The number $1$ may be linguistically described as "unity". This very number is the original source of various words in the terminology, like the "unit matrix" (a matrix behaving like the number $1$).

It is a convention to write down that dimensionless quantities like the Mach number have units $1$ because the multiplication by $1$ changes nothing about the result – this is the counterpart of the multiplication by another unit like ${\rm m/s}$. 

It just looks more coherent to write the unit $1$ into the tables. But verbally, one may also say that quantities with "this unit" have no units whatsoever. They are dimensionless. As long as one understands the logic, there's no problem in following these somewhat inconsistent conventions in which we sometimes say the units to be $1$ and sometimes we say that the units aren't there. 

In the tables, the "unit" column means "the ratio of the full quantity and its numerical value". With this definition, the result may be calculated as $1$ without any problems. It's similar to the task to compute budget deficits as the difference of revenue and expenses. If the latter two are equal, the difference is just $0$. One may write $0$ although he could also write it as $-$ and say that the difference "doesn't exist". The numbers $0$ and $1$ play the role of the "neutral objects" for addition and multiplication, respectively.

Is the number 1 a unit?
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