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Thread: Numbers

  1. #1

    Numbers


    Hindu-Arabic numeration system



    The following lists 4 main attributes of this numeration system


    First, it uses 10 digits or symbols that can be used in combination to represent all possible numbers
    The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.


    Second, it groups by tens, probably because we have 10 digits on our two hands. Interestingly enough, the word digit literally means finger or toes.

    In the Hindu-Arabic numeration system,
    ten ones are replaced by one ten,
    ten tens are replaced by one hundred,
    ten hundreds are replaced by one thousand,
    10 one thousand are replaced by 10 thousands,
    and so forth...


    Third, it uses a place value. starting from right to left,

    • the first number represents how many ones there are


    • the second number represents how many tens there are


    • the third number represents how many hundreds there are


    • the fourth number represents how many thousands there are


    • and so on...


    For example, in the numeral 4687, there are 7 ones, 8 tens, 6 hundreds, and 4 thousands

    Finally, the system is additive and multiplicative. The value of a numeral is found by multiplying each place value by its corresponding digit and then adding the resulting products


    Place values: thousand hundred ten one

    Digits 4 6 8 7

    Numeral value is equal to 4 × 1000 + 6 × 100 + 8 × 10 + 7 × 1 = 4000 + 600 + 80 + 7 = 4687


    Notice that the Hindu-Arabic numeration system require requires fewer symbols to represent numbers as opposed to other numeration system.


    Each Hindu-Arabic numeral has a word name. Here is short list:


    0: Zero 10: Ten

    1: One 11: Eleven

    2: Two 15: Fifteen

    3: Three 20: Twenty

    4: Four 30: Thirty-four

    5: Five 40: Fourty

    6: Six 100: One hundred

    7: Seven 590: Five hundred seventy

    8: Eight 5083: Five thousand eighty-three

    9: Nine 56000: Fifty-six thousand




    Numbers from 1 through 12 have unique names


    Numbers from 13 through 19 have "teens" as ending and the ending is blended with names for numbers from 4 through 9


    For numbers from 20 through 99, the tens place is named first followed by a number from 1 through



    Numbers from 100 through 999 are combinations of hundreds and previous names



    Hindu-Arabic numeration system








    Hindu-Arabic numerals, set of 10 symbols—1, 2, 3, 4, 5, 6, 7, 8, 9, 0—that represent numbers in the
    decimal number system. They originated in India in the 6th or 7th century and were introduced to Europe through the writings of Middle Eastern mathematicians, especially al-Khwarizmi and al-Kindi, about the 12th century. They represented a profound break with previous methods of counting, such as the abacus, and paved the way for the development of algebra.

    Hindu-Arabic numerals | History & Facts | Britannica.com



    Amalfi, Pisa, Genoa and Venice were the first doors and windows through which a permanent contact with the East was established. The sea towns were a means of cultural communication. Arabic numerals, which were to simplify and revolutionise the accounting of merchants, were introduced to the West by the Pisan Leonardo Fibonacci, author of a Liber abbaci, who lived in the late twelfth and early thirteenth centuries. The compass, already known to the Arabs, was adopted by Amalfitani.

    Some claim that it was in Spain that the use of this system began. Probably both Italy and Spain.

    Numbers - we just use them and don't think much about them.

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  7. #5

    Types of NUMBERS



    List of types of numbers - Wikipedia- take a look,28 V 2020.







    Perfect Number



    In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, σ₁(n) = 2n where σ₁ is the sum-of-divisors function.


    Perfect number - Wikipedia


    https://en.wikipedia.org/wiki/Perfect_number




  8. #6

    The mystique of mathematics: 5 beautiful math phenomena


    Mathematics is visible everywhere in nature, even where we are not expecting it. It can help explain the way galaxies spiral, a seashell curves, patterns replicate, and rivers bend.





    Fractals - patterns that repeat themselves on smaller scales - can be seen frequent


    Even subjective emotions, like what we find beautiful, can have a mathematical explanation.

    "Maths is not only seen as beautiful—beauty is also mathematical," says Dr. Thomas Britz, a lecturer in UNSW Science's School of Mathematics & Statistics. "The two are intertwined."

    Dr. Britz works in combinatorics, a field focused on complex counting and puzzle solving. While combinatorics sits within pure mathematics, Dr. Britz has always been drawn to the philosophical questions about mathematics.

    He also finds beauty in the mathematical process. "From a personal point of view, maths is just really fun to do. I've loved it ever since I was a little kid. "Sometimes, the beauty and enjoyment of maths is in the concepts, or in the results, or in the explanations. Other times, it's the thought processes that make your mind turn in nice ways, the emotions that you get, or just working in the flow—like getting lost in a good book."

    Here, Dr. Britz shares some of his favorite connections between maths and beauty.


    1. Symmetry—but with a touch of surprise



    Symmetry is everywhere you look.


    In 2018, Dr. Britz gave a TEDx talk on the Mathematics of Emotion, where he used recent studies on maths and emotions to touch on how maths might help explain emotions, like beauty. "Our brains reward us when we recognize patterns, whether this is seeing symmetry, organising parts of a whole, or puzzle-solving," he says. "When we spot something deviating from a pattern—when there's a touch of the unexpected—our brains reward us once again. We feel delight and excitement." For example, humans perceive symmetrical faces as beautiful. However, a feature that breaks up the symmetry in a small, interesting or surprising way—such as a beauty spot—adds to the beauty.

    "This same idea can be seen in music," says Dr. Britz. "Patterned and ordered sounds with a touch of the unexpected can have added personality, charm and depth."

    Many mathematical concepts exhibit a similar harmony between pattern and surprise, elegance and chaos, truth and mystery.

    "The interwovenness of maths and beauty is itself beautiful to me," says Dr. Britz.




    Each frond of a fern shoots off smaller versions of themselves.

    Each frond of a fern shoots off smaller versions of themselves. Sometimes, the frond pattern can even be seen in the leaves as well.


    2. Fractals: infinite and ghostly

    Fractals are self-referential patterns that repeat themselves, to some degree, on smaller scales. The closer you look, the more repetitions you will see—like the fronds and leaves of a fern.

    "These repeating patterns are everywhere in nature," says Dr. Britz. "In snowflakes, river networks, flowers, trees, lightning strikes—even in our blood vessels."

    Fractals in nature can often only replicate by several layers, but theoretic fractals can be infinite. Many computer-generated simulations have been created as models of infinite fractals.

    "You can keep focusing on a fractal, but you'll never get to the end of it," says Dr. Britz. "Fractals are infinitely deep. They are also infinitely ghostly. "You might have a whole page full of fractals, but the total area that you've drawn is still zero, because it's just a bunch of infinite lines."




    The Mandelbrot Set is arguably the most famous computer-generated fractal. Zoo


    3. Pi: an unknowable truth

    Pi (or 'π') is a number often first learned in high school geometry. In simplest terms, it is a number slightly more than 3.

    Pi is mostly used when dealing with circles, such as calculating the circumference of a circle using only its diameter. The rule is that, for any circle, the distance around the edge is roughly 3.14 times the distance across the center of the circle.

    But Pi is a lot more than this. "When you look into other aspects of nature, you will suddenly find Pi everywhere," says Dr. Britz. "Not only is it linked to every circle, but Pi sometimes pops up in formulas that have nothing to do with circles, like in probability and calculus."

    Despite being the most famous number (International Pi Day is held annually on 14 March, 3.14 in American dating), there is a lot of mystery around it.

    "We know a lot about Pi, but we really don't know anything about Pi," says Dr. Britz.

    "There's a beauty about it—a beautiful dichotomy or tension."




    Pi is tied to ocean and sound waves through the Fourier series, a formula used i

    Pi is infinite
    and, by definition, unknowable. No pattern has yet been identified in its decimal points. It's understood that any combination of numbers, like your phone number or birthday, will appear in Pi somewhere (you can search this via an online lookup tool of the first 200 million digits).

    We currently know 50 trillion digits of Pi, a record broken earlier this year. But, as we cannot calculate the exact value of Pi, we can never completely calculate the circumference or area of a circle—although we can get close.

    "What's going on here?" says Dr. Britz. "What is it about this strange number that somehow ties all the circles of the world together? "There's some underlying truth to Pi, but we don't understand it. This mystique makes it all the more beautiful."


    4. A golden and ancient ratio

    The Golden Ratio (or 'ϕ') is perhaps the most popular mathematical theorem for beauty. It's considered the most aesthetically pleasing way to proportion an object.

    The ratio can be shortened, roughly, to 1.618. When presented geometrically, the ratio creates the Golden Rectangle or the Golden Spiral.

    "Throughout history, the ratio was treated as a benchmark for the ideal form, whether in architecture, artwork, or the human body," says Dr. Britz. "It was called the "Divine Proportion."



    The Golden Spiral is often used in photography to help photographers frame the


    "Many famous artworks, including those by Leonardo da Vinci, were based on this ratio."

    The Golden Spiral is frequently used today, especially in art, design and photography. The center of the spiral can help artists frame image focal points in aesthetically pleasing ways.


    5. A paradox closer to magic

    The unknowable nature of maths can make it seem closer to magic.

    A famous geometrical theorem called the Banach-Tarski paradox says that if you have a ball in 3-D space and split it into a few specific pieces, there is a way to reassemble the parts so that you create two balls.

    "This is already interesting, but it gets even weirder," says Dr. Britz.

    "When the two new balls are created, they will both be the same size as the first ball."

    Mathematically speaking, this theorem works—it is possible to reassemble the pieces in a way that doubles the balls.




    "You can't do this in real life," says Dr. Britz. "But you can do it mathematically. "That's sort of magic. That is magic."

    Fractals, the Banach-Tarski paradox and Pi are just the surface of the mathematical concepts he finds beauty in.

    "To experience many beautiful parts of maths, you need a lot of background knowledge," says Dr. Britz. "You need a lot of basic—and often very boring—training. It's a bit like doing a million push ups before playing a sport.

    "But it is worth it. I hope that more people get to the fun bit of maths. There is so much more beauty to uncover."



    The mystique of mathematics: 5 beautiful math phenomena

    31 V 2020.


    "Everything is number" Pythagoras.

  9. #7

    The Most Irrational Number

    An irrational number can not be written in the form p/q, with a numerator and a denominator - square roots of non square numbers.




    The Most Irrational Number





    The golden ratio is even more astonishing than Dan Brown and Pepsi thought.


    BY JORDAN ELLENBERG



    This article is adapted from
    Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else by Jordan Ellenberg. © Penguin Press 2021.


    One of the great charms of number theory is the existence of irrational numbers—numbers like the square root of 2 or π / pi that can’t be expressed as the ratio of any two whole numbers, no matter how large. The legend goes—probably false, but hey, it makes a point—that the discovery of the irrationality of √2 was so disconcerting to the Pythagoreans, who wanted all numbers to be rational, that they threw the discoverer into the ocean.


    Among the mysteries of the irrationals, one number holds a special place: the so-called golden ratio. The golden ratio’s value is about 1.618 (but not exactly 1.618, since then it would be the ratio 1,618/1,000, and therefore not irrational) and it’s also referred to by the Greek letter φ, which is pronounced “fee” if you’re a mathematician and “fie” if you are in a fraternity. If you want an exact description, the golden ratio can be expressed as (1/2)(1+√5.)

    People have been making a fuss over this number for centuries. In Euclid, the proportion goes by the more mundane name of “division into the extreme and mean.” He needed it to construct a regular pentagon, since the golden ratio is the proportion between the diagonal of such a pentagon and its side. A golden rectangle is one whose length is φ times its width; it has the agreeable quality that if you cut it crosswise so that one of the two pieces is a square, the other one is a smaller golden rectangle.




    You can do the same thing to that smaller rectangle to make a new smaller golden rectangle, then cut a square off that little golden boy to make a golden rectangle yet smaller, and so on and so on; the result is a sort of rectangular spiral:





    Wikipedia


    The golden ratio doesn’t arise only in geometry; in the Fibonacci sequence, where each number is the sum of the two previous ones (1, 1, 2, 3, 5, 8, 13, 21, 34, …), the ratios between consecutive terms approach φ more and more closely as the terms get larger and larger. (But of course those ratios never arrive at φ, because, again, irrational!)

    There’s been a miasma of mysticism around the golden ratio for a long time. The number theorist George Ballard Mathews was already complaining about it in 1904, writing that “the ‘divine proportion’ or ‘golden section’ impressed the ignorant, nay even learned men like Kepler, with a sense of mystery, and set them a dreaming all kinds of fantastic symbolism.” Figures with lengths in golden proportion to one another are sometimes said to be inherently the most beautiful, though the claims that the Great Pyramid of Giza, the Parthenon, and the Mona Lisa were all designed on this principle aren’t well substantiated. An influential 1978 paper in the Journal of Prosthetic Dentistry suggests that a set of false teeth, for maximum smile appeal, should have the central incisor 1.618 times the width of the lateral incisor, which should in turn be 1.618 times as wide as the canine. There’s a small but persistent school of financial analysis which holds that the golden ratio governs fluctuations in the stock market; your Bloomberg terminal, should you be flush enough to have one, will draw little “Fibonacci lines” on the stock charts for you.


    One day in the ’90s, I had dinner with a friend of a friend at the Galaxy Diner in New York. He said he was making a movie about math and wanted to talk to a practitioner about what the mathematical life was really like. We ate patty melts, I told him some stories, I forgot about it, years went by. The friend of a friend was named Darren Aronofsky, and his movie Pi came out in 1998. The main character of the movie is a number theorist named Max Cohen who thinks extremely intensely and twirls his fingers in his hair a lot. He meets a Hasidic man who gets him interested in Jewish numerology, the practice called gematria where a word is turned into a number by adding up the numerical value of the Hebrew letters it contains. The Hebrew word for “east” adds up to 144, the Hasid explains, and “the Tree of Life” comes to 233. Now Max is interested, because those are Fibonacci numbers. He doodles some more Fibonaccis across the stock market pages of the newspaper. “I never saw that before,” says the impressed Hasid. Max feverishly programs his computer, which is named Euclid, and draws golden rectangle spirals, and stares for a good long while into the similar spirals of milk in his coffee. He computes a 216-digit number, which seems to be the key to forecasting stock prices and is also possibly God’s secret name. He plays a lot of Go with his thesis adviser. (“Stop thinking, Max. Just feel. Use your intuition.”) He gets a bad headache and twirls his hair even more intensely. The beautiful woman in the apartment next door is intrigued. I forgot to mention it, but this movie is in black and white. Somebody tries to kidnap him. Finally he drills a hole in his own skull to let some of the math pressure out and the movie arrives at what appears to be a happy ending.


    I don’t remember what I told Aronofsky about math, but it wasn’t that.


    Golden numberism really took off in 2003, when Dan Brown published his megahit novel The Da Vinci Code, the story of a “religious symbologist” and Harvard professor who uses the Fibonacci sequence and the golden ratio to unwind a conspiracy involving the Knights Templar and modern-day descendants of Jesus. After that, “put a φ on it” was just good marketing. You could buy jeans whose golden proportions were flattering to your rear (they go with your false teeth!). There was a “Diet Code,” which argued that Leonardo would have wanted you to lose weight by eating proteins and carbs in golden-ratio proportions. And there was perhaps the greatest work of mystical geometric hoo-hah ever produced: the Arnell Group marketing firm’s 27-page explanation of the new Pepsi “globe” logo it designed in 2008. The document was titled “BREATHTAKING.” The pitch explains that Pepsi and the golden ratio are natural partners, because, as you no doubt knew, “the vocabulary of truth and simplicity is a reoccurring phenomena in the brand’s history.” A timeline situates the unveiling of the new Pepsi logo as the culmination of 5,000 years of science and design including Pythagoras, Euclid, da Vinci, and somehow the Möbius strip. The new Pepsi logo would be built out of arcs from circles whose radii were in golden ratio to each other, a ratio that the pitch declares would now, in a truly impressive rebranding bid, be known as “the Pepsi Ratio.”


    But my favorite thing about the golden ratio has nothing to do with pentagons or Pepsi. It’s that the golden ratio, among all irrational numbers, is the most irrational one.


    What can that mean? Either a number is the ratio of two whole numbers or it isn’t*.


    It turns out that there are ways to talk about how irrational an irrational number is. After all, the fact that a number like φ is irrational doesn’t mean there aren’t rational fractions very close to it. Of course there are! A decimal expansion, after all, is a way of writing down fractions that are close to a number:

    16/10 = 1.6 (pretty close)

    161/100 = 1.61 (closer)

    1,618/1,000 = 1.618 (closer still)

    The decimal expansion gives you a fraction with denominator 1,000, which is within 1/1,000 of the golden ratio; if we let the denominator be 10,000 we can get within 1/10,000, and so on.

    We can do better than decimals! Remember, the ratios between Fibonacci numbers are also fractions that get closer and closer to the golden ratio:

    8/5 = 1.6

    13/8 = 1.625

    21/13 = about 1.615

    Go farther in the sequence and you get to:
    233/144 = 1.6180555555…

    … which is only about 2 in 100,000 away from the golden ratio, a substantially better approximation than 1,618/1,000, with a way smaller denominator.


    Some celebrity irrationals can be approximated even more closely. Zu Chongzhi, a fifth-century astronomer in Nanjing, observed that the simple fraction 355/113 is incredibly close to π /pi, only about 2 in 10 million away. He called it milü (“very close ratio”). Zu’s book on mathematical methods is lost, so we don’t know how he came up with the milü. But it was no simple find; it would be 1,000 years before the approximation was rediscovered in India, another 100 before it was known in Europe, and another century after that before it was conclusively proved that π/pi was actually irrational.


    One way to get a sense of how well a number can be approximated by rationals is by drawing a kind of picture I like to call a “barcode.” Here’s how you make one: If x is a number, look at the first, say, 300 multiples of x—x, 2x, 3x, 4x, and so on—and for each of those multiples, you make a little vertical tick mark at its “fractional part,” which is the part of the number after the decimal point. That is, it’s a number between 0 and 1; the fractional part of φ, for example, is about 0.618.


    Got that? Probably not. So here’s an example. Suppose you start with a rational number, like 1/7. I get a picture that looks like, well, seven bars; because no matter what I multiply 1/7 by, I get some number of sevenths, whose fractional part is either 0, 1/7, 2/7, 3/7, 4/7, 5/7, or 6/7.






    It’s the same for any rational number; we may consider more and more multiples, but the bars will be constrained to a finite collection, evenly spaced from 0 to 1.


    What about π? Here are its first 300 multiples:






    That’s a lot of bars. But not 300. In fact, if you were to count the bars visible here, you would see that there are exactly 113 of them. What you’re seeing here is the signature of the milü. Because π is so very close to 355/113, its first three hundred multiples are also very close to some number of 113ths, which means those bars are going to stay very close to the numbers 0, 1/113, 2/113, (pretend I wrote down all 113 possibilities), and 112/113. Since π isn’t exactly equal to the milü, its multiples don’t hit those fractions on the nose; the bars in the picture that look a little fatter and darker are those that are actually several bars clustered so close together they look like one on the page.


    Which brings us back to the golden ratio. The bar code formed by the first 300 multiples of φ is nicely spread out, not clustered like the π bars:






    Draw a thousand multiples, and it’s the same story, just with more bars:





    And no matter how many multiples of the golden ratio I take—a thousand, a billion, or more— those bars are never going to line up along a small set of evenly spaced positions, the way the bar code of a rational number does, or even cluster near those positions, the way the bar code for π /pi does. There is no milü for φ.


    Here’s a beautiful fact, a bit too hard to prove here: You won’t find any better rational approximations to φ than the ones the Fibonacci sequence provides. In fact, in a way that can be made quite precise (but not here), φ, out of all numbers, is the one that’s least well approximated by fractions; it is, in this sense, the most irrational number. That, to me, is golden.


    The golden ratio is the most irrational number.

    https://slate.com/technology/2021/06...phi-irrational...

    11 VI 2021.

    Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13. . . . each no is the sum of the previous two.

    π /pi + Circumference of a circle divided by the diameter,

    Circumference of a Circle:

    C = π /pi x diameter

    Area of a Circle:

    π (Pi) times the Radius squared: A = πr2

    or, when you know the Diameter: A = (π/4) × D2



    or, when you know the Circumference: A = C2 / 4π

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