. The resulting forecast for this couple is, tragically, a never-ending cycle of love and hate. At least they manage to achieve simultaneous love a quarter of the time.
The differential equations above should be recognizable to students of physics: Romeo and Juliet behave like simple harmonic oscillators. So the model predicts that R(t) and J(t)—the functions that describe the time course of their relationship—will be sine waves, each waxing and waning but peaking at different times.
The model can be made more realistic in various ways. For instance, Romeo might react to his own feelings as well as to Juliet’s. He might be the type of guy who is so worried about throwing himself at her that he slows himself down as his love for her grows. Or he might be the other type, one who loves feeling in love so much that he loves her all the more for it.
Add to those possibilities the two ways Romeo could react to Juliet’s affections—either increasing or decreasing his own—and you see that there are four personality types, each corresponding to a different romantic style. My students and those in Peter Christopher’s class at Worcester Polytechnic Institute have suggested such descriptive names as Hermit and Malevolent Misanthrope for the particular kind of Romeo who damps down his own love and also recoils from Juliet’s. Whereas the sort of Romeo who gets pumped by his own ardor but turned off by Juliet’s has been called Narcissistic Nerd, Better Latent Than Never, and a Flirting Fink. (Feel free to come up with your own names for these two types and the other two possibilities.)
n all cases, the business of theoretical physics boils down to finding the right differential equations and solving them. When Newton discovered this key to the secrets of the universe, he felt it was so precious that he published it only as an anagram in Latin. Loosely translated, it reads: “It is useful to solve differential equations.”
In mathematics we call this a three-body problem. It’s notoriously intractable, especially in the astronomical context where it first arose. After Newton solved the differential equations for the two-body problem (thus explaining why the planets move in elliptical orbits around the sun),
The other kind of derivative measures the curl of a vector field. Roughly speaking, it indicates how strongly the field is swirling about a given point.
The curl is extremely informative for scientists working in fluid mechanics and aerodynamics. A few years ago my colleague Jane Wang used a computer to simulate the pattern of airflow around a dragonfly as it hovered in place. By calculating the curl, she found that when a dragonfly flaps its wings, it creates pairs of counter-rotating vortices that act like little tornadoes beneath its wings, producing enough lift to keep the insect aloft. In this way, vector calculus is helping to explain how dragonflies, bumblebees, and hummingbirds can fly—something that had long been a mystery to conventional fixed-wing aerodynamics. With the notions of divergence and curl in hand, we’re now ready to revisit Maxwell’s equations. They express four fundamental laws: one for the divergence of the electric field, another for its curl, and two more of the same type but now for the magnetic field. The divergence equations relate the electric and magnetic fields to their sources, the charged particles and currents that produce them in the first place. The curl equations describe how the electric and magnetic fields interact and change over time. In so doing, these equations express a beautiful symmetry: they link one[…]
That was the first breakthrough—the theoretical prediction of electromagnetic waves. But the real stunner came next. When Maxwell calculated the speed of these hypothetical waves, using known properties of electricity and magnetism, his equations told him that they traveled at about 193,000 miles per second—the same rate as the speed of light measured by the French physicist Hippolyte Fizeau a decade earlier! How I wish I could have witnessed the moment when a human being first understood the true nature of light. By identifying it with an electromagnetic wave, Maxwell unified three ancient and seemingly unrelated phenomena: electricity, magnetism, and light. Although experimenters like Faraday and Ampère had previously found key pieces of this puzzle, it was only Maxwell, armed with his mathematics, who put them all together. Today we are awash in Maxwell’s once-hypothetical waves: Radio. Television. Cell phones. Wi-Fi. These are the legacy of his conjuring with symbols.
vector calculus, the branch of math that describes the invisible fields all around us
The greatest achievements of vector calculus lie in that twilight realm where math meets reality. Indeed, the story of James Clerk Maxwell and his equations offers one of the eeriest instances of the unreasonable effectiveness of mathematics. Somehow, by shuffling a few symbols, Maxwell discovered what light is.
what vector calculus is about, let’s begin with the word “vector.” It comes from the Latin root vehere, “to carry,” which also gives us words like “vehicle” and “conveyor belt.”
To a mathematician, a vector (at least in its simplest form) is a step that carries you from one place to another.
These arrows are vectors. They show two kinds of information: a direction (which way to move that foot) and a magnitude (how far to move it). All vectors do that same double duty
Vectors can be added and subtracted, just like numbers, except their directionality makes things a little trickier. Still, the right way to add vectors becomes clear if you think of them as dance instructions. For example, what do you get when you take one step east followed by one step north? A vector that points northeast, naturally.
The ball’s velocity relative to the court is the sum of two vectors: the ball’s velocity relative to you (a vector pointing down the line, as intended), and your velocity relative to the court (a vector pointing sideways, since that’s the direction you’re running).
Beyond such vector algebra lies vector calculus, the kind of math Mr. DiCurcio was using. Calculus, you’ll recall, is the mathematics of change. And so whatever vector calculus is, it must involve vectors that change, either from moment to moment or from place to place. In the latter case, one speaks of a “vector field.”
The direction and magnitude of the vectors in a magnetic field vary from point to point. As in all of calculus, the key tool for quantifying such changes is the derivative. In vector calculus the derivative operator goes by the name of del, which has a folksy southern ring to it, though it actually alludes to the Greek letter ∆ (delta), commonly used to denote a change in some variable.
As a reminder of that kinship, “del” is written like this: ∇. (That was the mysterious upside-down triangle Mr. DiCurcio kept writing on the napkin.)
It turns out there are two different but equally natural ways to take the derivative of a vector field by applying del to it. The first gives what’s known as the field’s divergence (the “div” that Mr. DiCurcio muttered).
That’s what the divergence of a vector field measures: how fast the area of a small circle of corks grows.
By embellishing the vector field with shading, we can now show where the curl is most positive (lightest regions) and most negative (darkest regions). Notice that this also tells us whether the flow is spinning counterclockwise or clockwise.
Using mathematical maneuvers equivalent to vector calculus—which wasn’t known in his day—Maxwell then extracted the logical consequences of those four equations. His symbol shuffling led him to the conclusion that electric and magnetic fields could propagate as a wave, somewhat like a ripple on a pond, except that these two fields were more like symbiotic organisms. Each sustained the other. The electric field’s undulations re-created the magnetic field, which in turn re-created the electric field, and so on, with each pulling the other forward, something neither could do on its own.
Yes, and even handier to know what a distribution is. That’s the first idea I’d like to focus on here, because it embodies one of the central lessons of statistics —things that seem hopelessly random and unpredictable when viewed in isolation often turn out to be lawful and predictable when viewed in aggregate.
The demo begins when hundreds of balls are poured into the top of the Galton board. As they rain down, they randomly bounce off the pegs, sometimes to the left, sometimes to the right, and ultimately distribute themselves in the evenly spaced bins at the bottom. The height of the stacked balls in each bin shows how likely it was for a ball to land there. Most balls end up somewhere near the middle, with slightly fewer balls flanking them on either side, and fewer still far off in the tails at either end. Overall, the pattern is utterly predictable: it always forms a bell-shaped distribution—even though it’s impossible to predict where any given ball will end up.
follow a beautifully simple distribution . . . as long as you look at them through logarithmic lenses.
In other words, suppose we regard the size differential between a pair of cities to be the same if their populations differ by the same factor, rather than by the same absolute number of people (much like two pitches an octave apart always differ by a constant factor of double the frequency).
Perhaps the most pulse-quickening topic of all is conditional probability—the probability that some event A happens, given (or conditional upon) the occurrence of some other event B. It’s a slippery concept, easily conflated with the probability of B given A. They’re not the same, but you have to concentrate to see why. For example, consider the following word problem.
The trick is to think in terms of natural frequencies—simple counts of events—rather than in more abstract notions of percentages, odds, or probabilities. As soon as you make this mental shift, the fog lifts.
Gigerenzer explores how people miscalculate risk and uncertainty. But rather than scold or bemoan human frailty, he tells us how to do better—how to avoid clouded thinking by recasting conditional-probability problems in terms of natural frequencies, much as my students did.
would tell the woman. After mulling the numbers over, he finally estimated the woman’s probability of having breast cancer, given that she has a positive mammogram, to be 90 percent. Nervously, he added, “Oh, what nonsense. I can’t do this. You should test my daughter; she is studying medicine.” He knew that his estimate was wrong, but he did not know how to reason better.
And what about 1? Is it prime? No, it isn’t, and when you understand why it isn’t, you’ll begin to appreciate why 1 truly is the loneliest number—even lonelier than the primes. It doesn’t deserve to be left out. Given that 1 is divisible only by 1 and itself, it really should be considered prime, and for many years it was. But modern mathematicians have decided to exclude it, solely for convenience. If 1 were allowed in, it would mess up a theorem that we’d like to be true. In other words, we’ve rigged the definition of prime numbers to give us the theorem we want.
among prime numbers, there are some that are even more special. Mathematicians call them twin primes: pairs of prime numbers that are close to each other, almost neighbors, but between them there is always an even number that prevents them from truly touching. Numbers like 11 and 13, like 17 and 19, 41 and 43. If you have the patience to go on counting, you discover that these pairs gradually become rarer.
Here I’d like to explore some of the beautiful ideas in the passage above, particularly as they relate to the solitude of prime numbers and twin primes. These issues are central to number theory, the subject that concerns itself with the study of whole numbers and their properties and that is often described as the purest part of mathematics.
Those algorithms rely on the difficulty of decomposing an enormous number into its prime factors.
But that’s not why mathematicians are obsessed with prime numbers. The real reason is that they’re fundamental. They’re the atoms of arithmetic. Just as the Greek origin of the word “atom” suggests, the primes are “a-tomic,” meaning “uncuttable, indivisible.” And just as everything is composed of atoms, every number is composed of primes.
Now that 1 has been thrown under the bus, let’s look at everyone else, the full-fledged prime numbers. The main thing to know about them is how mysterious they are, how alien and inscrutable. No one has ever found an exact formula for the primes. Unlike real atoms, they don’t follow any simple pattern, nothing akin to the periodic table of the elements.
To further underscore how disorderly the primes are, compare them to their straight-arrow cousins the odd numbers: 1, 3, 5, 7, 9, 11, 13, . . . The gaps between odd numbers are always consistent: two spaces, steady as a drumbeat. So they obey a simple formula: the nth odd number is 2n – 1. The primes, by contrast, m
Normally we think of symmetry as a property of a shape. But group theorists focus more on what you can do to a shape—specifically, all the ways you can change it while keeping something else about it the same. More precisely, they look for all the transformations that leave a shape unchanged, given certain constraints. These transformations are called the symmetries of the shape. Taken together, they form a group, a collection of transformations whose relationships define the shape’s most basic architecture. In the case of a mattress, the transformations alter its orientation in space (that’s what changes) while maintaining its rigidity (that’s the constraint). And after the transformation is complete, the mattress has to fit snugly on the rectangular bed frame (that’s what stays the same). With these rules in place, let’s see what transformations qualify for membership in this exclusive little group. It turns out there are only four of them. The first is the do-nothing transformation, a lazy but popular choice that leaves the mattress untouched. It certainly satisfies all the rules, but it’s not much help in prolonging the life of your mattress. Still, it’s very important to include in the group. It[…]
As these examples suggest, group theory bridges the arts and sciences. It addresses something the two cultures share—an abiding fascination with symmetry. Yet because it encompasses such a wide range of phenomena, group theory is necessarily abstract. It distills symmetry to its essence.
That’s one of the charms of group theory. It exposes the hidden unity of things that would otherwise seem unrelated
appen if they took their scissors and cut neatly down the midline all the way along the length of the strip? It will fall apart! It will make two pieces! they guessed. But after they tried it and something incredible happened (the strip remained in one piece but grew twice as long), there were even more squeals of surprise and delight. It was like a magic trick.
Topology shines a spotlight on a shape’s deepest properties—the properties that remain unchanged after a continuous distortion.
After one circuit, many of the students stopped and looked puzzled. Then they began shouting to one another excitedly, because their lines had not closed as they’d expected. The crayon had not come back to the starting point; it was now on the “other” side of the surface. That was surprise number one: you have to go twice around a Möbius strip to get back to where you started.
Then the shortest path between them is clearly a line segment, as shown above, and our elastic string would find that solution. So what’s new here? The cylindrical shape of the can opens up new possibilities for all kinds of contortions. Suppose we require that the string encircles the cylinder once before connecting to the second point. (Constraints like this are imposed on DNA when it wraps around certain proteins in chromosomes.) Now when the string pulls itself taut, it forms a helix, like the curves on old barbershop poles.
Here, an unmanned motorcycle rides along a geodesic highway on a two-holed torus, following the lay of the land. The remarkable thing is that its handlebars are locked straight ahead; it doesn’t need to steer to stay on the road. This underscores the earlier impression that geodesics, like great circles, are the natural generalization of straight lines. With all these flights of fancy, you may be wondering if geodesics have anything to do with reality. Of course they do. Einstein showed that light beams follow geodesics as they sail through the universe. The famous bending of starlight around the sun, detected in the eclipse observations of 1919, confirmed that light travels on geodesics through curved space-time, with the warping being caused by the sun’s gravity.
Sometimes when people say the shortest distance between two points is a straight line, they mean it figuratively, as a way of ridiculing nuance and affirming common sense. In other words, keep it simple. But battling obstacles can give rise to great beauty—so much so that in art, and in math, it’s often more fruitful to impose constraints on ourselves. Think of haiku, or sonnets, or telling the story of your life in six words. The same is true of all the math that’s been created to help you find the shortest way from here to there when you can’t take the easy way out. Two points. Many paths. Mathematical bliss
Neither argument seems more convincing than the other, so perhaps the sum is both 0 and 1? That proposition sounds absurd to us today, but at the time some mathematicians were comforted by its religious overtones. It reminded them of the theological assertion that God created the world from nothing.