On the Fibonacci Numbers and Nature page we saw that the ratio of two neighbouring Fibonacci numbers soon settled down to a particular value near 16:

In fact, the exact value is Phi and, the larger the two Fibonacci numbers, the closer their ratio is to Phi. Why? Here we show how this happens.

The basic Fibonacci relationship is

The graph shows that the ratio F(i+1)/F(i) seems to get closer and closer to a particular value, which for now we will call X.

If we take three neighbouring Fibonacci numbers, F(i), F(i+1) and F(i+2) then, for very large values of i, the ratio of F(i) and F(i+1) will be almost the same as F(i+1) and F(i+2), so let's see what happens if both of these are the same value: X.

But, using the The Fibonacci relationship we can replace F(i+2) by F(i+1)+F(i) and then simplify the resulting fraction a bit, as follows:

So, putting in this new format of F(i+2)/F(i+1) back into the equation for X, we have:

But the last fraction is just 1 + 1/X, so now we have an equation purely in terms of X:

Multiplying both sides by X gives:

But we have seen this equation before in A simple definition of Phi so know that X is, indeed, exactly Phi!

Remember, this supposed that the ratio of two pairs of neighbours in the Fibonacci series was the same value. This only happens "in the limit" as mathematicians say. So what happens is that, as the series progresses, the ratios get closer and closer to this limiting value, or, in other words, the ratios get closer and closer to Phi the further down the series that we go.

But there are two values that satisfy X2 = X + 1 aren't there?

Yes, there are. The other value, -phi which is -0 618... is revealed if we extend the Fibonacci series backwards. We still maintain the same Fibonacci relationship but we can find numbers before 0 and still keep this relationship:

i |
-10 |
-9 |
-8 |
-7 |
-6 |
-5 |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 | ||

Fib(i) |
-55 |
34 |
-21 |
13 |
-8 |
5 |
-3 |
2 |
-1 |
1 |
0 |
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 |

When we use this complete Fibonacci series and plot the ratios F(i)/F(i-1) we see that the ratios on the left-hand side of 0 are

— = -1 — = -0.5 — = -0.666 — = -0.6 — = -0.625, ... -1 2 -3 5 -8

Plotting these shows both solutions to X2 = X + 1:-

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