What is the square root of 1 cm

Question

If $1$cm = $.01$m then shouldn't the square root of $1$cm = the square root of $.01$m but the square root of $1$ = $1$ while the square root of $0.01$ = $0.1$ So my dilemma, is the square root of $1$ centimeter = $0.01$ meters or $0.1$ meters?

The square root of centi is easy, that's deci. But what the square root of metre could be, I have no idea. — Daniel Fischer, Nov 14 '13 at 19:28
$1$ cm or $0.01$ m is a length, we can not extract square root of it
But as $1$ cm $=0.01$ m $\implies1$ cm$^2=(0.01)^2$m$^2$ — lab bhattacharjee, Nov 14 '13 at 19:29

score 3 · Accepted Answer · answered Nov 14 '13 at 19:38

I think you need to think of the square root of a distance as a new unit. Just as a ${m^2}$ is an area, not a distance. So the following works:

$\sqrt {1m} = \sqrt {100cm} $

$\sqrt m = 10\sqrt {cm} $

$\sqrt {1cm} = \sqrt {cm} $

and

$\sqrt {.01m} = .1\sqrt m = .1\left( {10\sqrt {cm} } \right) = \sqrt {cm} $

So it works out either way.

score 2 · Answer 2 · answered Nov 14 '13 at 19:37

2

The square root of $0.01\,\mathrm{m}$ is $0.1\,\mathrm{m}^{1/2}$.

Such quantities whose units have fractional exponents don't have a ready physical interpretation, but can sometimes be useful as intermediates in calculations anyway.

From this calculation we can also see the necessary relation between the units $\mathrm m^{1/2}$ and $\mathrm{cm}^{1/2}$, namely $1\,\mathrm{cm}^{1/2} = 0.1\,\mathrm{m}^{1/2}$.

answered Nov 14 '13 at 19:37

hmakholm left over Monica

286,031

"don't have a ready physical interpretation, but can sometimes be useful as intermediates" - sounds the same as complex numbers (where btw, physical interpretation lagged the algebra). – alancalvitti Jan 31 '15 at 17:09

score 0 · Answer 3 · answered Sep 16 '22 at 01:40

In Quantum Mechanics the square root of the inverse meter arises as the unit of the value of (1D) wave function. Since the square of the wave function is the probability of finding a particle per unit length, the wave function itself has the unit of inverse unit length. And the physical meaning of the square root of an inverse meter itself is who knows what.

score 0 · Answer 4 · answered Nov 14 '13 at 19:31

0

The side of a square of area $0.01 \text{ m}^2$ is $0.1 \text { m}$. The side of a square of area $1 \text{ cm}^2$ is $1 \text { cm}$. There is no square root of a centimeter-there is (aside from fracals) nothing of dimension $\frac 12$, which is what would be required.

answered Nov 14 '13 at 19:31

Ross Millikan

374,822

In materials engineering, it is not unheard of to have units of MPa(m)^1/2, or $\textrm{MPa}\sqrt{\textrm{m}}$. – Emily Nov 14 '13 at 19:34

What is the square root of 1 cm

4 Answers4