TITLE: Can 1 be expressed with only irrationals in a non-trivial way?
QUESTION [1 upvotes]: There is an equation which I found interesting:
$$1=\phi+\frac{e^{i\pi}}{\phi}$$
where $\phi$ is the golden ratio (either its positive or negative value). Are there other ways to express 1 with only non-rational numbers, besides the obvious examples like $\pi / \pi$ or ${}-e^{i\pi}$?

REPLY [0 votes]: Note that $\phi$ can be defined as the (non-negative) solution to the equation
$$\phi^2 - \phi - 1 =0 \,.$$
Using this relationship, the equation you gave above basically is the same as $1 = -e^{i \pi}$. (Make the substitution for $1$, divide by $\phi$, rearrange terms...)
Consequently I don't understand why you consider one of those two equations to be "obvious" but do not also consider the other "obvious".
In any case, although $1 = -e^{i \pi}$ is well-known, I personally would not at all consider it "obvious". Ultimately all of this comes down to a matter of opinion it seems.
Anyway you could repeat the procedure used above with $\phi$ with many other irrational numbers, combine them with the $1 = -e^{i\pi}$ formula, and get similarly non-obvious relationships.
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