Is it possible to have a clear definition for the nearest perfect square number for a fractional number? For example, let us consider a number 0.004. What is another decimal number closest to it, that is also a perfect square? Is it 0.0025? We know 0.0025 is a perfect square (0.05*0.05) but is it the closest one? Is there any way to find out? (PS : please suggest some tags for questions like these)
2026-05-04 20:24:45.1777926285
finding nearest perfect square for a fractional number
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Note that $$0.004=\frac{4}{1000}.$$ Now, let's "zoom in" one digit. Change the denominator to $10000$, and bring the numerator to the nearest perfect square integer to $40$ - that'd be $36$. $$0.0036=\frac{36}{10000}=\left(\frac{6}{100}\right)^2.$$ This is closer to $0.004$ than $0.0025$, for sure. But is it the closest? Let's try it again. What if we look for stuff of the form $(x/1000)^2$? We need the closest perfect square to $4000$ - that's $63^2=3969$. Now we have $$0.003969=\frac{3969}{10000}=\left(\frac{63}{1000}\right)^2.$$ That's even closer. Let's go one deeper... $$0.00399424=\frac{399424}{100000000}=\left(\frac{632}{10000}\right)^2.$$ Can you see what's happening here?