Fractured Fractions

 

This document requires an HTML5-compliant browser. a = b = \[37/15\] \[ = [2,2,7]\] Settings Show all g(x) h(x)   f(x) g(x) h(x) a b App generated by Geometry Expressions

37/15 \[37/15 = \]\[[2,2,7]\]\[= 2 + \frac{1}{2 + \frac{1}{7}}\]\[= \frac{37}{15}\]\[= 2.466666667\] Continued Fraction RESET f(x) Table Play ♬     CAS Use the MathBox to enter expressions for graphing, tables and CAS: Define functions, simplify expressions, solve equations... For example, enter Solve(x^2-x-1=0) and press the CAS button. More?

Back to Top

Explore Further

Jump to GXWeb

 

---

---

 

Assessment

Show

Back to Top

Jump to Model

Jump to GXWeb

 

Feedback ✅ : Set Feedback OFF

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?

 

---

  37/15 = [2,2,7] = 37/15 = 2.466666667  

---

1. Try (34, 21), (55, 34) and (89, 55). Notice anything? What would be next?

Set my result for Question 1

 

Jump to model

Jump to GXWeb

 

2. If the continued fraction [1,2] is equivalent to 3/2, and [1,2,3] is 10/7, then what is the simplest fraction for [1,2,3,4]?

Set my result for Question 2

 

Jump to model

Jump to GXWeb

 

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

Set my result for Question 3

 

Jump to model

Jump to GXWeb

 

4. 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, ...]?

Set my result for Question 4

 

Jump to model

Jump to GXWeb

 

5. 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?

Set my result for Question 5

 

Jump to model

Jump to GXWeb

 

6. 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?

Set my result for Question 6

 

Jump to model

Jump to GXWeb

 

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?

Enter my response

 

Jump to model

Jump to GXWeb

 

---

  

My name: Enter your name:?Abel, NilsAgnesi, MariaBernoulli, DanielCantor, GeorgDescartes, ReneEuler, LeonhardFermat, PierreGalois, EvaristeGermain, SophieHypatiaJohnson, KatherineLiebniz, GottfriedLovelace, AdaMonge, GaspardNewton, IsaacNoether, EmmyPythagorasRamanujan, SrinivasaRiemann, BernhardSylvester, JamesTao, TerenceWeierstrass, KarlZeno My class: Teacher Email:

Email page Send to Web

Back to Top

Back to Model

 

---

  Show

Share using QR Codes

Back to Top

Jump to Model

Jump to GXWeb

Geometry Expressions Showcase

QR Codes are a great way to share data and information with others, even when no Internet connection is available. Most modern devices either come equipped with QR readers in-built, or freely available. The default link here is the Geometry Expressions Showcase, but you can use it as an alternative to sending your assessment data via email, web or Google Cloud. You can even use it to send your own messages! Share Your Continued Fraction Send Your Own QR Code Reset QR Code

 

---

---

  Show

More to Explore

Back to Top

Jump to Model

Try (34, 21), (55, 34) and (89, 55). Notice anything? What would be next?

What about something easy? (22, 7) and maybe (333, 106)

Set your decimal places to 1 and try (35.5,11.3) - look familiar?

What do you notice about the continued fractions of numbers of the form \(\frac{\sqrt{5}+2n-1}{2}\)? (These are called Noble Numbers).

Some more interesting Noble Numbers may be formed using tau (\(\tau: \frac{1}{\phi}\)), the reciprocal of phi (\(\phi\)), the Golden Ratio: For example, \(\frac{\tau}{1+3\tau}\), but what about \(\frac{2+5\tau}{5+12\tau}\)?

Seems like Farey neighbours make lovely continued fractions!

For interest, you might also want to check out the Metallic means: \(\frac{(n+\sqrt{4+n^2})}{2}\) (Golden, silver, bronze...), and while the continued fraction of Euler's number, \(e\) is interesting in itself, you should check out \(\frac{e^2+1}{e^2-1}\)!

Back to Top

Jump to Model

 

---

  Show

Extension 1: Bessel Family of Functions

Studying the last continued fraction featured (\(\frac{e^2+1}{e^2-1} = [1, 3, 5, 7, 9, 11...]\)) got me thinking about a continued fraction which should be even simpler: \([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\). Surely this should be given by a common and recognisable real number? Well, how wrong I was!

As we see by evaluating the continued fraction for the counting numbers, the real equivalent approaches something close to \(1.433127426722315\), but repeated Googling proved fruitless, until I discovered the wonderful OnLine Encyclopedia of Integer Sequences (OEIS).

The OEIS informed me that this most fundamental of continued fractions is defined, not by any simple irrational, but the ratio of two forms of a function family with important applications in Physics, the first two forms of the modified Bessel function of order 2, \(BesselI(0,2)\) and \(BesselI(1,2)\). A table of values may help to see the values of the ratio of the modified Bessel Function \(\frac{BesselI(0,2)}{BesselI(1,2)} = \frac{2.2795698924731185}{1.5906040268456376} \).

By the way, impress your friends at parties by reeling off the continued fraction forms for \(\frac{3}{2}\), \(\frac{10}{7}\), \(\frac{43}{30}\), \(\frac{225}{157}\) and \(\frac{1393}{972}\)!

Why not try for the record: \(\frac{9976}{6961}\), \(\frac{81201}{56660}\) and even \(\frac{7489051}{5225670}\)?

\[ x^2 \cdot \frac{d^2y}{dx^2} + x \cdot \frac{dy}{dx} + (x^2-v^2)\cdot y = 0\]

Bessel functions have been studied for hundreds of years, describing physics concepts from planetary perturbations to thermodynamics to oscillations, swinging chains and more! Graphs of the four principal types of Bessel functions (J, K, I and Y) reveal the varied motions they describe.

Just why one of these functions lies at the heart of that most elegant of continued fractions, however, remains a mystery - at least to me. Perhaps someone, some day might be able explain it...?

Back to Top

Jump to Model

 

---

  Show

Extension 2: Gamma Function

Interestingly, though, in delving into these curious matters, I discovered that underlying these very high level functions may be found one that is at least a little easier to get our heads around: the Gamma function.

The factorial function is well known - \(3! = 3 \cdot 2 \cdot 1 = 6, 4! = 4 \cdot 3 \cdot 2 \cdot 1 = 12, n! = n \cdot (n-1) \cdot (n-2)...\).

But what happens in between? The Gamma function, represented by the Greek capital, \(\Gamma(x)=(x-1)!\) includes all values, not just the integers, and has some interesting properties.

\[ \Gamma(z) = \int_0^{\infty}x^{z-1} \cdot e^{-x} dx, R(z) > 0\]

Consider in particular the "half values": \(\Gamma(\frac{1}{2})=\sqrt(\pi)\) and \(\Gamma(\frac{3}{2})=\frac{\sqrt(\pi)}{2}\).

What about \(\Gamma(\frac{5}{2})\) and \(\Gamma(\frac{7}{2})\)? You might apply a little CAS.

You could also use continued fractions to explore these special numbers: try \(\frac{\Gamma(\frac{5}{2})}{\sqrt(\pi)}\) and \(\frac{\Gamma(\frac{7}{2})}{\sqrt(\pi)}\).

Another path for exploration might involve a table of values.

Can you find the pattern, and predict the next few values in this sequence?

Back to Top

Jump to Model

 

---

  Show

Extension 3: Chaos Theory

If these functions have caught your interest, then you might be tempted down the path of chaos theory - first through recursive functions, and then through bifurcation and the logistic map!

Consider a population, say of fish in a pond. If the pond is fixed in size and limited in the amount of food which it can provide, then the population of fish cannot grow unbounded. In fact, the size of the population itself will limit the growth - as the number of fish (\(x\)) gets large, it will act to slow down the rate of population growth (\(r\)). A simple model of this situation over time is given by the relationship: \[f(x)=r\cdot x\cdot (1-x)\]

In this simple model, the values of \(x\) may range between 0 and 1, and \(r\) is the growth rate of the population. This is an example of the logistic equation for population growth, a useful model for studying real-world populations. (Notice that as \(x\) approaches the upper value of 1, this will cause the population to decrease).

The points where the graph of this function meets the identity function (\(I(x) = x\)) are called equilibrium points and these indicate points at which there will be zero population growth in a year. If the initial population is \(x\), then the population after one year is \(f(x)\). After two years, \(f(f(x))\), and so on. If \(f(x) = x\), then \(f(f(x)) = x\) and so on. For example, when \(r\) = 4 we would solve the equation \(4 \cdot x \cdot (1-x)=x\) to give two equilibrium points, 0 and 0.75.

More interesting is to consider the behaviour (or orbit) of a point which is NOT an equilibrium point. For varying values of (r), the behaviour of such orbits is quite different. (The initial population actually has little effect in the medium to long term - however, the simple cobweb model below allows you to explore the immediate effects of both initial value and growth rate. NOTE that you can use the initial value slider to estimate the equilibrium value for different growth rates.)

The value of \(r\) has great impact upon the population growth. For small values of \(r\) (< 3) the population quickly settles to equilibrium. Between 3 and 4, the growth moves from stability to instability - in fact, to chaos!

The two graphs shown here give capture this process in different ways: the first (called a cobweb graph) shows the orbit of a point over multiple iterations, and the second (a bifurcation map) beautifully illustrates the change known as period doubling.

Population Growth Rate r =

Start value =

Iterations (years) =

Bifurcation Map © Jeremy Likness

One of the characteristics of chaotic behaviour in complex systems appears to be a feature known as period doubling. When \(r\) = 3, you should observe that the population oscillates between two stable values - this is an example of a period 2 cycle. By the time \(r\) = 3.5, however, there is clearly a change - in fact, we have moved to a period 4 cycle (hence the term period doubling).

It appears that for values of \(r\) between 3 and 3.43... the cycle has period 2.

For 3.43... < \(r\) < 3.53... the cycle is of period 4.

By around 3.83... all cycles of the form \(2^n\) appear to have been exhausted, and 3-cycles take over. These occur even more rapidly and by the time \(r\) = 4 the system has reached a chaotic state.

But what are these mysterious values which trigger such changes? Perhaps you might explore using the tools available here: continued fractions, CAS, graphs and tables of values? Follow a little further down this rabbit hole by investigating the Feigenbaum Constants, the first of which \(\delta \approx\) \(4.669201609102990671853203820466\) is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map - just like the one considered here!

Perhaps at this point you are wondering what chaos has to do with continued fractions? You have seen that, if we exclude rationals, quadratic irrationals and the odd transcendental number (like \(e\)) then all continued fractions are inherently chaotic in their patterns.

But it goes deeper than this. Think backto the process of building a continued fraction. After extracting the whole number part, the next two steps are to take the reciprocal of the tail, and subtract the floor of this reciprocal (the whole number part once again. This may be thought of as a function: \(\frac{1}{x}-floor(\frac{1}{x})\). When viewed graphically what we observe is an infinite number of random hyperbolic branches - a perfectly chaotic function lies at the very heart of every continued fraction!

These are the elements of chaos, a field at the cutting edge of mathematics and human knowledge. There is much yet to be learned about its properties and applications, and yet it is accessible through such simple models as those studied here. Be patient and systematic and you may discover something new and unexpected!

Back to Top

Jump to Model

 

---

  Show

Extension 4: Why Not Listen to your Continued Fraction?

Suppose you had a list of numbers, for example, myList = [0, 2, 4, 5, 7, 9, 11, 12]. How might you turn each of those list elements into a musical tone - perhaps have 0 represent middle C, and each unit a semi-tone higher. 1 would be C#4, 2 D, and so on. Then myList should give the scale from middle C (C4) to the next C (C5).

The modern musical scale is based upon a geometric sequence - each successive note is produced my multiplying by the twelfth root of 2 (\( \sqrt [12] 2 \) or \( 2^{1 \over 12} \) - approximately 1.059463). Since each tone (k) in an octave of 12 notes (including sharps and flats) is twice the frequency of the corresponding note in the previous octave, this frequency will be given by the formula \[ frequency = {440 \cdot 2^{{myList[k] - 9} \over 12}} \]

In this formula, when myList[k] = 0, the tone played gives middle C (or C4), and myList[k] = 9 gives A4 with a frequency of 440 hz.

Think about this and play with the musical tools provided. See if you can find the real number value which gives the most pleasing melody!

Perhaps even write and play your own continued fraction music? Perhaps something more ambitious? You might try dividing a piece into parts and playing in turn with others!

If you would like to know more about the mathematics of music and harmony, play with your own Circle Piano, and visit Mathematics: A Search for Harmony.

Timing: Key Signature:

Play ♬

Switch Views: notes <=> frequencies

Back to Top

Jump to Model

 

---

  Show

Extension 5: Did You Know...?

There is a connection between the Euclidean algorithm for finding the greatest common divisor of two numbers and continued fractions?

Tap the image to explore.

Interested to learn more about continued fractions?

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

• Sacred Geometry: Continued Fractions

• Pianos and Continued Fractions

Back to Top

Jump to Model

 

---

---

 

Construct your own Model with GXWeb

Show

Back to Top

Jump to Model

---

Need help? Tap here for the construction steps

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)))

---

• 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))))))))))))

Back to Top

Jump to Model

 

---

---

Show GeoGebra

Symbolic computations on this page use the GeoGebra CAS engine.

---

©2020 Compass Learning Technologies ← Live Mathematics on the Web ← Geometry Expressions Showcase ← Fractured Fractions

 

---

---

3715​

GGB InputTableRESET

\frac{37}{15} (37)/(15)

Input