Joy of X by Strogatz

Ref: Steven Strogatz (2013). Joy of X. Mariner.

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Summary­

  • A practical history of Mathematics.

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Mathematics

  • Commutative Law: Any finite sum or product is unaltered by reordering its terms of factors (p+q = q + p,  pq = qp.

  • e~ 2.71828.

  • Normal Distribution: Arises whenever a large number of mildly random effects of similar size, all acting independently, are added together.

  • Great Circle Route: the straightest path between two points.

  • Algebra: Used when something changes steadily, at a constant rate.

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Calculus

  • Calculus: Mathematics of change; describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball.

  • Derivative: Tells you how fast something is changing; use derivatives to figure out where a function reaches its maximum or minimum, an issue that arises whenever we’re looking for the best or cheapest or fastest way to do something.

  • Integral: Tells you how much something is accumulating.

    • Symbol: Long-necked S, for “summation.”

  • Vector Calculus: describes the invisible fields all around us. Think of the magnetic field that twists a compass needle northward, or the gravitational field that pulls your chair to the floor, or the microwave field that nukes your dinner.

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Differential Equations

·       Differential Equations: the laws of inanimate things, which describe how interlinked variables change from moment to moment, depending on their current values.

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Fourier

  • When the basic building blocks are sine waves, the technique is known as Fourier analysis, and the corresponding sums are called Fourier series.

  • Gibbs phenomenon: the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.

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Binary Numbers

  • Binary: Instead of powers of 10, binary uses powers of 2. It still has a ones place like the decimal system, but the subsequent places now stand for twos, fours, and eights, because   2 = 21, 4 = 2 × 2 = 22, 8 = 2 × 2 × 2 = 23.   Of course, we wouldn’t write the symbol 2, because it doesn’t exist in binary, just as there’s no single numeral for 10 in decimal. In binary, 2 is written as 10, meaning one 2 and zero 1s. Similarly, 4 would be written as 100 (one 4, zero 2s, and zero 1s), and 8 would be 1000.

  • Morse Code was the technological forerunners of today’s zeros and ones.

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Nature

  • Planets move in elliptical orbits with the sun at one focus; or that comets sail through the solar system on elliptic, parabolic, or hyperbolic trajectories; or that a child’s ball tossed to a parent follows a parabolic arc.

  • 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.

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Utilitarian Mathematics

  • Exponential Growth: If your money grows at an annual interest rate of r, after one year it will be worth (1 + r) times your original deposit; after two years, (1 + r)²; and after x years, (1 + r)ˣ times your initial deposit.

  • If the interest was compounded infinitely often—this is called continuous compounding—your total after one year would be bigger still, but not by much: $2,718.28. The exact answer is $1,000 times e.

  • Generator: Converts rotary motion into alternating current (AC).

  • By plotting y versus x to visualize how one variable affects another, all scientists translate their problems into the common language of mathematics. The rate of change that really concerns them—a viral growth rate, a jet’s velocity, or whatever—then gets converted into something much more abstract but easier to picture: a slope on a graph. Like slopes, derivatives can be positive, negative, or zero, indicating whether something is rising, falling, or leveling off.

  • Functions can be tools; with y= -x2 +4, the x² bends the piece of the x-axis, and the 4 lifts it. On the other hand, they’re building blocks: the 4 and the –x² can be regarded as component parts of a more complicated function, 4 – x², just as wires, batteries, and transistors are component parts of a radio. Once you start to look at things this way, you’ll notice functions everywhere.

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Maxwell’s Equations

  • Maxwell’s equations express a beautiful symmetry through four laws: they link one field’s rate of change in time to the other field’s rate of change in space, as quantified by its curl.

    • The divergence of the electric 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 of an electric field; the curl equations describe how the electric and magnetic fields interact and change over time.

  • 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.

  • 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!

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Misc Quotes

  • In Einstein’s General Theory of Relativity, gravity is no longer viewed as a force, but rather as a manifestation of the curvature of space.

  • I have given up newspapers in exchange for Tacitus and Thucydides, for Newton and Euclid; and I find myself much the happier.-Jefferson to Adams, 1812.

  • In the OJ Simpson trial, both sides were asking the court to consider the probability that a man murdered his ex-wife, given that he previously battered her. But as the statistician I. J. Good pointed out, that’s not the right number to look at. The real question is: What’s the probability that a man murdered his ex-wife, given that he previously battered her and she was murdered by someone? That conditional probability turns out to be very far from 1 in 2,500. To see why, imagine a sample of 100,000 battered women. Granting Dershowitz’s number of 1 in 2,500, we expect about 40 of these women to be murdered by their abusers in a given year (since 100,000 divided by 2,500 equals 40). We also expect 3 more of these women, on average, to be killed by someone else (this estimate is based on statistics reported by the FBI for women murdered in 1992; see the notes for further details). So out of the 43 murder victims altogether, 40 of them were killed by their batterers. In other words, the batterer was the murderer about 93% of the time.

  • Google’s PageRank Algorithm: The approach taken by Larry Page and Sergey Brin, the grad students who cofounded Google, was to let webpages rank themselves by voting with their links. If you have 3 websites: x, y, and z and pages X and Y both link to Z, then Z is the only page with two incoming links. So it’s the most popular page, however popularity means nothing on its own. What matters is having links from good pages. Google’s algorithm assigns a fractional score between 0 and 1 to each page. That score is called its PageRank; it measures how important that page is relative to the others by computing the proportion of time that a hypothetical Web surfer would spend there. Whenever there is more than one outgoing link to choose from, the surfer selects one at random, with equal probability. Under this interpretation, pages are regarded as more important if they’re visited more frequently.

  • A 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.

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Chronology

  • 1919: A solar eclipse confirms Einstein’s theory that light travels on geodesics through curved space-time, with the warping being caused by the sun’s gravity.-Joy of X by Strogatz.

  • 1865: Maxwell’s Equations are published linking electricity and magnetic fields, the properties of electromagnetic waves, and confirming the speed of light.-Joy of X by Strogatz.

  • 24 May, 1844: The USA’s first telegraph line, constructed by Cornell as a link from Baltimore to Washington, D.C officially opens. Morse sent the first message down the wire: “What hath God wrought.”-Joy of X by Strogatz.

  • 1792: The Prime Number Theorem, ln(N) as N approaches infinity, is first noted by Carl Gauss (when he was 15).-Joy of X by Strogatz.

  • 1650’s: Derivatives are first developed in England and Germany.-Joy of X by Strogatz.

  • 12c: The Hindu-Arabic Numeral system based on ten symbols (0-9) is first used in Europe and the Middle East.-Joy of X by Strogatz.

  • 820: Al-Jabr is published by Mathematician Muhammad ibn Musa al-Khwarizmi in Baghdad. The book highlighted the usefulness of restoring a quantity being subtracted by adding it to the other side of an equation. He called this process al-jabr (Arabic for “restoring”), which later morphed into “algebra.” One practical impetus at that time was the challenge of calculating inheritances according to Islamic law.-Joy of X by Strogatz.

  • 7c: The Numeral system based on ten symbols (0-9) is first developed in India.-Joy of X by Strogatz.

  • 250 BCE: The integral is first developed in Greece.-Joy of X by Strogatz.

  • 300 BCE: The Elements is published by Euclid in Greece laying down a deductive approach to logical reasoning in all walks of life.-Joy of X by Strogatz.

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