Try the Continued Fraction JigSaw Quiz! Just rearrange the squares to fill the given rectangle, and then press INPUT (or the JIGSAW button again) to enter your answer.
Then go on and explore some of the wonderful connections between continued fractions and some surprising and important corners of mathematics.
Hint: When entering mathematical expressions in the math boxes below (f(x), g(x) and h(x)), use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.
Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot3" with a backslash (\)): for example: "\sqrt(2)[space][space]". Learn More?
The powerful Wolfram Alpha online CAS engine will answer almost anything you care to ask - within reason! From the continued fraction of pi to Solve x^2=x+1 to the population of Australia!
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:\[x=a + \cfrac{1}{1-a+x} => (2\cdot (x-a)+1)^2 = 5 => \frac{(\sqrt(5)+2a-1)}{2}\].
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...
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...
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!
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 \(\frac{\Gamma(x)}{\sqrt(\pi)}\).
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 \[f(x)=r\cdot x\cdot (1-x)\]
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!
QR Codes are a great way to share data and information with others, even when no Internet connection is available. Most modern devices either come equipped with QR readers in-built, or freely available.
The default link here is the GXWeb Jigsaws and Quizzes, but you can use it as an alternative to sending your assessment data via email, web or share with others in your class. You can even use it to send your own messages!
