I am wondering if there is a basic unit in math.

$$3\ cm \times 2\ cm = 6\ cm^2$$

The $cm$ is a unit of measurement, but what about:

$$3\ cm \times 2 = 6\ cm$$

Should the $2$ have a unit of counting? Like I have $2$ (amount of stuff) $3\ cm$ strings. 

Should it be something like:

$$3\ cm \times 2\ mu^0 = 6\ cm$$ (where $mu$ is a "*mathematical*" unit)

The unit is to the power of $0$ because if it was not it would do this:

$$3\ mu \times 2\ mu = 6\ mu^2$$

And I don't think this works right, but $mu^0$ seems to:

$$3\ mu^0 \times 2\ mu^0 = 6\ mu^0$$

Is any of this correct, or is it all wrong?