Continued Fractions

©2019 Compass Learning Technologies ← Live Mathematics on the Web ← Learn How to Create your own Live Web Pages →Fractured Fractions

Fractured Fractions

Introduction

Some More Examples

Build Your Own Continued Fraction

Explore Continued Fractions

Why Not Listen to Your Continued Fraction?

And then Add... Matrices

Visualise Your Continued Fraction with GeoGebra

Assessment

Tap for more...

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

Back to Top

The Golden Ratio (1 + sqrt(5))/2: \({1 + \sqrt 5} \over 2 \)
\[ \phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cdots}}}} \]

sqrt(2) = \[ \sqrt 2 = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cdots}}}} \]

pi = \[ \pi = 3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{292 + \cdots}}}} \]

Mathematics has been described as a search for patterns and relationships. What patterns and relationships did you find in these few examples? Use the button which follows as a tool to explore your ideas further, and take a few moments to answer the questions that follow.

Once you have finished the questions, feel free to come back and continue to play with and learn more about these extraordinary numbers. They can be a gateway to interesting and important topics in mathematics, nature and computing.

Build Your Own Continued Fraction

Back to Top

To convert a real number (n) to a continued fraction is easy!

Step 1: Take the integer part of the number - int[n] or Math.floor(n): This is your first index (call it \(a_0 \)), and first convergent.

Step 2: Subtract the integer part to leave the decimal part - the tail. \( Tail = n - int(n) \)

Step 3: Calculate the reciprocal of the tail (\( 1 \over Tail \)) - this should give a number greater than or equal to 1.

Step 4: Take the integer part of this reciprocal value (\(a_1 = int({ 1 \over Tail }) \)): This is your next index - call it \(a_1 \), and \({a_0 + {1 \over a_1}} \) the next convergent.

Step 5: Return to Step 2 and repeat the process...

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!).

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

Back to Top

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?

1. What irrational number is given by the continued fraction [1, 1, 2, 1, 2, 1, 2, ...]?

Jump to model

2. Study the continued fractions for \( \sqrt 2 \) and \( \sqrt 3 \), and then for \( \sqrt 5 \) and \( \sqrt 6 \). What irrational would you predict for [3, 3, 6, 3, 6, 3, 6, ...]?

Jump to model

3. We know that numbers such as \( \sqrt 5 \,, \pi \) and Euler's number \(e\) are irrational, and their decimal representations are infinite and non-recurring - they have no pattern and are unpredictable. However, continued fractions are different - some irrationals (known as quadratic irrationals) have simple recurring forms; some others are known as transcendental numbers, and usually remain elusive and refuse to give up their secrets. Of the three examples given (\( \sqrt 5 \,, \pi \,, e\)), which is least predictable?

Jump to model

4. Both \( \pi \) [3,7,15,1,292,1,1,1,2,1,...] and \( e \) [2,1,2,1,1,4,1,1,6,1,...] are examples of transcendental numbers, but even these strange mathematical creatures may surprise us. \( \pi \) is clearly chaotic, whether in decimal or continued fraction form. But \( e \) seems more well-behaved. What would you predict should be the 15th index (where index #1 is 2, index #2 is 1 and index #3 is 2) of the number known as Euler's number, after the great Swiss mathematician, Leonhard Euler?

Jump to model

How difficult did you find this activity? Please assign a rating value from 1 (very easy) to 5 (extremely difficult). Comments? Suggestions? What did you learn from this activity?

Jump to model

My name:

My class:

Teacher Email: