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Imaginary Golden Ratio

Original question on math.stackexchange by MathTrain

While playing with the results of defining a new operation, I came across a number of interesting properties with little literature surrounding it; the link to my original post is here:

Finding properties of operation defined by x⊕y=1/(1/x + 1/y)? ("Reciprocal addition" common for parallel resistors)

and as you can see, the operation of interest is x⊕y = 1/(1x+1y) = (xy)/(x+y).

In wanting to find a condition such that x⊕y = x−y, I found that the ratio between x and y mus be φ=1.618... the golden ratio, for this to work!

x⊕y = x−y

1/(1/x + 1/y) = x−y

(xy)/(x+y) = x−y

xy = x^2 − y^2

0 = x^2 − xy − y^2

and, using the quadratic formula,

x = (y ± √(y^2 + 4y^2)) / 2

x = y((1±√5)/2)

x = φy

This result is amazing in and of itself. Yet through the same basic setup, we find a new ratio pops out if we try x⊕y = x+y and it is complex.

x⊕y = x+y

1/(1/x + 1/y) = x+y

(xy)/(x+y) = x+y

xy = x^2 + 2xy + y^2

0 = x^2 + xy + y^2

x = (−y ± √(y^2−4y^2))/2

x=y((1±√−3)/2)

x= y((1±(√3)i)/2)

and this is the "imaginary golden ratio"!

φ_i = (1+(√3)i)/2

It has many properties of the golden ratio, mirrored. This forum from 2011 is the only literature I could dig up on it, and it explains most of the properties I also found and more.

http://mymathforum.com/number-theory/17605-imaginary-golden-ratio.html

This number is extremely cool, because its mathematical properties mirror φ but also have their own coolness.

φ_i = 1 − (1/φ_i)

φ^2_i = (φ_i) − 1

and generally

φ^n_i = φ^(n−1)_i − φ^(n−2)_i

This complex ratio also lies on the unit circle in the complex plane, and has a representation as a power of e!

φ_i = cos(π/3) + isin(π/3) = e^(iπ/3)

|φ_i| = 1

It is also a nested radical, because of the identity φ2i+1=φ_i

φ_i = √(−1 + √(−1 + √(−1 + √(−1 + ...))))

Since the only other forum which I could find that has acknowledged the existence of the imaginary golden ratio (other than the context of it as a special case imaginary power of e) I’d like to share my findings and ask if anybody has heard of this ratio before, and if anybody could offer more fine tuned ideas or explorations into the properties of this number. One specific qustion I have involves its supposed connection (according to the 2011 forum) to the sequence

f_n = f_(n−1) − f_(n−2)

f_0 = 0

f_1 = 1

0, 1, 1, 0, −1, −1, 0, 1, 1, ...

could somebody explain to me how this sequence is connected to φ_i? The forum states there is a connection, but I can’t figure out what it is based on the wording. What am I missing?

Thanks for your help with my question/exploration.

Archivist’s note

Though I’ve seen multiple people talking about this “imaginary golden ratio” in many places across the internet (StackExchange, the forum post linked above, and Reddit, to name a few), no one seems to have made the observation that the number (1+(√3)i)/2 is, in fact, one of the cube roots of −1.