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# GXWeb Fractured Fractions Collection

Saltire Software, home of Geometry Expressions and GXWeb

Symbolic computations on this page use Nerdamer Symbolic JavaScript to complement the in-built CAS of GXWeb

An Introduction to Continued Fractions by Dr Ron Knott

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

Bates, B., Bunder, M. and Tognetti, K. (2010): Linking the Calkin–Wilf and Stern–Brocot trees

With thanks to Dr Keith Tognetti for pointing me in the direction of continued fractions many years ago and igniting in me a life-long passion, and to Dr Bruce Bates (both of the University of Wollongong) for revealing to us all the beautiful Stern-Brocot Continued Fraction. With colleague Dr Martin Bunder they have brought to light many wonderful and often surprising connections between binary trees and continued fractions.

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, that fractions grow on trees?

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## More to Explore: the Continued Fractions Collection

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

### Golden Numbers and More

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### Fractured Functions: Generalised Continued Fractions

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### Continued Logarithms

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### Continued Fractions and Farey Trees

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

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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 mediant to 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 $$\frac{1+1}{1+2} = \frac{2}{3}$$

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 $\frac{0}{1}$ $\frac{1}{1}$
 $\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) or right (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 LL or LR. Begin your continued fraction with zero (since all our fractions - for now! - lie between 0 and 1) and then count each repeated element. Then LL will give the continued fraction $$[0, 2]=0+\cfrac{1}{2}$$ and LR gives $$[0, 1, 1] = 0 + \cfrac{1}{1+\cfrac{1}{1}}$$.

In the same way, to get to $$\frac{1}{3}$$ from $$\frac{1}{1}$$, step LL. Add another L to get LLL ($$[0,3]$$) or LLR ($$[0,2,1]$$). For $$\frac{2}{3}$$ from $$\frac{1}{1}$$, step LR. Add another L to get LRL ($$[0,1,1,1]$$) or LRR ($$[0,1,2]$$).

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

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 $\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

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### Explore the Stern-Brocot Tree

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### Explore the Calkin-Wilf Tree

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### Bessel Functions

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### Gamma Functions

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### Chaos Theory and Cobweb Functions

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### Musical Continued Fractions

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### And more...

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## Construct your own Model with GXWeb

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