Complex Fibonacci Numbers?
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as an accessible, visual follow-up when questions touch on extending Binet's formula to negative, rational or complex arguments.
Recommended by 2 people across 2 different months, 2020 to 2025.
What it is
Matt Parker of Stand-up Maths explores what happens when Binet's formula for Fibonacci numbers is fed non-integer and complex inputs, producing curves and surfaces in the complex plane, including a complex zero. The 20-minute video uses GeoGebra visualisations built by Ben Sparks.
What people said
Recommends K.defaoite · 2020-12 · Mathematics Stack Exchange · on “Convention for analytic continuation of Binet formula”
Hi, I have not yet taken a course in complex analysis, but Matt Parker has a rather nice video on this exact topic, though it doesn't contain the technical details you're looking for - I just think it's interesting. Video [this video]
Recommends greg · 2021-12 · Mathematics Stack Exchange · on “Binet's Formula for Negative Fibonacci Sequences”
Yes, the formula is mind-blowing. As a follow up, you should watch this video [this video] at Stand-up Maths YouTube channel.
Cites Niklas Bäck · 2025-12 · Mathematics Stack Exchange · on “How does Binet’s formula work for rational inputs with an odd denominator?”
From watching Matt Parker’s YouTube video [this video] on Complex Fibonacci numbers again today, I was left to wonder about getting real values from rational inputs to Binet’s formula [LINK] . This would have to use powers with odd denominators, because the conjugate of the golden ratio is negative and that would lead to complex numbers. I was able to get a plot with python where using math.cbrt() I took 81st roots and with that plotted powers with the denominator as 81 and I got this graph. For some reason there seems to be two lines. Similar thing happens with 27th roots. $$\text{Plot up to 6}$$ The difference between the two lines seems to decrease. Do other odd denominator powers get the same