Continued Fraction Explorer

In the MathBox below, try wrapping cf(...) around your fractions, decimals, irrationals - even sums and integrals. This is a great way to calculate both exact values and approximations of rationals and irrationals - but, be warned - you may never look at numbers quite the same way again...

Tap each example below to write to the MathBox, and visualise the Continued Fraction. Then press Input ⇒ CAS to evaluate.

The Golden Ratio \(phi: \phi = \frac{1 + \sqrt{5}}{2} \) and its reciprocal \(tau: \tau = \frac{\sqrt{5}-1}{2} \)

\(Sin(\frac{\pi}{4}) \)

Cube root of m (\(\sqrt[3]{m}\))

Integral of (1/x) from 1 to m \(\int_{1}^{m}\left(\frac{1}{x}\right)dx \): Use the m-slider to study the area under the curve, and then...

Continued Fraction of Integral of (1/x) from 1 to m \(ContinuedFraction\left(\int_{1}^{m}\left(\frac{1}{x}\right)dx\right) \): visualise the continued fraction, and then press Input ⇒ CAS .

Continued Fraction of Integral of nth root of x from 0 to m \(ContinuedFraction\left(\int_0^{m}\left(\sqrt[n]{x}\right)dx\right) \): visualise the continued fraction, and then press Input ⇒ CAS .

The continued fraction of Euler's Number \(e \left(\exp(1)\right)\) is interesting enough, but what about this surprising result from \(\frac{e^2+1}{e^2-1}\)?

Try changing the value of the n-slider to explore the continued fractions of \(\frac{n+\sqrt{4+n^2}}{2}\). These are known as the metallic means (golden, silver, bronze...).

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

Some more interesting Noble Numbers may be formed using tau (\(\tau: \frac{1}{\phi}\)): You should recognise \(\frac{\tau}{1+3\tau}\) but what about \(\frac{2+5\tau}{5+12\tau}\)?

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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.

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Interested to explore continued fractions further?

Build your own continued fractions...

View different representations of continued fractions...

Listen to the music of continued fractions...

Discover the relationship between continued fractions and matrices...

Try the online Continued Fraction assessment task.

Tap the image to explore.

Continued Fraction Explorer

Try These Continued Fractions

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