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GXWeb Continued Fraction JigSaw

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More to Explore: the Continued Fractions Collection

Interested to learn more? Delve more deeply into Continued Fractions with GXWeb Fractured Fractions and then on to the following...

 
 
  • Golden Numbers and More: Try \(\frac{34}{21}\), \(\frac{55}{34}\) and \(\frac{89}{55}\). Notice anything? What comes next?

    This might just begin a search for the most beautiful (and most irrational) of numbers...! And make sure you take a moment to explore the archimidean spiral along the way:

    Perhaps take another moment or two to also explore the metallic means: \[x=a + \cfrac{1}{x} => x^2=a\cdot x+1 => \frac{(a+\sqrt(4+a^2)}{2}\] and the noble numbers: \[x=a + \cfrac{1}{1-a+x} => (2\cdot (x-a)+1)^2 = 5 => \frac{(\sqrt(5)+2a-1)}{2}\].

     
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  • Fractured Functions: Generalised Continued Fractions

    Most of the continued fractions you will come across are likely to be of the simple variety, but the many generalised forms offer so many interesting patterns and possibilities for further exploration! Try a few and see what you discover...

    Quadratic Continued Fractions

    Euclid's Number

    Pieces of Pi

    Not So AbSurd After All

    \[ x^2+b\cdot x + c = 0\]

    \[ x \cdot (x+b) = -c\]

    \[ x = \frac{-c}{b+x}\]

    \[ x = \cfrac{-c}{b-\cfrac{c}{b-\cfrac{c}{x}}}\]

     
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  • Continued Fractions and Farey Neighbours

    There are some wonderful connections between continued fractions and Farey neighbours - adjacent terms of Farey Sequences.

    For example, \(\frac{2}{5}\) and \(\frac{5}{12}\) are neighbours in Farey(12), as are \( \frac{6}{19}\) and \(\frac{7}{22}\) in Farey(25). Check their continued fraction forms and try more of your own...

     
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  • Bessel Functions: Studying this continued fraction (\(\frac{e^2+1}{e^2-1} = [1, 3, 5, 7, 9, 11...]\)) got me thinking about a continued fraction which should be even simpler: \([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\). Surely this should be given by a common and recognisable real number? Well, how wrong I was! Welcome to the world of Bessel functions!

     
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  • The Gamma Function: The factorial function is well known -
    \(3! = 3 \cdot 2 \cdot 1 = 6,\)
    \(4! = 4 \cdot 3 \cdot 2 \cdot 1 = 12,\)
    \(n! = n \cdot (n-1) \cdot (n-2)...\)
    But what happens in between? The Gamma function, represented by the Greek capital, \(\Gamma(x)=(x-1)!\) includes all values, not just the integers, and has some interesting properties - especially when you consider the half values of \(\frac{\Gamma(x)}{\sqrt(\pi)}\).

     
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  • Chaos Theory: Consider a population, say of fish in a pond. If the pond is fixed in size and limited in the amount of food which it can provide, then the population of fish cannot grow unbounded. In fact, the size of the population itself will limit the growth - as the number of fish (\(x\)) gets large, it will act to slow down the rate of population growth (\(r\)). A simple model of this situation over time is given by the relationship \[f(x)=r\cdot x\cdot (1-x)\]

    But what happens next is far from simple!

     
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  • Musical Continued Fractions: Suppose you had a list of numbers, for example, myList = [0, 2, 4, 5, 7, 9, 11, 12]. How might you turn each of those list elements into a musical tone - perhaps have 0 represent middle C, and each unit a semi-tone higher. 1 would be C#4, 2 D, and so on. Then myList should give the scale from middle C (C4) to the next C (C5).

    Perhaps even write and play your own continued fraction music? You might try dividing a piece into parts and playing in turn with others!

     
  • Did You Know...? There is a connection between the Euclidean algorithm for finding the greatest common divisor of two numbers and continued fractions?

    Tap the image to explore.

    Interested to learn more about continued fractions?

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