
Can the golden ratio accurately be expressed in terms of e and π? - peter_d_sherman
https://math.stackexchange.com/questions/454333/can-the-golden-ratio-accurately-be-expressed-in-terms-of-e-and-pi
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peter_d_sherman
This potential identity looks interesting:

phi = e^((i(pi))/5) + e^((-i(pi))/5)

(From user A. Rex on the referenced Math StackExchange web page...)

I don't have a calculator with enough precision, or a mathematical proof to
know if this potential identity is exactly correct however... any comments?

Also, are there any simpler identities of phi, in terms of e only, but without
i or pi?

Or maybe an identity of e, in terms of phi... I'm particularly interested in
the relationship between phi and e, or vice-versa...

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qubex
If it’s not algebricaly proven don’t call it an ‘ _identity_ ’... even if a
calculator were to confirm it to some arbitrary number of digits, that’d just
be a more-or-less accurate approximation. ‘Identity’ denotes a far more
profound relationship and the term should be reserved for use in such cases.

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peter_d_sherman
That's what I'm trying to ascertain... is this an identity or not? (Note, I
changed my comment to 'potential identity' rather than 'identity').

That, and are there any simple(r) identities for phi in terms of e, and e in
terms of phi...?

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qubex
As somebody in that thread points out, using division you can trivially
construct an identity by taking the definition of φ and replacing any integer
with expressions such as π÷π for 1 and (e+e+e+e+e)/e for 5.

