patterns in the decimal expansions of adjacent square and cube roots

Question

For fun I made a table in Excel which evaluated the square and cube roots of whole numbers in ascending order. Then of the result, I extracted the first, second and third decimal place digits, then color-coded them (using resistor color code).

Interestingly, I found several 'mirrored' (or ascending-descending) patterns which I can only explain via the attached screenshots (try it for the cube roots - similar patterns) ...

Can someone figure out what's going on here? Am I asking a wrong question from the get-go? Is it simply emergent from the way Excel evaluates the decimal values?

Thank you!

Answer 1

Long story short:

• The fact that $\sqrt{x}$ is almost linear when $x$ does not change much results in the repetitive pattern: $\sqrt{a},\sqrt{a+1},\sqrt{a+2},\dots,\sqrt{a+n}$ behaves like an arithmetic sequence for $n\ll a$.
• The second degree correction to the linear approximation around $a$ is symmetric around $a$, which gives the mirrored (symmetric) pattern.

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Let's take for example the longest pattern, that one around $\sqrt{400}$. Making the first degree assumption for $\sqrt{x}$ around $x=400$ yields $$ \sqrt{x} \approx 20 + \frac{x-400}{2\sqrt{400}} = 20 + \frac{x-400}{40}. $$ If we plug $x=382$ into that approximation, we get $$ \sqrt{382} \approx 19.55, $$ which is quite close to the real value. In other words, the linear approximation is rather accurate for the values in this pattern. Let's see what values the linear approximation gives: $$ \begin{align*} 400 &&& 20.0 \\ 401 &&& 20.025 \\ 402 &&& 20.05 \\ 403 &&& 20.075 \\ 404 &&& 20.1 \\ 405 &&& 20.125 \\ 406 &&& 20.15 \\ 407 &&& 20.175 \\ 408 &&& 20.2 \\ 409 &&& 20.225 \\ 410 &&& 20.25 \\ \end{align*} $$ Tada! There's the pattern: the second decimal is $0,2,5,7,0,2,5,7,\dots$ ad infinitum. Because the actual values are a bit smaller than what the linear approximation gives, you see the pattern $9,2,4,7,9,2,\dots$ until the error between the real value and the approximation grows large enough.

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But that only explains the repetition. Why the mirrored pattern?

Let's make a 2nd degree approximation around $x=a$! $$ \sqrt{x} \approx \sqrt{a} + \frac{x-a}{2\sqrt{a}} - \frac{(x-a)^2}{8\sqrt{a}^3} $$ Now the correction term $- \frac{(x-a)^2}{8\sqrt{a}^3}$ is symmetric around $x=a$. So when the linear approximation stops giving the right decimals, the correction term changes the values the same amount in the same direction for $x=a+b$ and $x=a-b$. This causes the symmetry when the repetitive pattern near $x=a$ fails.