©2023 Compass Learning Technologies ← GXWeb Showcase ← Fractured Fractions → Exploring Continued Fractions with GX

 

Exploring Continued Fractions with GXWeb

Saltire Software, home of Geometry Expressions and GXWeb

Download a Geometry Expressions Model for Fractured Fractions

Take Continued Fractions to the Next Level

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

 

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• Introduction

• Visualise Continued Fractions with GXWeb

• Need a Little CAS?

• Build Your Own Model with GXWeb

• Behind the Scenes

 

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Every real number, rational and irrational, can be represented by a continued fraction - the rational ones are finite, of course, but both finite and infinite offer some wonderful patterns and opportunities to explore!

Continued fractions have many practical applications but one of the most important lies in their ability to offer VERY good approximations to irrational numbers - the more convergents, the better the approximation. Indeed, each convergent gives the Best Approximation of the First Kind (look that up!) AND if a large value turns up in the list, then cutting the continued fraction just before that large value gives an extremely accurate approximation!

Step-by-Step Guide to building your own continued fractions

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Drag point P or use the INPUT button to explore different numbers and their continued fractions. INPUTRESET This document requires an HTML5-compliant browser. App generated by GXWeb

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Need a Little CAS?

Sometimes you just need a little CAS to answer that question, or explore that problem more deeply. 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!

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Construct your own Model with GXWeb

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Need help? Tap here for the construction steps

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You can copy and paste the following recursive definitions within your GXWeb model to construct the first five convergents for your graphical continued fraction for a/b:

• Z0 = (abs(a-b*floor(a/b)))
• Z1 = (abs(b-z0*floor(b/z0)))
• Z2 = (abs(z0-z1*floor(z0/z1)))
• Z3 = (abs(z1-z2*floor(z1/z2)))

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• Z0 = (abs(a-b*floor(a/b)))
• Z1 = (abs(b-(abs(a-b*floor(a/b)))*floor(b/(abs(a-b*floor(a/b))))))
• z2 = (abs((abs(a-b*floor(a/b)))-(abs(b-(abs(a-b*floor(a/b)))*floor(b/(abs(a-b*floor(a/b))))))*floor((abs(a-b*floor(a/b)))/(abs(b-(abs(a-b*floor(a/b)))*floor(b/(abs(a-b*floor(a/b)))))))))
• z3 = (abs((abs(b-(abs(a-b*floor(a/b)))*floor(b/(abs(a-b*floor(a/b))))))-(abs((abs(a-b*floor(a/b)))-(abs(b-(abs(a-b*floor(a/b)))*floor(b/(abs(a-b*floor(a/b))))))*floor((abs(a-b*floor(a/b)))/(abs(b-(abs(a-b*floor(a/b)))*floor(b/(abs(a-b*floor(a/b)))))))))*floor((abs(b-(abs(a-b*floor(a/b)))*floor(b/(abs(a-b*floor(a/b))))))/(abs((abs(a-b*floor(a/b)))-(abs(b-(abs(a-b*floor(a/b)))*floor(b/(abs(a-b*floor(a/b))))))*floor((abs(a-b*floor(a/b)))/(abs(b-(abs(a-b*floor(a/b)))*floor(b/(abs(a-b*floor(a/b))))))))))))

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Behind the Scenes

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Learn How: Create your own GXWeb Assessment Task

 

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View JavaScript Continued Fractions Code (gxwcf.js)

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©2023 Compass Learning Technologies ← GXWeb Showcase ← Exploring Continued Fractions with GXWeb

 

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