Imaginary gold

This morning Andrew Stacey posted a beautiful identity I’d never seen before relating the golden ratio ϕ and the imaginary unit i:

Here’s a proof:

By De Moivre’s formula,

and so

8 thoughts on “Imaginary gold”

1. Andrew Rasmussen says:

Cool post! I love these identities. I think there are two typos in the proof: exp(ilog(phi)) ought to be exp(log(phi)), and the second to last line ought to be i(phi – 1/phi).

2. John says:

Thanks. I fixed the typos.

3. jonathan says:

That’s just beautiful. Thanks.

4. Steve says:

Very neat.

I’d never heard of De Moivre’s before, and even after reading through the Wikipedia page, wondered why you reference it instead of simply referencing Euler’s identity, from which your second equality follows pretty naturally.

5. Peter Fry says:

I don’t see why (1/phi – phi) should equal -1 ?

What have I missed here?

6. _cronos2 says:

@Peter Fry
Phi is a solution to the equation x^2 – x – 1 = 0, so clearly 1 – phi^2 = -phi

7. John Estes says:

Incredibly elegant. Thanks for sharing. This is amazing.

8. GlennF says:

This is a beautiful identity.. I haven’t seen it before either.

There’s imaginary silver, too. If we introduce:

\phi_n = n + \frac{1}{\phi_n}

So that \phi_1 is the ordinary golden ratio, \phi_2 the “silver ratio”, etc, then:

2 \sin \imath \log \phi_n = n \imath