I am a student studying mathematics at university. I have come across the expression $$1-\dfrac{1}{x}$$ in several different classes and I'm wondering about its significance.
The first place that I noticed it was in mathematical biology studies, where the percent of a population that needs to be vaccinated to induce herd immunity is given the formula, where $x$, in this case, is called the basic reproduction number, $R_0$.
The second place that I have noticed it is in Electricity and Magnetism, where surface charge densities are related by the above equation, and $x$, in that case, is $K$, the dielectric constant of a material.
I remembered this Numberphile video, where they use an equation of a similar form $\left(x = 1+\frac{1}{x}\right)$ to find the golden ratio, which makes me this that somehow the first equation is related to the golden ratio, but I'mt not sure if that is correct. Why does this same formula appear in two seemingly unrelated fields? What makes it so relevant and (presumably) useful?
Rewriting as $\dfrac{x-1}x$, this is a ratio of two quantities, one being a "modified" version of the other. More generally,
$$\frac{a\pm b}{a}=1\pm\frac ba$$ expresses a relative change.
Regarding the Golden Ratio, we can remark that it is the limit of the ratio of successive Fibonacci numbers,
$$\frac{F_{n+1}}{F_n}=\frac{F_n+F_{n-1}}{F_n}=1+\frac1{\dfrac{F_n}{F_{n-1}}},$$ which explains the relation
$$\phi=1+\frac1\phi.$$