When will function output have a specific decimal component

Question

Given a function f(x), is there any way to predict when the function will give a specific decimal part without brute-force iteration over possible x-values? Ex. f(x)=100/a^2 with arbitrary a, when will 100/a^2 be (any integer value here).8? Thanks in advance.

Answer 1

What you are asking for is a way to predict the fractional part of $f(x)$. There is no way to do this in general, but there are some nice results on that link. Searching for 'fractional part' in google or wikipedia may return additional results.

In the case of your example, it is particularly easy - you wish $\left\{\frac{100}{a^{2}}\right\}=0.8$, so $\frac{1000}{a^{2}}$ is an integer ending in $8$. Further, $$\frac{1000}{a^{2}}=10n+8 \implies a=\pm\sqrt{\frac{1000}{10n+8}}$$