Coincidence, Underlying Reason, it doesnt matter, Its pretty Amazing...

[Ragib]

O My God. The Golden Ratio. Its quite an interesting subject. Its defined as:

 

[1+(Sqaure root of 5) ]/2. In words, 1 plus root 5, all over 2.

In symbols, the value is represented by the greek letter, phi.

Lets cut to the chase, I just recently attained a calculator that computes to 5011 digits. The value of the golden ratio is 1.618033988749894848204586834366 to 30 decimal places. Anyway, I noticed that the recipricol, or phi^(-1), is phi - 1. Yes, 0.618033988749894848204586834366. And its not a coincidence to 30 decimal places, wen computed to 5011 decimal places, same result, phi^(-1) =phi - 1. Can anyone explain why or how..or what ever.

[timo]

Anyway, I noticed that the recipricol, or phi^(-1), is phi - 1. Yes, 0.618033988749894848204586834366. And its not a coincidence to 30 decimal places, wen computed to 5011 decimal places, same result, phi^(-1) =phi - 1. Can anyone explain why or how..or what ever.

Just take pen&paper and calculate it out: [math] \left(\frac{1+\sqrt{5}}{2}\right)^{-1} = \frac{2}{1+\sqrt{5}} = \frac{2}{1+\sqrt{5}} \cdot \frac{1-\sqrt{5}}{1-\sqrt{5}} = ... = \frac{1+\sqrt{5}}{2} -1 [/math]

[matt grime]

Or more algebraically it satisfies the equation x^2=x+1, which can be used to demonstrate simply all of the properties that people think are mystical but really are very straight foward. I see no one ever gets as worked up by that fact that 1/1 =1 , which is very odd: surely that is more amazing....

[bascule]

The golden ratio is:

 

[math]\frac{a}{b} = \frac{b}{a+b}[/math]

 

So substitute [math]a = \phi - 1[/math] and [math]b = 1[/math]:

 

[math]\phi-1 = \frac{1}{\phi}[/math]

[Ragib]

O, thank you all so much I really should have thought of that method Athesist showed..also bascule, thats is also quite clever. Matt Grime, please excuse my ignorance, why is 1/1 =1 so amazing..

[Ragib]

OO not only that, but the successive powers of φ obey the Fibonacci recurrence:

 

φ−2 = − φ + 2,

φ−1 = φ − 1,

φ0 = 1,

φ1 = φ,

φ2 = φ + 1,

φ3 = 2φ + 1,

φ4 = 3φ + 2,

φ5 = 5φ + 3,

φn = F(n)φ + F(n − 1),

[the tree]

please excuse my ignorance, why is 1/1 =1 so amazing.

It isn't, that was his point.

 

It certainly is a pretty cool number, I'll give you that.

[matt grime]

Oh, please, just look at the damn recurrence relation before posting more 'isn't this amazing' crap. phi satisfies x^2=x+1 (and you could at least try to learn to typeset a little more clearly your maths [powers etc]), and that's all there is to it. Now what is amazing is why this pattern explicably occurs in sunflowers and so on (owing to convergence of continued fractions). That is an amazing theory: that we have an explanation for this phenomenon, not that it happens.

[the tree]

Urm, Matt, I think that having a career centered around maths has made you forget what it was like when you were younger and everything above the very basics was like some fantastic mystery. Give the kid a break

And do you suggest that an explanation could come about before noting what happens anyway?

[matt grime]

I simply don't see why the fact that if you invert a number you get the same digits after the decimal expansion is 'amazing'. There are infintely many numbers with this property: any roots of 1/x=x+n where n is any integer (-1 in this case), and I'm fed up with all the nonsense that people come up with about the golden ratio at the best of times (classical greek pottery); though it should be pointed out that this is not nonsense. Certainly the Fibonacci sequence occurs here: phi is one root of x^2=x+1 which explains all of those observations made about powers, andthe fibonacci sequence satisfies F(n+2)=F(n+1)+F(n).

 

There are some truly amazing things in mathematics (it is not jaded cynicism on my part), but this isn't one of them: someone writes a computer program to compute phi to 5011 places but doesn't look at the equation that defines phi? It takes 2 seconds with a pen and paper to figure out what's going on here.

 

I truly hope that the OP does continue to investigate mathematics (and I admit I have been too harsh, in retrospect), and I will gladly give up free time to help people's investigations.

 

If the OP wants to investigate it some more, consider the Fibonacci sequence. It is a so called difference equation ( like a differential equation) and just like a (homogeneous) differential equation we can do the following:

 

Suppose that we can write F(n) as a function of n, then what form must it take.

 

Well, experience tells us that in this case we should look for constants a,b,r and s such that

 

F(n)= a(s^n)+b(r^n)

 

Now, can you find r and s? They must satisfy a certain equation given by the recurrence F(n+2)=F(n+1)+F(n), or x^2=x+1....

 

a and b you choose so that F(1)=F(2)=1

 

So you see, phi actually enables you to write the n'th Fibonacci number the sums of n'th powers.

[Ragib]

umm..sorry, i guess i am ignorant, im better at physics than maths...but well, for a 14 year old, i thought i wasn't that bad..i guess i should have researched more before i posted..and about the typing about the powers, in wikipedia it came out properly and i just copied and pasted, i thought it would be the same..

[the tree]

As a rule, don't bother with calculators until you really need to: they cause rounding errors and turn elegant expressions into ugly numbers. As you go through school you will come to apreciate the brilliance of pen and paper and there will be whole terms where you'll find your calculator to be completely useless.

 

You're not going to come up with much to impress Matt, he's been doing this longer than the both of us combined.

[matt grime]

Fortunately we have a routine running in the back ground that lets you typeset mathematics on here.

 

[math] \phi^2=\phi+1[/math]

 

html doesn't do maths very well.

 

Click on a piece of maths to see what the code is. There's a also a tutorial somewhere on the forums.

 

The point is that making qualitative judgements about what is 'pretty amazing' is frequently going to lead to a difference of opinion, especially when it is clear from the definition of phi either as an honest to goodness number or simply the larger root of some polynomial why the properties you noticed holds. But it is good that you figured out it would be Fibonacci numbers appearing.

 

See if you can figure out why the continued fraction (google it) of phi is [1;1,1,....]. This in some sense accounts for the appearance of Fibonacci like patterns in nature since this is in the correct interprettion means that phi has the slowest converging rational approximations.

 

You could also try to figure out what the continued fractions are of all of those numbers that have the same part after the decimal point as their reciprocals (remember they satisfy 1/x=x+n for some integer n, and n=-1 gives phi).