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GXWeb Fractured Functions

Saltire Software, home of Geometry Expressions and GXWeb

Download a Geometry Expressions Model for Fractured Fractions

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

Wikipedia: Generalized Continued Fractions

Continued Fraction Arithmetic by Ross Dempsey

An Introduction to Continued Fractions by Dr Ron Knott

GXWeb Fractured Fractions OR try a simpler version of this task.

Online Mathematics Research: The Ramanujan Machine

 
 
   

 

Introduction

As we have seen, continued fractions come in more than one flavour. Finding a finite (closed) form for the real number represented by a continued fraction can be very useful in understanding the patterns and applications involved.

For rational numbers, this is easy, since the continued fraction is finite. For most irrationals, the closed form is much harder, but not for quadratic irrationals, which are periodic.

For any periodic continued fraction, we need to isolate the periodic part and use this to form a quadratic equation. For example...

[1,2,3,1,2,3,1,2,3...]=1+12+13+11+12+13+... x=1+12+13+1x=4+(37)7

This may also be expressed in another form - using matrices!

 

 

Now we dig more deeply into the non-simple form - generalised continued fractions.

π41+123+225+327+429+5211+6213+

e3142536475869+

But where do these remarkable fractions come from? Calculating simple continued fractions is relatively easy, but these monsters?

Here, too, is where we see the true power of continued fractions in helping us to better understand the nature of irrational numbers. Even for simple continued fractions, the sequences which define all but the quadratic irrationals remain as random and unpredictable as their decimal counterparts - such numbers throughout history have been described as essentially unknowable.

Generalised continued fractions, however, reveal that most if not all irrationals ARE knowable - they can be expressed in forms that are predictable and can readily be calulated to any desired degree of accuracy. Students learning about irrational numbers should be introduced to continued fractions so that they may understand them better, not be put off with a wave of the hand and reference to them as, more or less, simply mysterious entities.

Once again, if mathematics is to be taught and learned as a search for patterns and relationships, then the multitude of patterns and relationships to be found within continued fractions should be part of that journey.

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Explore the Magic Table

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Negative (or Reversal) Continued Fractions

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Some Matrix Fraction Magic

Index

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Quadratic Continued Fractions

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Euler and his Numbers

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Continued Fraction Arithmetic

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Mathematical ToolBox

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Continued Logarithms

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Pieces of Pi

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Not so AbSurd After All

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Visualise Continued Fractions with GXWeb

About the MathBoxes...

Use the Continued Fractions button or enter coordinates in the text boxes to explore (and even ♬ listen to!) your own continued fractions.

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Continued Fraction Spreadsheet Explorer

This browser-based spreadsheet uses the handsontable JavaScript library.

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Share with Different Senses

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Construct your own Model with GXWeb

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Behind the Scenes

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