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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.

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