/sci/ - Science & Math

[Anonymous No.3867055]

Any of you /sci/clones know anything interesting about the Fibonacci sequence or the golden ratio?

Seems like they should have lots of intrinsic properties in mathematics and the natural world..

[Marion Bartoli is my Waifu No.3867084]

<span class="math">x^2 - x - 1 = 0[/spoiler]
<span class="math">x^2 = x+1[/spoiler]
<span class="math">x = 1+\frac{1}{x} [/spoiler]
<div class="math">x = 1 + \frac{1}{1+\frac{1}{1+\cdots}}</div>

[TN5 !/sci/TN5.. No.3867109]

They do. People have been working on that for centuries... I mean, just read about wall sun sun primes, and you'll see how strange things can be with fibonacci. Some class of prime numbers that might exist or not, and whose existence would prove fermat's theorem? I mean wtf...

[Anonymous No.3867133]

f(n) = f(n-1) + f(n-2)
f(0)=0,f(1)=1
f(100)=f(99)+f(98)=2*f(98)+f(97)=3*f(97)+2*f(96)=5*f(96)+3*f(95) ...

f(n) = f(p)*f(q)+f(p-1)*f(q-1), p+q=n+1

[Anonymous No.3867137]

>>3867084
jesus fucking christ what is this

pic related

[Marion Bartoli is my Waifu No.3867203]

>>3867137
<span class="math">\varphi[/spoiler]

[Anonymous No.3867209]

>>3867084
>>3867203

howd you go from step 3 to the last step...

[Anonymous No.3867214]

>>3867209
he replaced x with what x equals, making it recursive.

X = 1 + 1/x, so he replaced x on the right with 1+1/x

[Marion Bartoli is my Waifu No.3867227]

>>3867209
If <span class="math">x=1+\frac{1}{x}[/spoiler] then
<span class="math">x=1+\frac{1}{1+\frac{1}{x}}[/spoiler]
and
<div class="math">x=1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}</div>

Every time you see an <span class="math">x[/spoiler], replace it with <span class="math">1+\frac{1}{x}[/spoiler]

[Anonymous No.3867237]

The most straightforward implementation of function calculating fib(n) takes about fib(n) steps to compute.

[Marion Bartoli is my Waifu No.3867247]

>>3867227
A sidenote:

As this is an infinite continued fraction, its rational approximations (terminating the continuation) come arbitrarily close to the underlying value, in this case the golden ratio.

It also happens to be the *worst* continued fraction around, in terms of the rate of convergence.

As it is a regular (numerators are all 1) repeating (the denominators are repeating terms), this shows that the continued fraction represents some square root (plus a constant). All continued fractions with repeating denominator terms are square roots (possibly plus a constant).

[Anonymous No.3867255]

I heard somewhere, maybe i read a macrothesis on it, that phi is the MOST irrational number.

Is this true?

I know e is the MOST efficient radix for a base.

[TN5 !/sci/TN5.. No.3867269]

>>3867255
Well, the "most irrational number" means nothing, but I'd at least expect it to be transcendental, and it isn't.

[Marion Bartoli is my Waifu No.3867287]

>>3867255
The "most irrational" bit is due to the convergence of the best rational approximations.

[Marion Bartoli is my Waifu No.3867298]

>>3867287
http://www.ams.org/samplings/feature-column/fcarc-irrational1

Good page. The four links at the bottom should be read in that order, they elaborate the idea.

[Anonymous No.3867329]

x = 1x
x = 1 (1x)
x = 1 (1(1x))
So on, and so forth.