Why square root of $1$ is $1$ but square root of $0.1$ is $0.316$, again square root of $0.01$ is $0.1$?

Question

I would like to understand the reason behind this pattern:

$$\begin{align} \sqrt 1 &= 1 \\[4pt] \sqrt{0.1} &= 0.31622 \\[4pt] \sqrt{0.01} &= 0.1 \\[4pt] \sqrt{0.001} &=0.03162 \\[4pt] \sqrt{0.0001}&=0.01 \\[4pt] \sqrt{0.00001}&=0.003162 \end{align}$$

I expected $\sqrt{0.1}$ to "behave" in a similar way to $\sqrt 1$... Why this intermittent pattern? What does $3162\ldots$ represent? Does it represent an irrational number like $\pi$ or a ratio like Fibonacci?

Edit:

As some comments have kindly let me see, it all comes from $\sqrt{10} =3.16227766,$ so my question becomes: what does this number represent? I notice it's really close to $\pi,$ are the two things related? Also still not clear why this intermittent pattern as to why for example $\sqrt{1} = 1$ and $\sqrt{0.01}=0.1$ and so on and they are not something like $0.31622...$

Answer 1

Don’t be discouraged by the comments.

The numbers you are considering are of the form $\sqrt{10^{-n}} = 10^{\frac{-n}{2}}$ where $n$ is a natural number. If $n$ is even then (like for $0.01, 0.0001$) then $\frac{n}{2}$ is also a natural number, call it $k$. Then $10^{\frac{-n}{2}}=10^{-k}$ which is $0.00..1000$ with 1 at k-th position. Now if $n$ is odd write $n=2k+1$ (like $0.1,0.001$) then $10^{-\frac{2k+1}{2}}=10^{-k-\frac{1}{2}} = 10^{-k}10^{-\frac{1}{2}} = 10^{-k} \cdot 3.16..$

So this is why the numbers seem “alternating”. The fact that $\sqrt{10}$ is 3.16… (which happens to be close to $\pi$) is just because that happens to be the number that when squared is $10$

Answer 2

yes I know the definition of square roots, I just don't get basically why $\sqrt{10}$ is completely different from $\sqrt{1}$ and $\sqrt{0.01},$ where one would expect a similar result

As some comments have kindly let me seen, it all comes from $\sqrt{10}$

\begin{align} \sqrt 1& =\sqrt\frac{1^2}{1^2} &&=\frac11&&&= 1\\ \sqrt{0.1}&=\sqrt\frac{1^2}{1^2\times10}&&=\frac1{\sqrt{10}} &&&= 0.31622 \\ \sqrt{0.01}&=\sqrt\frac{1^2}{10^2}&&=\frac1{10} &&&= 0.1 \\ \sqrt{0.001}&=\sqrt\frac{1^2}{10^2\times10}&&=\frac1{10\sqrt{10}} &&&=0.03162 \\ \sqrt{0.0001}&=\sqrt\frac{1^2}{100^2}&&=\frac1{100} &&&=0.01 \\ \sqrt{0.00001}&=\sqrt\frac{1^2}{100^2\times10}&&=\frac1{100\sqrt{10}} &&&=0.003162 \end{align}

Line 2 outputs a square root from an input that isn't a perfect square, so the result $\displaystyle\frac1{\sqrt{10}}=\frac1{\sqrt2\sqrt5}$ is irrational; since each successive even-numbered line merely divides the previous one by $\sqrt{100}=10,$ every even-numbered line has the same string of significant digits.

The odd-numbered lines behave similarly but with perfect-square inputs and rational outputs.

This is why the irrationality, as evidenced by the presence of $\displaystyle\frac1{\sqrt{10}}$ in the third column, alternates through the sequence.

Answer 3

Square roots (of positive real numbers) are equivalent to halving a power: $\sqrt {a^n} = a^{n/2}$.

Therefore, it's particularly convenient when you have a radicand that's expressible as an even power. E.g., $1 = 1^2$, $0.01 = \frac{1}{100} = \frac{1}{10^2}$, $0.0001 = \frac{1}{10000} = \frac{1}{10^4}$, and so forth. For cases like these, $\sqrt {\frac{1}{10^2}} = \frac{1}{10} = 0.1$, $\sqrt {\frac{1}{10^4}} = \frac{1}{10^2} = 0.01$, etc.

When you have a radicand that's an odd power of ten, then you get a number of the other form.