To keep this at its very simplest, we will illustrate here with respect to the number 2.
Now as will become quickly apparent, every number in this context is defined with respect to both a base and dimensional number aspect that are complementary in quantitative and qualitative terms.
Thus once again in the expression ab, a is the base and b the dimensional number accordingly!
So it is important to appreciate how the base number (representing a quantity) is complementary to its (default) dimensional number, which relatively - has a qualitative meaning.
1) Thus when I refer to "2" as a number quantity, this is necessarily defined with respect to a (default) dimensional number of 1.
Therefore it is more properly written as 21.
Thus 2 is thereby a specific actual number that is defined with respect to an overall linear dimension that potentially relates to all real numbers.
Thus when we use 1 to represent this dimension it strictly carries a qualitative - as opposed to quantitative - meaning.
The very notion of quantitative implies independence (from all other numbers).
However to enable such a number to be then related with other numbers, we require the corresponding notion of general number interdependence which is - relatively - of a qualitative nature
And this number interdependence is provided by the dimensional notion of number (which in this default case represents the 1st dimension i.e. the number line).
So I represent 2 as a specific number quantity, by highlighting this number (which is emphasised here in an explicit manner) in black, while the default dimensional number of 1 is shown in light grey (to show that it remains merely implicit in this instance).
2) We next look at the reverse notion of "2", now written as 21.
The number "1" which now represents the dimensional aspect of 1, carries an actual quantitative meaning i.e. as applying to all actual numbers (in this context natural numbers) on the number line.
The number now "2" represents the qualitative aspect of 2, where this number is now understood as in common with all other classes of 2 objects. For example if we have 3 columns with 2 items in each row, then we can strike a one-to-one correspondence as between the three columns (where each contains "2" items).
What is vital however to appreciate here is "2" is not now being used in a quantitative sense (where it is viewed as independent of other numbers) but rather qualitatively, whereby "2" is now seen to be interdependent with the members of each column (i.e. each column is similar in containing 2 members)
This in fact represents the fundamental difference as between addition and multiplication.
With addition two number quantities are combined (without change of qualitative dimension).
So 21 + 31 = 51.
However when we multiply thee numbers we strictly combine both quantitative and qualitative meaning.
So 21 * 31 = 61 * 12.
Thus both a quantitative change in units, as well as a qualitative change in the dimensional nature of the units takes place. Thus is we have a small table with length 3 ft. and width 2 ft. respectively we can immediately recognise that its are is 6 square feet. However in the conventional treatment of number multiplication the qualitative dimensional aspect of transformation is simply ignored.
Therefore referring again to the rows and columns, we must recognise initially the independence of the rows and columns (as separate).
However equally in then multiplying 3 by 2, we recognise the common quality of twoness with respect to each of the 3 columns.
So once again 21 represents the situation where the base number "2" is now of a qualitative nature and "1" as dimensional number assumes a quantitative identity as applying to any actual (natural) number on the (1-dimensional) line. In this way we can multiply 2 by 1, 2, 3, 4,.......
However with multiplication a change takes place to the qualitative aspect.
So in this context "2" refers to the two-dimensional plane which now potentially stretches in two directions in an infinite manner
Thus in moving from 1) to 2), the meaning of both base and dimensional numbers switch.
In 1), the base number "2" is quantitative and the dimensional number "1" is qualitative; however in 2), the base number "2" is now qualitative, and the dimensional number "1" is quantitative.
3) We now look at the interpretation of 12.
"2" as dimensional number is now used in a qualitative sense, representing the simple multiplication operation 1 * 1. Once again the length and width of the unit square are not independent of each other but related to each other in a specific manner. Therefore "2" is here qualitative.
However "1" as base number has a quantitative meaning representing the one (2-dimensional) object that results. So just as in 1) we defined an actual number i.e. "2" with respect to the number line (as potentially infinite), here we are defining an actual object i.e. 1 object with respect to the 2-dimensional plane that is potentially infinite.
Thus in this context, the base number is quantitative and the dimensional number qualitative respectively.
4) Finally we look at the interpretation of 12.
Here "2" representing dimension carries an actual meaning, whereby it can be applied to classes of 1 object. So for example if we were comparing areas of different fields, these would all be of a 2-dimensional nature, that would apply in the case of each field. In this sense each field (as a unit) would be in common with each other field so that "1" would now have a qualitative meaning.
And "2" now representing the dimensional number (with an actual finite significance) would be thereby quantitative in nature.
Thus again in 3) the base number "1" is quantitative and the dimensional number "2" is qualitative; however in 4) the base number "1" is now qualitative and the dimensional number "2" is quantitative.
Thus in the dynamics of experience, both base and dimensional numbers (which psychologically are represented through corresponding perceptions and concepts) keep switching as between both their quantitative and qualitative meanings respectively in a dynamic complementary manner.
Therefore in the simplest possible case the number "1" can have a base meaning (corresponding to the actual rational perception of "1" that is of an independent quantitative nature.
However equally "1" can have a base meaning (corresponding to the potential intuitive perception of "1") that is of a common qualitative nature as applying to all classes of 1 object).
Then "1" can have a dimensional meaning (corresponding to the potential intuitive concept of "1" that is of an common qualitative nature (i.e. the number line as potentially applying to all numbers).
Finally "1" can have a dimensional meaning (corresponding to the actual rational concept of "1" that is of a common quantitative nature (i.e. the number line as actually applying to specific numbers).