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  • feedwordpress 09:01:59 on 2018/11/20 Permalink
    Tags: , , Euler, , , , , ,   

    “There are 10 kinds of people in the world: those who understand binary numerals, and those who don’t”*… 


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    Guide to Computing

    From a collection of vintage photos of computing equipment by “design and tech obsessive” James Ball…

    Guide to Computing

    More at Docubyte

    [TotH to Kottke]

    * vernacular joke, as invoked by Ian Stewart in Professor Stewart’s Cabinet of Mathematical Curiosities

    ###

    As we rewind, we might spare a thought for Christian Goldbach; he died on this date in 1764.  A mathematician, lawyer, and historian who studied infinite sums, the theory of curves and the theory of equations, he is best remembered for his correspondence with Leibniz, Euler, and Bernoulli, especially his 1742 letter to Euler containing what is now known as “Goldbach’s conjecture.”

    In that letter he outlined his famous proposition:

    Every even natural number greater than 2 is equal to the sum of two prime numbers.

    It has been checked by computer for vast numbers– up to at least 4 x 1014– but remains unproved.

    (Goldbach made another conjecture that every odd number is the sum of three primes; it has been checked by computer for vast numbers, but also remains unproved.)

    Goldbach’s letter to Euler (source, and larger view)

    (Roughly) Daily is headed into a Thanksgiving hiatus; regular service will resume when the tryptophan haze clears…  probably around Monday, November 26.  Thanks for reading– and have Happy Holidays!

     
  • feedwordpress 08:01:27 on 2018/08/06 Permalink
    Tags: , , Euler, , Johann Bernoulli, , , ,   

    “Once is happenstance. Twice is coincidence. Three times, it’s enemy action.”*… 


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    complexity

    A couple of weeks ago, we considered the human urge to find significance, meaning in everyday occurrences: “All mystical experience is coincidence; and vice versa, of course.” Today, we consider the same phenomena from a more mathematical point-of-view…

    Was it a chance encounter when you met that special someone or was there some deeper reason for it? What about that strange dream last night—was that just the random ramblings of the synapses of your brain or did it reveal something deep about your unconscious? Perhaps the dream was trying to tell you something about your future. Perhaps not. Did the fact that a close relative developed a virulent form of cancer have profound meaning or was it simply a consequence of a random mutation of his DNA?

    We live our lives thinking about the patterns of events that happen around us. We ask ourselves whether they are simply random, or if there is some reason for them that is uniquely true and deep. As a mathematician, I often turn to numbers and theorems to gain insight into questions like these. As it happens, I learned something about the search for meaning among patterns in life from one of the deepest theorems in mathematical logic. That theorem, simply put, shows that there is no way to know, even in principle, if an explanation for a pattern is the deepest or most interesting explanation there is. Just as in life, the search for meaning in mathematics knows no bounds…

    Noson Yanofsky on what math can teach us about finding order in our chaotic lives.

    * Ian Fleming

    ###

    As we consider the odds, we might send carefully-calculated birthday greetings to Johann Bernoulli; he was born on this date in 1667.  A member of the mathematically-momentous Bernoulli family, Johann (also known as Jean or John) discovered the exponential calculus and (with Leibniz and Huygens) the equation of the catenary.  Still, he be best remembered as teacher and mentor of Leonhard Euler.

    220px-Johann_Bernoulli2 source

     

     
  • feedwordpress 09:01:46 on 2015/11/20 Permalink
    Tags: conjecture, Euler, , , Kids say the darnedest things, , ,   

    “I’m gonna put a curse on you and all your kids will be born completely naked”*… 


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    More at “Rejected Titles for Kids Say the Darnedest Things.” (Younger readers click here for explanatory background.)

    * Jimi Hendrix

    ###

    As we remark that an acorn never falls far from the tree, we might spare a thought for Christian Goldbach; he died on this date in 1764.  A mathematician, lawyer, and historian who studied infinite sums, the theory of curves and the theory of equations, he is best remembered for his correspondence with Leibniz, Euler, and Bernoulli, especially his 1742 letter to Euler containing what is now known as “Goldbach’s conjecture.”

    In that letter he outlined his famous proposition:

    Every even natural number greater than 2 is equal to the sum of two prime numbers.

    It has been checked by computer for vast numbers– up to at least 4 x 1014– but remains unproved.

    (Goldbach made another conjecture that every odd number is the sum of three primes; it has been checked by computer for vast numbers, but remains unproved.)

    Goldbach’s letter to Euler (source, and larger view)

     
  • feedwordpress 09:01:31 on 2014/12/04 Permalink
    Tags: , , , Euler, Euler's Identity, , , , ,   

    “I know numbers are beautiful. If they aren’t beautiful, nothing is”*… 


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    Euler’s identity: Math geeks extol its beauty, even finding in it hints of a myste­rious connected­ness in the universe. It’s on tank tops and coffee mugs [and tattoos]. Aliens, apparently, carve it into crop circles (in 8-bit binary code). It’s appeared on The Simpsons. Twice.

    What’s the deal with Euler’s identity? Basically, it’s an equation about numbers—specifically, those elusive constants π and e. Both are “transcendental” quanti­ties; in decimal form, their digits unspool into infinity. And both are ubiquitous in scientific laws. But they seem to come from different realms: π (3.14159 …) governs the perfect symmetry and closure of the circle; it’s in planetary orbits, the endless up and down of light waves. e (2.71828 …) is the foundation of exponential growth, that accelerating trajectory of escape inherent to compound interest, nuclear fission, Moore’s law. It’s used to model everything that grows…

    Now, maybe you’ve never thought of math equations as “beautiful,” but look at that result: It combines the five most fundamental numbers in math—0, 1, e, i, and π—in a relation of irreducible simplicity. (Even more astonishing if you slog through the proof, which involves infinite sums, factorials, and fractions nested within fractions within fractions like matryoshka dolls.) And remember, e and π are infinitely long decimals with seemingly nothing in common; they’re the ultimate jigsaw puzzle pieces. Yet they fit together perfectly—not to a few places, or a hundred, or a million, but all the way to forever…

    But the weirdest thing about Euler’s formula—given that it relies on imaginary numbers—is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trig problems (you know, sines, secants, and so on) into more tractable algebra—like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digi­tize music, and it tames all manner of wavy things in quantum mechanics, electron­ics, and signal processing; without it, computers might not exist…

    More marvelous math at “The Baffling and Beautiful Wormhole Between Branches of Math.”

    [TotH to @haarsager]

    Paul Erdős

    ###

    As we wonder if Descartes wasn’t right when he wrote that “everything turns into mathematics,” we might spare a thought for Persian polymath Omar Khayyam; the mathematician, philosopher, astronomer, epigrammatist, and poet died on this date in 1131.  While he’s probably best known to English-speakers as a poet, via Edward FitzGerald’s famous translation of the quatrains that comprise the Rubaiyat of Omar Khayyam, Omar was one of the major mathematicians and astronomers of the medieval period.  He is the author of one of the most important works on algebra written before modern times, the Treatise on Demonstration of Problems of Algebra, which includes a geometric method for solving cubic equations by intersecting a hyperbola with a circle.  His astronomical observations contributed to the reform of the Persian calendar.  And he made important contributions to mechanics, geography, mineralogy, music, climatology, and Islamic theology.

     source

     
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