GXWeb Mathematical Explorer

MathBox Examples: Some to try...

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Examples Index

Graphing
Calculus

Numbers
Playing with Primes
Sums

Something Different
- Binary and other bases
- Birthday Buddies
- Bezier Curves,
- Kissing Circles, and
- Dragon Curves, Paperfolds and more...
- LOGO Turtle Graphics

Continued Fractions

Classical and Reversal/Negative Continued Fractions
Linear Fractional Transformations and Continued Fractions
Explore the Magic Table for Continued Fractions
Continued Fraction Arithmetic
Continued Logarithms
Some Matrix Fraction Magic
FractionTrees
Quizzes

Graphing

Examples Index

Define and plot \(ax^2+bx+c\)

Enter ax^2 [space]+bx+c

Complete the Square for \(ax^2+bx+c\)

Nerdamer: sqcomp

Enter sqcomp(ax^2 [space]+bx+c)

Factor \(2x^2-6x+4\)

Nerdamer: Factor

Enter factor(2x^2 [space]-6x+4)

Solve \(x^2=x+1\)

Nerdamer: solve

Enter x^2 [space]=x+1

Graph expressions in \(x\) and \(y\) \((x-2)^2+(y-2)^2=4\)

Enter (x-2)^2 [space]+(y-2)^2[space]=4

Graph parametric expressions in \(t\) \(curve(2t-1,(t-1)^2)\)

Enter curve(2t-1,(t-1)^2)

Calculus

Examples Index

Derivative of \(y= x^2-x-1\)

Nerdamer: diff

Enter d/dx[space](x^2-x-1)
(Be sure to include parentheses around the expression!)

Integral of \(y=x^2-x-1\) between 0 and 3

Nerdamer: Integrate or defint

Enter \int_0[space]^3[space][space][space](x^2[space][space][backspace]-x-1)[space][space]dx
(Be sure to include parentheses around the expression!)

Tangent of \(y= x^2-x-1\) at \(x = a\)

Enter tangent(x^2-x-1) or tangent(a, x^2-x-1)

Normal of \(y= x^2-x-1\) at \(x = a\)

Enter normal(x^2-x-1) or normal(a, x^2-x-1)

Numbers

Examples Index

Cube root of 8

Nerdamer: nthroot
Enter \nthroot3[space]8[space]

GCD(24,32)

Nerdamer: gcd
Enter gcd(24,32)

Sums

Examples Index

Sum of \(({\frac{1}{4}})^x\) from x = 0 to 500
NOTE: This is Archimedes Infinite Sum!

Nerdamer: sum(1/(4^x),x,1,500)

Enter \sum(x=1)[space][space]500[space][space](1/4[space][space]^x)

Theorem of Nicomachus: The sum of the first \(n\) cubes is equal to the SQUARE of the sum of the first \(n\) integers!
\[\sum_{k=1}^{n}{k^3} = (\sum_{k=1}^{n}{k})^2\]
Type \(nico()\) or try \(nico(9)\) for some fun facts about the year 2025!

Playing with Primes

Examples Index

Prime factors of 288

Nerdamer: pfactor
Enter pfactor(288)

Is 289 a prime number?

Enter isprime(289)

You can also simply use willan

Enter willan(289)

Like to know the 100th prime number? Or even the 1000th?

Enter prime()

You might also use Willan's Formula to see if a number is prime - \[isPrime(1117) = \lfloor(cos(\pi \cdot (\frac{(1116)!+1}{1117}))^2\rfloor = 1\]\[isPrime(1119) = \lfloor(cos(\pi \cdot (\frac{(1118)!+1}{1119}))^2\rfloor = 0\]

Willan's Formula returns 1 if the argument is prime and 0 if it is composite!

This allows us to find the nth prime. For example, to find the 5th prime, count from the left:

n 0 1 2 3 4 5 6 7 8 9 10 11
\[\sum_{i=0}^n willan(i)\] 0 0 1 1 0 1 0 1 0 0 0 1
sum ⇒ prime - - 1 2 - 3 - 4 - - - 5

Enter isPrime() OR sum(i=0)[space][space](11) [space][space] (willan(i)) \[\sum_{i=0}^{(11)} (willan(i))\]

Something Different

Examples Index

Convert 110111 base 2
(or other bases) to decimal.

Enter 1100111_2 or b2n(1100111) or bin2num(1100111)

Express 43 in binary form (or other bases).

Enter 43_>2 or n2b(43) or num2bin(43)

For something different, try Birthday Buddies

Enter bbud(23) or bbud(x)

Bezier Curves are also very cool.

Enter bezier(1,2,x) or bezier2(1,3,2,x)
Then try curve(bezier(1,2,t),bezier2(1,3,2,t))

Make sure that you explore the amazing Apollonian Gaskets, the the Arbelos, the Pappus Chain, the Farey Sequence and Ford Circles - and even a Farey Star! (use the (a)-slider to play!). These feature in the beautiful Kissing Circles Collection

Possibly the coolest mathematical creatures may be Dragon Curves and their associated paperfolds.

Enter dragon() or pfold() or just spf() to explore different aspects of dragon curves, paperfold sequences and even summed paperfolding sequences and series!

LOGO (or Turtle Graphics) is an educational programming language designed by Wally Feurzeig, Seymour Papert and Cynthia Solomon in the 1960s. Here we implement the simple LOGO commands using GXWeb. To enter LOGO mode, just enter "turtle" (or "logo") in the MathBox and follow the prompts.

If using the MathBox to enter a series of commands, separate these with a colon (:) rather than the space you would normally use: turtle(repeat:6:fd(5):lt(60)).

Simple commands such as FD (forward), BK (back), LT (left), RT (right), PU (pen up), PD (pen down) and REPEAT are used to drive a "turtle" around the screen and to trace the path it follows. Variables num and angl may serve to create more interesting patterns.

Some shortcuts have also been defined. For example, to draw a polygon, just enter POLY and enter the values as prompted (or, for a pentagon, for example, enter POLY(5,5,72) - or even PENT(5)!). Other commands include SQUARE, TRI, HEX... and even FLOWER and SPI or THEO (for the Spiral of Theodorus).

You might even explore the wonderful Dragon Curve, arising from simple folding of paper - enter DRAGON (or just dragn) when prompted followed by the number of folds!

You may change view settings (show/hide axes, color, line thickness) by preceding your LOGO command with a question mark. For example, enter logo or turtle in the MathBox, then at the prompt, ?theo.

Try a few commands to create different constructions.

For example, what might you add to FD(5) LT(45) FD(5) LT(135) FD(5) to complete the parallelogram?
(Tap for result)

To turn a parallelogram into a star?

Or take a spiral and turn it into something weird?

And then imagine designing, building and even remotely controlling your own robot! It is actually easier than you might think! Alternatively, grab a TI Innovator Robot already made and follow the directions to hack the hub! Then use your LOGO commands to drive your robot.

Continued Fractions

Examples Index

Continued Fraction for \(\sqrt{a^2+b}\)

Nerdamer: continued_fraction

Enter \sqrt(a^2+b[space]

What is special about this continued fraction?

Solve \(x=a+\frac{b}{a+x}\)

Nerdamer: solve

Enter x = a + b/a + x

Notice the connection with the continued fraction above?

And try entering 2,<2,4> and even something like 4,<1,3,1,8> - what do you notice? Try more of your own.

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Examples Index

Did you know that there are TWO classical forms for continued fractions?

The simple or general form is built by taking the floor (or whole number part) of the fraction at each level and adding the reciprocal of the tail.
\[\frac{10}{7} = [1,2,3] = 1 + \cfrac{1}{2 + \cfrac{1}{3}}\]
Enter 10/7 or 1,2,3

The reversal or negative form takes instead the ceiling or next whole number value above, and subtracts the tail!
\[[\frac{10}{7}]^{-} = [2,2,4]^{-} = 2 - \cfrac{1}{2 - \cfrac{1}{4}}\]
Enter [10/7]^- or [2,2,4]^- or [10/7]{neg} or [2,2,4]{neg}

Try a few examples and see if you can see a pattern...

HINT: It helps if your simple continued fraction form has an EVEN number of terms!

Examples Index

The Linear Fractional Transformation of a Continued Fraction
\[ \frac{a\cdot x+b}{c\cdot x + d}\]
explores how a continued fraction \(x\) is transformed by the operation
\[x_1 = \frac{a\cdot x+b}{c\cdot x + d}\]

Enter lft([1,4,0,7],\sqrt(37)) or lft(\frac{[1,4]}{[0,7]},\sqrt(37)))

Can you find any patterns?

Examples Index

Introduction: What is a Magic Table?

Introducing the Magic Table: Dimension 1

Explore the Magic Table in 1 Dimension

Introducing the Magic Table: Dimension 2

Explore the Magic Table in 2 Dimensions

Teasing Out the Resultant Continued Fraction

Step by Step

General Continued Fractions: Some Examples

Try More Examples

Try Some Yourself

Introduction: What is a Magic Table?

To convert a real number to a continued fraction is easy! Try this yourself?

But what if you want to go in the other direction? If you have a continued fraction, and would like to know the real number that it represents?

For some rational numbers (and approximations of irrationals), this is (relatively) easy, since the continued fraction is finite. For most, though, it can be quite arduous.

The most commonly used method involves starting from the bottom and working upwards. This approach, however, does not reveal the successive approximations, the convergents, which in many cases may be just as useful as the final result.

\[ \frac{10}{7}\]\[= 1+\frac{3}{7}\]\[= 1 + \frac{1}{\frac{7}{3}}\]\[= 1 + \cfrac{1}{2 + \cfrac{1}{3}} \]\[= [1,2,3]\]

\[ [1,2,3]\]\[= 1 + \cfrac{1}{2 + \cfrac{1}{3}} \]\[= 1 + \frac{1}{\frac{7}{3}}\]\[= 1+\frac{3}{7} \]\[= \frac{10}{7} \]

Alternatively, you might try the Magic Table method!

The magic table¹^,² (first described by Gosper (1972) - credit for the name goes to Associate Professor Terry Gagen of the University of Sydney) offers an efficient and (relatively) simple way to generate the convergents of a continued fraction - it can even be used, in 1 or 2 dimensions, on linear fractional transformations of continued fractions and, in 3 dimensions, forms the basis for Gosper's algorithm for continued fraction arithmetic.

For example, the continued fraction of
\[\sqrt(37) => \]\[a(n) = [6,12,12,12,12,12,12,...]\]\[b(n) = [1,1,1,1,1,1,1...]\] \[6 + \cfrac{1}{12 + \cfrac{1}{12 + \cfrac{1}{12 + \cfrac{1}{12 + \cfrac{1}{12 + \cfrac{1}{12 + \cfrac{1}{12... }}}}}}}\]
Using the approaches described here is much simpler than, for example, matrix methods to calculate a result such as
\[\frac{\sqrt{37} + 4}{7} = \begin{bmatrix} 1 && 4 \\ 0 && 7 \end{bmatrix} \cdot \sqrt(37) = \begin{bmatrix} 1 && 4 \\ 0 && 7 \end{bmatrix} \cdot \begin{bmatrix} \sqrt(37) && 0 \\ 0 && 1 \end{bmatrix} \]
where the linear fractional transformation is the matrix
\[\begin{bmatrix} 1 && 4 \\ 0 && 7 \end{bmatrix} \ and \ \begin{bmatrix} 1 && 0 \\ 0 && 1 \end{bmatrix}\]
represents the identity matrix.

Magic Table Index

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\[...\] \[12\] \[12\] \[12\] \[12\] \[12\] \[6\] \[a(n)\]
\[...\] \[1\] \[1\] \[1\] \[1\] \[1\] \[(1)\] \[b(n)\]
\[...\] \[213442\] \[17665\] \[1462\] \[121\] \[10\] \[1\] \[4\]
\[...\] \[148183\] \[12264\] \[1015\] \[84\] \[7\] \[0\] \[7\]

Introducing the Magic Table: Dimension 1

Gosper's original (dimension 1) version of the magic table was presented horizontally, reading from right to left. A simplified form (which delivers each convergent in turn) might be represented as shown above.

Note that Gosper dealt only with simple continued fractions for which all b(n) (numerator) terms may be assumed to be 1. We go a step further here, adding a b(n) row to support general continued fractions.

Can you see how the pairs of numbers below the continued fraction terms are calculated? (Notice that the first convergent is calculated directly from the terms of the linear fractional transformation.)

Tap on these pairs to see, or tap on the red continued fraction terms - from right to left - to see Gosper's Magic Table in action!

Magic Table Index

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Introducing the Magic Table: Dimension 2

More importantly, Gosper was not really interested in simply outputting the convergents of continued fractions.

His real goal was to take a continued fraction, operate on it in some way - initially, using a linear fractional transformation, but leading to arithmetic operations with another continued fraction - and output the result directly as a continued fraction: true continued fraction arithmetic!

For this, he needed more dimensions for his magic table!

Teasing Out the Resultant Continued Fraction

It should be noted that drawing out each term of the resultant continued fraction is not always a straightforward operation.

Consider the first TWO convergents of
\[\frac{\sqrt{37} + 4}{7} = \begin{bmatrix} 1 && 4 \\ 0 && 7 \end{bmatrix} \cdot \sqrt(37)\] \[=> \frac{10}{7} \ and \ \frac{121}{84}\]
Taking the floor or integer part of both delivers the same result (\(|\frac{10}{7}| = |\frac{121}{84}| = 1\)), so we can be confident in the first term of our resultant continued fraction.

To continue, we need the dimension 2 table, since we now move one step down and one step to the left (in the table above, or to the right in the table below). The next two convergents to consider are \(\frac{84}{37}\) and \(\frac{1015}{447}\). If the floor values of both agree (which they do in this case), we continue this process, moving down and horizontally forward.

If NOT, then we simply move one step horizontally forward, and try the next pair of convergents, until we find two that agree, and this process gives each subsequent term of our resultant continued fraction.

For a more interesting example, consider Gosper's case of \(e\) with linear fractional transformation, [1,-1,1,1]:
\[tanh(\frac{1}{2}) = \frac{e-1}{e+1}\]
Here we see that the first convergent pair (\(\frac{1}{3}\) and \(\frac{2}{4}\)) both deliver floor values of 0. The second pair (\(\frac{11}{5}\) and \(\frac{4}{2}\)) give us the next resultant term of 2.

Subsequently, though, \(|\frac{5}{1}| = 5\) and \(|\frac{7}{1}| = 7\). We must continue to the next terms, \(\frac{12}{2}\) and (with a little calculation) \(\frac{55}{9}\) (\(4 \cdot 12 + 1 \cdot 7 = 55\) and \(4 \cdot 2 + 1 \cdot 1 = 9\)) to confirm the floor result of 6.

The floor values of pairs of adjacent convergents must differ by no more than 1 to deliver the next term of the resultant continued fraction.

Magic Table Index

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For ease of presentation, in our versions, the terms are displayed vertically, with the terms of the linear fractional transformation found at the top of the numerator and denominator columns (rotated -90°).

Magic Table Index

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Step by Step

Tap to view:

Simple Convergent Form 1

Simple Convergent Form 2

Gosper Magic Table 1

Gosper Magic Table 2

Gosper Magic Table 3

Gosper Magic Table 4

Gosper Magic Table 5

In its simplest form, Gosper's Magic Table initially offers a convenient way to generate the convergents for a given continued fraction, along with a linear fractional transformation, expressed as a 2x2 matrix.
(Tap for an example) ⇑

Magic Table Index

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General Continued Fractions: Some Examples

Note that the Magic Table presented here represents an extension of Gosper's method, supporting general continued fractions in addition to simple ones. For example, did you know that
\[\pi \approx 0 + \cfrac{4}{1 + \cfrac{1}{3 + \cfrac{4}{5 + \cfrac{9}{7 + \cfrac{16}{9 + \cfrac{25}{11 + \cfrac{36}{13 + \cfrac{49}{15 + \cfrac{64}{17 + \cfrac{81}{19...}}}}}}}}}}\]\[a(n) = [0,1,3,5,7,9,11,13,15,17,19]\]\[b(n) = [4,1,4,9,16,25,36,49,64,81]\]

And what about the Farey and Stern-Brocot Continued Fractions: if you wanted to start a list of every rational number, then these amazing continued fractions are perfect for that!

\[0 + \cfrac{1}{3 - \cfrac{1}{1 - \cfrac{1}{3 - \cfrac{1}{1}}}} = 0 + \cfrac{1}{3 + \cfrac{1}{-1 + \cfrac{1}{3 + \cfrac{1}{-1}}}}\] \[⇒ [0,\frac{1}{3},\frac{1}{2},\frac{2}{3},1]\]
Farey Sequence Level 3:
mtable([[0,3,-1,3,-1],[1,1,1,1,1]],[1,0,0,1])
OR

mtable([[0,3,1,3,1],[1,-1,-1,-1,-1]],[1,0,0,1])

\[0 + \cfrac{1}{4 - \cfrac{1}{1 - \cfrac{1}{3 - \cfrac{1}{1 - \cfrac{1}{5 - \cfrac{1}{1 - \cfrac{1}{3 - \cfrac{1}{1}}}}}}}}\] \[⇒ [0,\frac{1}{4}, \frac{1}{3}, \frac{2}{5},\frac{1}{2}, \frac{3}{5}, \frac{3}{4}, \frac{2}{3},1]\]
Farey Sequence Level 4:
mtable([[0,4,-1,3,-1,5,-1,3,-1],[1,1,1,1,1,1,1,1,1]],[1,0,0,1])

OR

mtable([[0,4,1,3,1,5,1,3,1],[1,-1,-1,-1,-1,-1,-1,-1,-1]],[1,0,0,1])

\[0 + \cfrac{1}{3 - \cfrac{1}{1 - \cfrac{1}{3 - \cfrac{1}{1 - \cfrac{1}{5 - \cfrac{1}{1 - \cfrac{1}{3 - \cfrac{1}{1}}}}}}}}\] \[⇒ [0,\frac{1}{3},\frac{1}{2},\frac{2}{3},1,\frac{3}{2},\frac{2}{1}, \frac{3}{1},\frac{1}{0}]\]
Stern-Brocot Sequence Level 3:
mtable([[0,3,-1,3,-1,5,-1,3,-1],[1,1,1,1,1,1,1,1,1]],[1,0,0,1])

OR

mtable([[0,3,1,3,1,5,1,3],[1,-1,-1,-1,-1,-1,-1,-1]],[1,0,0,1])

Stern-Brocot Sequence Level 4:
\[0 + \cfrac{1}{4 - \cfrac{1}{1 - \cfrac{1}{3 - \cfrac{1}{1 - \cfrac{1}{5 - \cfrac{1}{1 - \cfrac{1}{3 - \cfrac{1}{1 - \cfrac{1}{7 - \cfrac{1}{1 - \cfrac{1}{3 - \cfrac{1}{1 - \cfrac{1}{5 - \cfrac{1}{1 - \cfrac{1}{3 - \cfrac{1}{1}}}}}}}}}}}}}}}}\] \[⇒ [0,\frac{1}{4}, \frac{1}{3}, \frac{2}{5},\frac{1}{2}, \frac{3}{5}, \frac{3}{4}, \frac{2}{3},1, \frac{3}{2}, \frac{4}{3}, \frac{5}{3},\frac{2}{1}, \frac{5}{2}, \frac{3}{1}, \frac{4}{1}, \frac{1}{0}]\]

mtable([[0,4,-1,3,-1,5,-1,3,-1,7,-1,3,-1,5,-1,3],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]],[1,0,0,1],0,17)

OR

mtable([[0,4,1,3,1,5,1,3,1,7,1,3,1,5,1,3],[1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1]],[1,0,0,1],0,17)

Magic Table Index

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Try More Examples

Simple (Convergent)
Magic Table
Dimension 1 Gosper
Magic Table
Dimension 2

Enter mtable(), or
mtable(sqrt(37)), or
mtable(sqrt(37),[1,4,0,7]), or even
mtable(sqrt(37),[1,4,0,7],0,12)

Try one of Gosper's examples:
mtable(sqrt(2),[0,2,-1,3],0,12)

You can also enter arrays: mtable([1,2,3,4,5],[1,0,0,1],0,12) or
mtable([[1,2,3,4],[4,3,2,1],[1,0,0,1],0,12)

Try mtable([[2,1,2,3,4,5,6,7,8,9,10],[1,1,2,3,4,5,6,7,8,9,10]],[1,0,0,1],0,12)

Enter gmt(), or
gmt(sqrt(37)), or
gmt(sqrt(37),[1,4,0,7]), or even
gmt(sqrt(37),[1,4,0,7],0,12)

Try one of Gosper's examples:
gmt(sqrt(2),[0,2,-1,3],0,12)

You can also enter arrays: gmt([1,2,3,4,5],[1,0,0,1],0,12) or
gmt([[1,2,3,4],[4,3,2,1],[1,0,0,1],0,12)

Try gmt([[2,1,2,3,4,5,6,7,8,9,10],[1,1,2,3,4,5,6,7,8,9,10]],[1,0,0,1],0,12)

mtable([[0,1,3,5,7,9,11,13,15,17,19],[4,1,4,9,16,25,36,49,64,81,100]],[1,0,0,1],0,12).

Perhaps even compare
mtable([[4,2,2,2,2,2,2,17,294],[-1,-1,-1,-1,-1,-1,-1,-1,-1]],[1,0,0,1]) with
mtable([[4,-2,2,-2,2,-2,2,-17,294],[1,1,1,1,1,1,1,1,1]],[1,0,0,1])?

Magic Table Index

Examples Index

Have you ever considered the idea of Continued Fraction Arithmetic?
Bill Gosper did in 1972: 'Contrary to everybody, this self contained paper will show that continued fractions are not only perfectly amenable to arithmetic, they are amenable to perfect arithmetic.' (Bill Gosper, 1972)
Consider applying
\[ \frac{3\cdot x+4}{2\cdot x - 1} [+] \frac{y+3}{2\cdot y - 1} \ or \ \frac{3\cdot x+4}{2\cdot x - 1}[\cdot ] \frac{y+3}{2\cdot y - 1}\]
to continued fraction forms of \(x\) and \(y\), such as \(x = \sqrt(2)\) and \(y = \sqrt(3)\) - or perhaps start with something simpler, such as \(x[*]y\) with \(x = \frac{10}{7}\) and \(y = \frac{22}{5}\)!

Note that [+], [-], [*] and [/] are custom operations defined for continued fraction arithmetic using either Gosper's original process, or a conjectured matrix process.

Simpler still might be a third approach, computing arithmetic operations on the corresponding convergents of two continued fractions, and so arrive at increasingly accurate results through successive approximation.

Enter [x[*]y],[10/7],[22/5]

sqrt(2)[*]sqrt(3)

x[*]y

[(3x+4)/(2x-1)][*][(y+3)/(2y-1)]

Examples Index

Bill Gosper's historic unpublished 1972 paper not only laid the foundation for realistic continued fraction arithmetic but, in the last few pages, introduced a new "mutation on continued fractions" - (binary) continued logarithms!

In the newly dawning age of personal computers, Gosper recognised both the limitations of continued fractions in dealing with very large and very small numbers, and the possibilities for a new type of binary representation, reducing such numbers to highly accurate arrays of zeros and ones. Gosper described it as a "sort of recursive version of scientific notation" (1972).

While we might generally count ourselves lucky to know of one or two generalized continued fraction forms for most reals, this approach reveals the relatively simple processes which deliver up to five distinct forms for most numbers - in any base of your choosing!

Continued Logarithms (base \(b\)) (Gosper, 1972) are defined by
\[g(x) = \left\{ \begin{array}{ll} \frac{x}{b} & x\geq b \\ \frac{b-1}{x-1} & 1\lt x\lt b \\ terminate & x=1 \\ \end{array} \right. \]
\[19 = [4,2,1,1]_{cl(2)}\] \[⇒ 2^4+\cfrac{2^4}{2^2+\cfrac{2^2}{2^1+\cfrac{2^1}{2^1}}}\]
\[\frac{21}{5} = [1,1,1]_{cl(3)}\] \[⇒ 3^1+\cfrac{2\cdot 3^1}{3^1+\cfrac{2\cdot 3^1}{3^1}}\]

Enter cl() (⇒ cl(input, base, steps, show))

\(cl(\frac{10}{7})\)

\(cl(\frac{10}{7},2)\)

\(cl(\pi,2,10)\)

\[...\]	\[12\]	\[12\]	\[12\]	\[12\]	\[12\]	\[6\]	\[a(n)\]
\[...\]	\[1\]	\[1\]	\[1\]	\[1\]	\[1\]	\[(1)\]	\[b(n)\]
\[...\]	\[213442\]	\[17665\]	\[1462\]	\[121\]	\[10\]	\[1\]	\[4\]
\[...\]	\[148183\]	\[12264\]	\[1015\]	\[84\]	\[7\]	\[0\]	\[7\]

Simple (Convergent) Magic Table Dimension 1	Gosper Magic Table Dimension 2
Enter mtable(), or mtable(sqrt(37)), or mtable(sqrt(37),[1,4,0,7]), or even mtable(sqrt(37),[1,4,0,7],0,12) Try one of Gosper's examples: mtable(sqrt(2),[0,2,-1,3],0,12) You can also enter arrays: mtable([1,2,3,4,5],[1,0,0,1],0,12) or mtable([[1,2,3,4],[4,3,2,1],[1,0,0,1],0,12) Try mtable([[2,1,2,3,4,5,6,7,8,9,10],[1,1,2,3,4,5,6,7,8,9,10]],[1,0,0,1],0,12)	Enter gmt(), or gmt(sqrt(37)), or gmt(sqrt(37),[1,4,0,7]), or even gmt(sqrt(37),[1,4,0,7],0,12) Try one of Gosper's examples: gmt(sqrt(2),[0,2,-1,3],0,12) You can also enter arrays: gmt([1,2,3,4,5],[1,0,0,1],0,12) or gmt([[1,2,3,4],[4,3,2,1],[1,0,0,1],0,12) Try gmt([[2,1,2,3,4,5,6,7,8,9,10],[1,1,2,3,4,5,6,7,8,9,10]],[1,0,0,1],0,12)

Some Matrix Fraction Magic

Examples Index

Start with something simple: Number ⇒ matrix - n2mat()

Enter n2mat(10/7)

Work with matrices \([\frac{[1,4]}{[0,7]}]\cdot \sqrt{37}\)

Nerdamer: matrix

Enter [[1,4]/[0,7][space][space]*\sqrt(37)[space]

Have you ever thought about calculating the square root of a matrix?

Enter \sqrt([[2,1][space]/[2,3]
Square Root of a Matrix: YouTube (3:24)

Explore some matrix magic in the tridiagonal matrix (tdm()) form of continued fractions!

Some expected relationships...

\[1+\cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{4}}}= \]\[f(x) = \begin{bmatrix} 1 & 1 & 0 & 0 \\ -1 & 2 & 1 & 0 \\ 0 & -1 & 3 & 1 \\ 0 & 0 & -1 & 4 \end{bmatrix}\]\[g(x) = inv(\begin{bmatrix} 1 & 1 & 0 & 0 \\ -1 & 2 & 1 & 0 \\ 0 & -1 & 3 & 1 \\ 0 & 0 & -1 & 4 \end{bmatrix})\]\[=\frac{1}{43} \begin{bmatrix} 30 & -13 & 4 & -1 \\ 13 & 13 & -4 & 1 \\ 4 & 4 & 12 & -3 \\ 1 & 1 & 3 & 10 \end{bmatrix}\]\[g(x)\cdot f(x) =\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\]

And even some unexpected ones!

\[[1,2,3,4,5]= \]\[1+\cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{5}}}}= \frac{225}{157}\]\[\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 4 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 5 & 1 \\ 1 & 0 \end{bmatrix} \]\[= \begin{bmatrix} 225 & 43 \\ 157 & 30 \end{bmatrix}\] \[det(\begin{bmatrix} 1 & 1 & 0 & 0 & 0 \\ -1 & 2 & 1 & 0 & 0 \\ 0 & -1 & 3 & 1 & 0 \\ 0 & 0 & -1 & 4 & 1 \\ 0 & 0 & 0 & -1 & 5\end{bmatrix}) = 225\]\[det(\begin{bmatrix} 2 & 1 & 0 & 0 \\ -1 & 3 & 1 & 0 \\ 0 & -1 & 4 & 1 \\ 0 & 0 & -1 & 5\end{bmatrix})=157\]

Tap on the matrix examples above and follow the directions. Press \(h\) for the final result.

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Fraction Trees

Examples Index

Farey/Stern-Brocot and Calkin-Wilf Fraction Trees

Enter trees(19)

Enter tree(19)

Find the Stern-Brocot Tree and Calkin-Wilf paths for any number

Enter path(10/7)

OR Find the rational number for a given Stern-Brocot Tree or Calkin-Wilf path for any number

Enter LLRRR

About the Farey/Stern-Brocot Continued Fraction

Did you know that the Farey/Stern-Brocot Continued Fraction is related to the paperfold sequence and the dragon curve fractal?
Enter dragon(4)

Farey/Stern-Brocot Continued Fraction
Enter sbcf(4)

You could also enter some like the following directly into the MathBox to find the convergents for the Farey/Stern-Brocot Continued Fraction

0,3,-1,3,-1 (Farey Sequence of Order 3)

0,3,-1,3,-1,5,-1,3 (Stern-Brocot Sequence of Order 3)

0,4,-1,3,-1,5,-1,3,-1 (Farey Sequence of Order 4)

0,4,-1,3,-1,5,-1,3,-1,7,-1,3,-1,5,-1,3 (Stern-Brocot Sequence of Order 4)

0,5,-1,3,-1,5,-1,3,-1,7,-1,3,-1,5,-1,3,-1 (Farey Sequence of Order 5)

0,5,-1,3,-1,5,-1,3,-1,7,-1,3,-1,5,-1,3,-1,9,-1,3,-1,5,-1,3,-1,7,-1,3,-1,5,-1,3 (Stern-Brocot Sequence of Order 5)

Can you see the pattern? Can you explain why there is a trailing -1 for the Farey Sequence but not the Stern-Brocot?

Farey/Stern-Brocot Continued Fraction Quiz
Enter sbquiz(0)

Quizzes

Examples Index

Test Yourself: Continued Fraction Quiz: From Number to Continued Fraction: cfQuiz1()

Test Yourself: Continued Fraction Quiz 2: From Continued Fraction to Number: cfQuiz2()

Test Yourself: Continued Fraction Quiz 1 and 2: cfQuiz()

From Fraction to Complete SB Tree - cf2tree()

Test Yourself: Matrix Multiplication Quiz: mquiz()

Test Yourself: Fraction Tree Quiz: From Tree Number to Fraction: ftreeQuiz1()

Test Yourself: Fraction Tree Quiz: From Fraction to Tree Number: ftreeQuiz2()

Test Yourself: Fraction Tree Quiz 1 and 2: ftreeQuiz()

GXWeb Mathematics Explorer

Introduction

Continued Fraction Spreadsheet Explorer

MathBox Examples: Some to try...

Graphing

Calculus

Numbers

Sums

Playing with Primes

Something Different

Continued Fractions

Introduction: What is a Magic Table?

Introducing the Magic Table: Dimension 1

Introducing the Magic Table: Dimension 2

Teasing Out the Resultant Continued Fraction

Step by Step

General Continued Fractions: Some Examples

Try More Examples

Some Matrix Fraction Magic

Fraction Trees

Quizzes

Construct your own Model with GXWeb

Share Your Results

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Behind the Scenes

n	0	1	2	3	4	5	6	7	8	9	10	11
\[\sum_{i=0}^n willan(i)\]	0	0	1	1	0	1	0	1	0	0	0	1
sum ⇒ prime	-	-	1	2	-	3	-	4	-	-	-	5

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