2
$\begingroup$

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)

  • 0
    Yes I am only talking about terminating decimals, otherwise we will probably have infinite choices2012-12-05

1 Answers 1

4

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?

  • 0
    @user13267 Exactly. This is what Andre meant above when he said that these squares are "dense": if we pick a small number $\epsilon$, no matter how tiny, there will always be some number of the form $(x/10^n)^2$ in the interval $0.04-\epsilon$ to $0.04+\epsilon$.2012-12-05