Why isn't the golden ratio defined as the points where $f(x)=x^3-x^2-x$ are zero?

Question

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.

Link: https://www.desmos.com/calculator/qmmlhbtwog

Answer 1

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.