Basic Examples (1) 

10 terms in the Engel expansion of π:

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Scope (8) 

Expand a rational number:

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Compute the original value using the definition of the expansion:

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Engel expansion of the number 1.175:

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First 5 terms of the Pierce expansion of the number 1/√2:

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Sylvester expansion of the rational number 3/19:

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Cantor expansion of the number 384:

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First 5 terms of the Cantor product expansion of the irrational number π:

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Lüroth expansion of the rational number 5/13. It returns an infinite expansion series with periodicity 3:

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Lüroth expansion of the golden ration reciprocal:

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Oppenheim expansion of 1/π. See Details and Options for the extra parameters in the Oppenheim expansion:

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Applications (2) 

Fractions of the form a/3^k are conjectured to always have a finite Lüroth expansion. This can be investigated with the following:

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Whether all terms of the Sylvester series (Sylvester expansion of the number 1) are square-free is an open problem. All known terms till now are square-free. This urges us to further investigate which terms might have all square-free terms. The Sylvester expansions for 1/Pi and 1/GoldenRatio have square terms:

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From this, it might seem numbers greater than or equal to 1 might have all square-free terms. But the Sylvester expansion of ϕ seems to have a square term:

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Properties and Relations (1) 

The resource function FromNumberExpansion is effectively the inverse of NumberExpansion:

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Possible Issues (5) 

Expanding an irrational number requires specifying a finite length:

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Since the function uses arbitrary-precision arithmetic, computing a large number of terms might be very slow depending on the user's hardware, or might run past the limit imposed by $MaxExtraPrecision:

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