The Fibonacci sequence is one of the most well known sequences of numbers: 1,1,2,3,5,8... Most people also know how to generate it: starting with 1 and 1, take the the last two terms and add them. Starting out, you would go 1+1=2, 1+2=3, 2+3=5, as far out as you wanted to go. But how, for example, would you find the 1000th Fibonacci number. You could do the iterative 'add the last two terms' approach 1000 times, but that would be tedious and full of errors. Well, as it turns out, there is actually a better way: Binet's formula.
Binet's formula is where ϕ (phi) is (approximately 1.618), and n is the index of the Fibonacci number you want to find (n=1 is the first, n=2 is the second, etc.). The result is the nth Fibonacci number.
Try it out! Plugging in 1 for n gives 1, and plugging in 6 gives 8, as expected. But this formula also raises more questions. What happens when you ask for the 1.5th Fibonacci number? You might notice that in Binet's formula, n is a power, and when raising a number to a fractional power, you actually end up taking a root (ie. ). But in the second part of the formula, , the base of the root is a negative number. That raises another question- what exactly is the root of a negative number?
I'm assuming most of the people here know what imaginary numbers are, but if you don't, Khan Academy's videos on them explain them much better than I (ha ha) could.
So, going back to our last question, what happens when you ask for the 1.5th Fibonacci number. Well plugging in 1.5 to the formula gives us 0.920-0.217i.And we can plot that point on the complex plane. What if we keep doing that, plugging in numbers into Binet's formula, then plotting them on the complex plane. If we do, a really interesting shape emerges:
These kinds of beautiful formulas and visualizations are what I love most about math. It's really amazing that the function "knows" that it needs to go through 1 twice and does that interesting spiral. I really enjoyed learning about Binet's formula and how you can get complex Fibonacci numbers, and I hope someone else out there thinks these are neat as well.
Here is a project I made that graphs Binet's formula in the complex plane.
Sorry for the wall of text...
Things get even more interesting when you plug in complex numbers to Binet's formula...