Is there a basic "unit" of measurement in math 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?
 A: Let's approach this in a different sense. Imagine you are discovering math as it was in earlier times. The first problem you encounter is the necessity of counting. We go hunting, and you see a group of cattle (what did people used to hunt...?) and, upon returning to the hunting party, you want to notify the group on how many cattle there are. In this sense you say a number $n$ and then notify the group that it is $n$ cattle. The unit here would therefore be the head of cattle.
This is an interesting problem for a mathematician. We had to have come up with a counting system that is systematic throughout all units. For example, 1 cattle and 1 apple have the same count, but they have different units. As pure mathematics grew, the units became tools to relate these numbers. Essentially, we could use the natural numbers, $\mathbb{N}$, to express the count of objects in a set. This is referred to as the cardinality of a set.
Later, people realized that these counting numbers could also represent lengths. But another problem arose: how do you express a number in the middle of $0$ to $1$ or $1$ to $2$? In this sense we expanded the definition of the number system to include the positive rational numbers, $\mathbb{Q^+}$. As mathematics grew, and more people applied mathematics to the world around it, more units were added to give meaning to these numbers.
In an applied world, this is good. Finding meaning behind math and mathematical objects is a very natural idea, and as you have pointed out, this is very reasonable to see the relation between units and numbers. However, numbers extend further than our scope of universal intuition. In a pure sense, numbers are unitless objects. Mathematical objects are "ideas" such that in every sense we could think of the number "1" acts like the number "1". No matter it's shape or form.
As we expanded our set of numbers to include $\mathbb{C}, \mathbb{H}, \mathbb{O}$ it becomes difficult to relate these numbers to a universal system of measurement because what meaning does it make to say that we have $1+i$ cattle? It makes very little sense at all! If you are interested in this subject however, this (as I see it) is a good way to view the creation of set theory. Also, if you want to learn more about this type of thing, mathematics is the right place to look!
A: The number $1$ is the "basic unit of measurement".  All other numbers are scalar multiples of it.
A: There is no unit of measurement in math. When you say the "area" is 5, the area is 5, there is no measurement for area. There is measurement for area outside of math, and those are the units we use, but within math, "1" is 1, it is not like 1 some unit. 
A: It is unitless. Please read about "dimensionless quantity"
A: The basic unit of quantity
is the "publication".
The basic unit of quality
is the velocity of
quantity
or "$\dfrac{publications}{year}$".
A: How about looking at things this way...
We measure from 0 and we usually count by and from 1. So in that case 1 can be seen usually as both a unit of measure and as a unit of counting. 
That being said, we can use a unit interval of 1/2 or 2 as our unit of measurement, instead. If we do then our unit of counting is, respectively, by either  1/2 or 2 and from either 1/2 or 2. Fwiw, we cannot use 0 as a unit of measure nor count by and from 0, unless and until we are ready to see, e.g., that 0/2 is analogous to 1/2 and 0 divided by 1/2 is analogous to 1 divided by 1/2. Which would introduce a concept of parts and multiples of 0 as an extension to the real number line's parts and multiples of +1 and -1. But who other than a math explorer would want to sail in that scary direction?
