I mentioned that around the age of 10, I formed serious misgivings regarding the nature of multiplication. To my way of thinking, even at that young age, a comprehensive explanation required a qualitative - as well as quantitative - aspect. However - quite literally - the qualitative dimension was effectively removed from conventional interpretation. And this proved far from a passing concern. In fact I spend some considerable time thereafter in the attempt to "capture" the qualitative dimension. Thus to focus clearly on this missing aspect, I considered a square with side of 1 unit (i.e. expressed in 1-dimensional terms). The area of the square is then given as 1 square unit (i.e. now expressed in a 2-dimensional manner). Therefore, though the quantitative result has remained unchanged as 1, clearly a qualitative dimensional change has taken place. 1 2 thereby represents the quantity of 1 (expressed in 2-dimensional units). Then as all real number quantit...