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## GXWeb Fractured Fractions Collection

Saltire Software, home of Geometry Expressions and GXWeb

Symbolic computations on this page useNerdamer Symbolic JavaScriptto complement the in-built CAS of GXWeb

Chaos in Numberland: The secret life of continued fractions by John D. Barrow

Explore Bill Gosper (1972): Continued Fraction Arithmetic

Calkin and Wilf (1999): Recounting the Rationals

Bruce Bates (2014): The Stern-Brocot Continued Fraction

Bates, B., Bunder, M. and Tognetti, K. (2010): Linking the Calkinâ€“Wilf and Sternâ€“Brocot trees

With special thanks toDr Keith Tognettifor pointing me in the direction of continued fractions many years ago and igniting in me a life-long passion, and toDr Bruce Bates(both of the University of Wollongong) for revealing the beautiful Stern-Brocot Continued Fraction

Welcome to the Fractured Fractions Collection!Try the

Continued Fraction JigSaw! 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. Did you know, for example, thatfractions grow on trees?

Mathematical ToolBox

More to Explore: the Continued Fractions Collection

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

Golden Numbers and More

Fractured Functions: Generalised Continued Fractions

Continued Fractions and Farey Trees

## Climb Around the Farey (Stern-Brocot) Tree

In addition to the Farey Sequence, there are other beautiful ways to display the rationals between 0 and 1, including one which is even more closely related to

continued fractions.The Farey Tree (a subset of the Stern-Brocot Tree) begins with the endpoints of 0 and 1 and uses the

mediantto find a fraction between each. As with the Farey sequence, this simply involves forming a fraction by adding the two numerators and the two denominators.So \(\frac{0}{1}\) and \(\frac{1}{1}\) give \(\frac{0+1}{1+1} = \frac{1}{2}\)

Then \(\frac{0}{1}\) and \(\frac{1}{2}\) give \(\frac{0+1}{1+2} = \frac{1}{3}\) and \(\frac{1}{1}\) and \(\frac{1}{2}\) give \(\frac{1+1}{1+2} = \frac{2}{3}\)

\[\frac{0}{1}\] \[\frac{1}{1}\]

\[\frac{0+1}{1+1}\]\[\frac{1}{2}\]

\[\frac{0+1}{1+2}\]\[\frac{1}{3}\] \[\frac{1+1}{1+2}\]\[\frac{2}{3}\] Each subsequent row of fractions is built by continuing this process, stepping either to the

left (L)orright (R), beginning from 1. Each \(n\)-th row begins with \(\frac{1}{n}\) and ends with \(\frac{n-1}{n}\).From \(\frac{1}{1}\), to get to \(\frac{1}{2}\), move down to the Left (L). To form the continued fraction(s), simply add another L or R (it does not matter which because both deliver correct but slightly different results). So for \(\frac{1}{2}\), we could have

LLorLR. Begin your continued fraction with zero (since all our fractions - for now! - lie between 0 and 1) and then count eachrepeatedelement. ThenLLwill give the continued fraction \([0, 2]=0+\cfrac{1}{2}\) andLRgives \([0, 1, 1] = 0 + \cfrac{1}{1+\cfrac{1}{1}}\).In the same way, to get to \(\frac{1}{3}\) from \(\frac{1}{1}\), step

LL. Add another L to getLLL(\([0,3]\)) orLLR(\([0,2,1]\)). For \(\frac{2}{3}\) from \(\frac{1}{1}\), stepLR. Add another L to getLRL(\([0,1,1,1]\)) orLRR(\([0,1,2]\)).

Any continued fraction ending in 1 can becompactedby adding that trailing one to the previous term.

\[\frac{0}{1}\]\[[0]\] \[\frac{1}{1}\]\[[1]\]

\[\frac{1}{2}\]\[L+L\]\[[0,2]\]\[L+R\]\[[0,1,1]\]

\[\frac{1}{3}\]\[LL+L\]\[[0,3]\]\[LL+R\]\[[0,2,1]\] \[\frac{2}{3}\]\[LR+R\]\[[0,1,2]\]\[LR+L\]\[[0,1,1,1]\]

## Follow a Binary Path to Continued Fractions

## Explore the Stern-Brocot Tree

## Explore the Calkin-Wilf Tree

Bessel Functions

Gamma Functions

Chaos Theory and Cobweb Functions

Musical Continued Fractions

And more...

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