how do they work, does anyone know how to explain it cause i need to do a portfolio and give it in tomorow, if not: death
It is relying on a theorem which is not so easy to show that all continued fractions converge (the ones with integers).
So we know that $\displaystyle x=[1;1,1,...,]$
Converges to some number.
But,
$\displaystyle x=[1;x]$
For it repeats.
Thus,
$\displaystyle x=1+\frac{1}{x}$
Thus,
$\displaystyle x^2=x+1$
Thus,
$\displaystyle x^2-x-1=0$
Thus,
$\displaystyle x=\frac{1\pm \sqrt{5}}{2}$
This tells us that this is one of these two possible values.
It cannot be negative thus,
$\displaystyle x=\frac{1+\sqrt{5}}{2}=\psi$
The Divine Proportion (My body is shaped in Divine Proportion).
Hello, Cilia!
This is a vast and intricate topic which could take months to explain.
If you know the very basics, I can show you a tiny sliver of the whole sprectrum.
Your example:
$\displaystyle t_1 \:=\:1 + 1\qquad t_2 \:=\:1 + \frac{1}{1+1}\qquad t_3 \:=\:1 + \frac{1}{1 + \frac{1}{1+1}} $
By inspection ("eyeballing" it),
. . we see that each term is: one plue one over the preceding term.
That is: .$\displaystyle t_{n+1} \;= \; 1 + \frac{1}{t_n} $
Assuming that the fraction goes on forever: .$\displaystyle 1 + \frac{1}{1 + \frac{1}{1+\frac{1}{1 + ...}}}$
. . can we determine its value?
$\displaystyle \text{Let }x\:=\:1 + \frac{1}{1 + \frac{1}{1+\frac{1}{1 + ...}}}
\begin{array}{ccc} \\ \\ \bigg\}\text{ this is }x \end{array}
$
Hence, we have: .$\displaystyle x \:=\:1 + \frac{1}{x}\quad\Rightarrow\quad x^2 - x - 1 \:=\:0$
Quadratic Formula: .$\displaystyle x \:=\:\frac{\text{-}(\text{-}1) \pm \sqrt{(\text{-}1)^2 - 4(1)(\text{-}1)}}{2(1)} \;=\;\frac{1 \pm \sqrt{5}}{2}$
Since $\displaystyle x$ is positive, we have: .$\displaystyle x \:=\:\frac{1 + \sqrt{5}}{2} \:= \:1.618033989... $
. . which happens to be the Golden Mean, $\displaystyle \phi$.
$\displaystyle \text{Therefore: }\;1 + \frac{1}{1 + \frac{1}{1+\frac{1}{1 + ...}}} \;=\;\phi$
A useful link: Continued fraction - Wikipedia, the free encyclopedia