Fibonacci numbers part 2

In a previous post, we studied about Fibonacci numbers :
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
(add two consecutive numbers to get the next one).

I noted that those numbers appear in nature in many places, and asked if your child/student know about that.

My opinion is yes, he or she should know. But why? After all, that stuff is not needful in daily life.

I think it's important that youngsters learn a few math topics that show how math appears in nature. It is about math appreciation - (or better yet, appreciation of the Master Mathematician of the Universe...).

Kids learn Art Appreciation - so they can appreciate human works of art... Oh, how much better you can appreciate the "artworks" in nature such as flower petals or seedheads when you understand a little bit of the math behind them!

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Here's another amazing thing about these numbers:
Let's study the RATIOS when you take a Fibonacci number divided by the previous Fibonacci number, and make a list:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, ...

So what's so great about that? you ask. (And your student might ask too when you do this with him.)
It's not so visible when you see the ratios written as fractions, but let's take the decimal expansions of these (and you should have the student or students do this on their own):

1, 2, 1.5, 1.6666..., 1.6, 1.625, 1.615384615..., 1.619047619..., 1.617647059..., 1.618181818...

Do you notice something about that sequence?

It's something special. If you continue calculating the decimal expansions of the ratios, they will keep getting closer and closer to a certain number... they never reach it totally but they keep getting closer and closer and closer each time.

The ratios keep approaching the number (√5 + 1)/2 which is approximately 1.6180339887... if you write out some of its decimal expansion. I couldn't write out all of the decimal expansion because this number in itself is IRRATIONAL and it has the name Phi.

(Oh I will some day write more about irrational numbers; I think the concepts of rational versus irrational numbers is very fascinating - they're kind of elusive and as if they weren't from this world)

So what we did was: take the ratios of a Fibonacci number per the previous one, look at decimal expansions, and notice they keep getting close to something - and I told you (without proving it) that something is Phi, and it's an irrational number and it's exactly (√5 + 1)/2.

See also The Ratio of neighbouring Fibonacci Numbers tends to Phi - this page has a graph and also a proof for this fact.

So far so good... I hope I'm not confusing you. I think I will stop here with pure math and continue some other time - yes, there's still more to come! This Phi is an exciting number in many mathematical senses that you may not know about, but you might have already heard about Phi as the GOLDEN RATIO... (to be continued)