Complex Fibonacci Numbers?
Stand-up Maths
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#### Published On Jul 01, 2020

Huge thanks to Jane Street! https://www.janestreet.com/join-jane-street/

Check out Ben Sparks's GeoGebra files.
Binet formula 2D complex output: https://www.geogebra.org/m/twvvzpga
3D imaginary output of Binet formula: https://www.geogebra.org/m/z6dy9cj5
3D plot of absolute output of Binet formula: https://www.geogebra.org/m/pb7hmxyd

My four-part series on Numberphile videos about Fibonacci Numbers (from 2014) starts here.

Here is me going on about the square root of five (Numberphile 2018).

This was the Fibonacci puzzle video from Matt Parker's Maths Puzzles.

https://arxiv.org/pdf/1707.09151.pdf

The zero I found was at -9.14202391817 + 2.80064954276i and you can see the exact form here: https://www.wolframalpha.com/input/?i=%2810*pi%29%2F%28pi+%2B+i*%28log%28sqrt%285%29+-+1%29+-+log%281+%2B+sqrt%285%29%29%29%29

Try it for yourself and put the Binet Formula (((1+sqrt(5))/2)^n - ((1-sqrt(5))/2)^n)/sqrt(5) in the Wolfram roots calculator: https://www.wolframalpha.com/widgets/view.jsp?id=b858339e64fa997454dd12f77cb1ece1

This site has everything you'll ever need to know about Fibonacci Numbers.
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormula.html

Buttercup - The original buttercupchallenge

CORRECTIONS
This was a long video and in hindsight there are a few things I wish I had phrased better. Here they all are:
- I misspoke around 01:13 when I said "negative one, zero" as it is clearly "negative one, one, zero".
- At 07:53 I mean the negative values -5 to 0. I said it a weird way.
- My language at about the 1D input to 2D plot from 09:17 is a bit sloppy. The real values going into the Binet function are not the horizontal axis shown; the plot onscreen is solely the output.
- I say "axis" when I mean "plane" or even "complex plane". The big flat thing.
Let me know if you spot anything else!

Thanks again, as always, for Jane Street being my principal sponsor.
https://www.janestreet.com/

Thanks to my Patreon supports who do support these videos and make them possible. Here is a random subset:

Loren Thomas
Richard Dickins
Barry Salter
Susan Moury
Sarah Gerweck
Ulrich Kempken
Piotr
Gary Martin
Euler
Daniel DeJarnatt

Support my channel and I can make more videos:
https://www.patreon.com/standupmaths

Filming and editing by Matt Parker
Music by Howard Carter (excluding Buttercup)
Design by Simon Wright and Adam Robinson

MATT PARKER: Stand-up Mathematician
Website: http://standupmaths.com/
US book: https://www.penguinrandomhouse.com/books/610964/humble-pi-by-matt-parker/
UK book: https://mathsgear.co.uk/products/5b9fa76f230ffa140094dc43
Nerdy maths toys: http://mathsgear.co.uk/