\(f\)

\(f(x)\)

\(g\)

\(g(x)\)

\(h\)

\(h(x)\)

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ToolKit Continued
Fractions

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\(a = \)					▶
		-1	5	0
\(b = \)					▶
		-1	5	0
\(c = \)					▶
		-1	5	0
\(d\)				▶
	-1	0	5
\(e\)				▶
	-1	0	5

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Continued Fraction Spreadsheet Explorer

This browser-based spreadsheet uses the handsontable JavaScript library.

 

10/7 43/30 Phi Pi Pi+ e e+

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MathBox Examples

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Some to try...

Cube root of 8 Nerdamer: nthroot Enter \nthroot3[space]8[space] GCD(24,32) Nerdamer: gcd Enter gcd(24,32) Prime factors of 288 Nerdamer: pfactor Enter pfactor(288)

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?

 

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

• Nerdamer: matrix

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

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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. Then just press \(h\) (at the end, you may need to press the TextBoxes button to lay out the result!)

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Explore continued fractions further?

 

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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

 

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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)

 

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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)

 

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Farey/Stern-Brocot and Calkin-Wilf Fraction Trees

Enter tree(19)

About the Farey/Stern-Brocot Continued Fraction

Farey/Stern-Brocot Continued Fraction

Enter sbcf(4)

Farey/Stern-Brocot Continued Fraction Quiz

Enter sbquiz(0)

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Enter your name:	? Abel, Nils Agnesi, Maria Bernoulli, Daniel Cantor, Georg Descartes, Rene Euler, Leonhard Fermat, Pierre Galois, Evariste Germain, Sophie Hypatia Johnson, Katherine Liebniz, Gottfried Lovelace, Ada Monge, Gaspard Newton, Isaac Noether, Emmy Pythagoras Ramanujan, Srinivasa Riemann, Bernhard Sylvester, James Tao, Terence Weierstrass, Karl' Zeno
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