# Binet's Formula and Extensions of the Fibonacci Numbers

**Binetâ€™s Formula:** \(F_{n} = \frac{1}{\sqrt{5}} \left(\left(\frac{1 + \sqrt{5}}{2}\right) ^{n} - \left(\frac{1 - \sqrt{5}}{2}\right)^{n}\right)\)

where \(F_{n}\) is the \(n^{th}\) Fibonacci number (the recursive definition for the Fibonacci numbers is \(F_{n} = F_{n-1} + F_{n-2}\) where \(F_{1} = 1\) and \(F_{2} = 1\). We can also include \(F_{0} = 0\))

Typically, Binetâ€™s formula over \(\mathbb{N}\) gives us \(F_{1} = 1, F_{2} = 1, F_{3} = 2\) â€¦, but what happens when we use Binetâ€™s formula to find the â€ś\(0.5^{th}\) Fibonacci numberâ€ť or the â€ś\(\pi^{th}\) Fibonacci numberâ€ť (if they even exist)? Well, if we try to find \(F_{\pi}\), what we end up with is roughly \(2.11702 + 0.04244i\). We end up with complex numbers because trying to find \(F_{n}\) where \(n \not\in \mathbb{N}\) leads to complex outputs. So, letâ€™s take a look at the outputs of Binetâ€™s formula over some continuous, real domain (e.g. \(\left[0, 5\right]\)).

Notice that the only places where Binetâ€™s formula has real outputs on this interval are at the natural numbers, where the outputs are the typical Fibonacci numbers. What about the â€śnegative Fibonacci numbersâ€ť? Letâ€™s see what the outputs of Binetâ€™s formula look like on the interval \(\left[-5, 0\right]\).

We end up with \(F_{-1} = 1, F_{-2} = -1, F_{-3} = 2, F_{-4} = -3\) â€¦ This large spiral thatâ€™s travelling around the complex plane actually intersects the real line at the usual Fibonacci numbers with alternating signs! There is actually a generalization of the typical recurrence relation that allows us to have negative values for \(n\):

\[F_{-n} = \left(-1\right)^{n+1}F_{n}\]Extending discrete mathematical structures, such as the Fibonacci sequence, to have continuous properties often leads to interesting results. In this example, we saw how Binetâ€™s formula allows us to find complex and negative â€śFibonacci numbersâ€ť. The field of math that seeks to solve discrete problems about integers using tools from analysis is known as *analytic number theory*, and it has provided number theorists with other interesting results, such as bounds for the prime counting function and solutions to Diophantine equations.