I was messing around on desmos, and when I plugged in $f(x) = x^2 - x - 1$, I get two points where $f(x)$ is zero, which are answers to the golden ratio. Why is this not used in the definition? It seems so much clearer to me.
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1Why not consider $g(x) = f(x)/x = x^2 - x - 1$ instead? $f(0) = 0$ doesn't seem relevant here. – GDumphart Jan 13 '17 at 14:19
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@GDumphart Wow Thanks! – Jan 13 '17 at 14:21
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1I normally think of it as the solution to $x^2 = x + 1$, which makes clear one of its biggest properties, that it is the number that you can square by adding one. – Kaynex Jan 13 '17 at 14:33
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It is more clear to you, maybe. There are LOTS of equations whose solutions may be the golden ratio.
But it's definition comes from geometry, like many other mathematical constants like $\pi$ and $\sqrt{2}$.
The golden ratio is defined in this way: it's the ratio of two numbers which is also equal to the ratio between their sum and the larger of the two, that is naming $a$ and $b$ with $a >b$,
$$\frac{a+b}{a} = \frac{a}{b} = \phi$$
This definition comes in handy because it shows many interesting properties of the golden ratio such as:
$\phi^{-1} = 1 - \phi$
$ \phi^2 = \phi + 1$
It's also straightforward to derive the golden ration from this definition since it's.. the definition!
Your equation cannot be solved that easily by hands, whereas the definition for $\phi$ is immediate.
$$\frac{a+b}{a} = \frac{a}{b} = \phi$$
hence
$$\frac{a+b}{a} = 1 + \frac{b}{a} \longrightarrow 1 + \frac{1}{\phi}$$
That is
$$1 + \frac{1}{\phi} = \phi$$
That is
$$\phi^2 - \phi - 1 = 0$$
Form where the golden ration can be easily calculated.
Enrico M.
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