Fractured Fractions

 


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If you have not come across continued fractions in your mathematical travels, then it is high time you did!

Every real number, rational and irrational, can be represented as a continued fraction. While normal fractions can only represent rational numbers, continued fractions are different. Rational numbers produce finite continued fractions, while irrationals become infinite continued fractions. Study the examples which follow and see what you notice.

\[ {37 \over 15} = 2 + \cfrac{1}{2 + \cfrac{1}{7}} \]

 

Some More Examples

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Build Your Own Continued Fraction

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Explore Continued Fractions

Enter a mathematical value to convert to a continued fraction (eg 1.625, 37/15, sqrt(2), cbrt(2), pi, e, phi or any JavaScript Math term, such as Math.sin(pi/4) or even Math.random() - but watch out! Case is important!).

 

\( { \cfrac {37 }{ 15} = 2 + \cfrac {7 }{ 15}} \)

\( {2 + \cfrac {1}{2 + \cfrac {1}{7}}} \)

\( => \begin{bmatrix} 2 & 1 \\ 1 & 0 \\ \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 1 & 0 \\ \end{bmatrix} \begin{bmatrix}7 & 1 \\ 1 & 0 \\ \end{bmatrix}\)

\( = \begin{bmatrix} 37 & 5 \\ 15 & 2 \\ \end{bmatrix} \) => 2.466666666666667 => \( 37 \over 15\)

 

 

  
 

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Assessment

Hint: When entering mathematical expressions in the math boxes below, 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]". More?

 

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If you have TI-Nspire™ for handheld, computer software or iPad App, then download the Continued_Fractions.tns document to explore these ideas (apart from the musical ones!)

 


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