[MUSIC] I'm standing next to a statue of Fibonacci, created in 1863 by Giovanni Paganucci, a sculptor in Florence where the work was commissioned. At the time, Italy was trying to unify as a country, and the government wanted to promote Italian culture. And of course Fibonacci was considered a great son of Italy, being one of the, or maybe the best mathematician of the middle ages. The statue of Fibonacci here was made in Florence but was actually placed in an ancient cemetery in Pisa where Fibonacci was born. It's interesting that the likeness of Fibonacci in this statue and also in his iconic portrait probably looks nothing like Fibonacci, since no true drawings of him exist from 850 years ago. But nevertheless, Italy still honors him with this sculpture. Now, in this video I want to return to the Fibonacci numbers and Fibonacci recursion relation. And show you, how they are related to the golden ratio. So let's do some mathematics. So what is the recursion relation? We have the n + 1 Fibonacci number is equal to the n Fibonacci number + F n-1 Fibonacci number. If I want to show you what the relationship is between the Fibonacci numbers and the golden ratio, we need to look at ratios so we can divide this equation by F sub n. And we'll get F sub n + 1 divided by F sub n = F sub n divided by F sub n is 1 + F sub n- 1 divided by F sub n. This equation by itself is difficult to solve. But we can look at the limit of this equation when n becomes large, when n goes to infinity. So what we need to do is we need to make an assumption here. That the limit, we say the limit, as n goes to infinity, when n is large, of the ratio of two consecutive Fibonacci numbers. F sub n +1 over F sub n, we say that if we make the assumption that, that limit exists, we can call that limit alpha, okay? With that assumption, we can take the limit of this equation, so, we can take this equation and consider the limit. As n goes to infinity. If we do that, we end up with alpha equals 1+ and then this is also the ratio of consecutive Fibonacci numbers. But it's the smaller one over the larger one. So in the limit the smaller one over the larger one, will be equal to 1 over the larger one over the smaller one. So this will become 1 over Alpha. So alpha = 1 + 1 over alpha. This was just the equation that we derived for the golden ratio. Capital phi = 1 + 1 over capital phi, right? 1 over capital phi is the golden ratio conjugate. So what we have here is that this ratio is equal to the golden ratio, okay? So this one here is the golden ratio. So the limit of two consecutive Fibonacci numbers, when the numbers go to infinity is equal to the golden ratio. That's the key relationship between the Fibonacci numbers and the golden ratio. We can look at some numerical values. Here I list n from 1 to 10, and we look at the n plus 1 Fibonacci number divided by the nth Fibonacci number. 1 / 1, 2 / 1, 3 / 2, 5 / 3, 8 / 5, 13 / 8, etc. We can look at the numerical value, 1, 2, 1.5, 1.66, 1.6, 1.625, 1.6154, 619, 617, 618. We're getting convergence, very slow convergence to the golden ratio. Remember the golden ratio is 1.618 something. The convergence is going positive, negative, positive, negative on both sides of the Golden Ratio. We can look at the difference between the ratio of the two consecutive Fibonacci numbers and the Golden Ratio. It's negative then positive, negative, positive, negative, positive, negative, positive, negative, positive. And these numbers are getting closer and closer to zero. So, the ratio of consecutive Fibonacci numbers is eventually converging to the Golden Ratio. That's the clearest connection we have between the Fibonacci numbers and the Golden Ratio. In the next lecture, I can actually show you what are the Fibonacci numbers? We can find a formula for the Fibonacci numbers that contains the Golden Ratio. That will be a very interesting formula called Binet's formula. I'll see you in the next lecture.