Try the continued fractions for \(\frac{34}{21}\), \(\frac{55}{34}\) and \(\frac{89}{55}\). Notice anything? What would be next?
This might just begin a search for the most beautiful (and most irrational) of numbers...! And make sure you take a moment to use the a-slider to explore the archimidean spiral along the way:
\[x(t) = sin(t) \cdot (a^{\frac{t}{\pi}})\]
\[y(t) = cos(t) \cdot (a^{\frac{t}{\pi}})\]
What about something easy? \(\frac{22}{7}\) and maybe \(\frac{333}{106}\)?
Set your decimal places to 1 and try \(\frac{35.5}{11.3}\) - look familiar?
Continued fractions are awesome. Every real number, rational and irrational, can be represented by a continued fraction - the rational ones are finite, of course, but both finite and infinite offer some wonderful patterns and opportunities to explore!
Use the f(x) MathBox or the x-value box above to enter a value to express as a continued fraction. You can even share your current continued fraction as a QR Code or use them to encrypt short messages!
For example, try \(43 \over 30\), \(\pi\), \(\phi\) or even \(\frac{225}{157}\).
Use the CAS Evaluate option from the dropdown menu above to solve an expression or equation in the \(f(x)\) MathBox. Click on the following to try:
Solve(x^2=x+1,x)
Solve(x^2=x+1,x),x
Notice the difference between these two results? Adding an extra \(,x\) at the end forces an \(exact\) rather than a \(numeric\) result!
And some more to try (just click on each line):
pFactor(320) (Prime factors)
sqcomp(x^2-4*x+3,x),x (Complete the Square)
nthroot(10,3) (Enter into the MathBox as \ nthroot [space] 3 [space] 10 [space])
Use the Solve option from the dropdown menu above to solve an expression or equation in the \(f(x)\) MathBox, or enter solve(an equation) in \(f(x), g(x)\) or \(h(x)\).
Use the Derivative option from the dropdown menu above to differentiate a function in the \(f(x)\) MathBox, or enter \(d/dx\)(a function) in \(f(x), g(x)\) or \(h(x)\).
Use the Integral option from the dropdown menu above to integrate a function in the \(f(x)\) MathBox, or enter \(int\)(a function) in \(f(x)\).
For definite integrals, perform the integration as described using the dropdown, and enter end values when requested. Try \(\int(x^2-x-1,0,x)\) for example. Use the \(x\) box or slider to vary this.
Display the tangent (or Normal) of \(f(x)\) for the current value of \(x\) using the Tangent/Normal button, or enter tangent (or tngnt) or normal directly into the graph mathBoxes - set the desired x-value first.
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 data via email, web or Google Cloud. You can even use it to send your own messages!
