Graphic Illustration of the Operation of Multiplication

Graphic Illustration of the Operation of Multiplication

By M. Stark

Example 1. 3 x 2 =6

x =

                 è
            ¾    éÊ3    in the operation

è
éÊ3 x =

__________________________º ___________

Legend

¾ means becomes

è
éÊ3 means the factor with an intrinsic value of 3; a natural number which becomes a factor; or 3 can be considered a group

3 = factor or multiplier; 2 = multiplicand; 6 = product

_______________________________ º _________________

Both terms in the multiplication are symmetrical to one (1) and therefore they share a mutual symmetry with one (1) and therefore to each other. When the operation of multiplication occurs, the identity of the factor, in this example 3, is transferred by its intrinsic value onto the group of 2 as pictured. Before the operation was performed these two terms were symmetrical to one another, which had conserved each of their intrinsic values, of course. Once the operation takes place, 3 becomes partially symmetrical with the multiplicand 2. The factor 3 now takes on the intrinsic value of 2 as its unit, so that the symmetry to one (1) as the unit of 3 is non-conserved. 3 orders the group or value of 2 to add unto itself 3 times.

è
éÊ3 x = + +

è
¾ éÊ3 in the operation of multiplication

As the diagram shows 3 of 2 are added to one another. As basic as this simple equation is, such simplicity is useful in defining the nature of symmetry so as to understand more complex mathematical concepts and operations. Furthermore, the fundamentals are requisite to higher mathematical treatments.

The clarity of the diagram of Example 1 above allows a conceptual window to the nature of symmetry. The word symmetry etymologically means ‘to measure together.’ When the multiplying factor determines in the operation of multiplication that the group of 2 should be tripled in this example, or added up three times total, it has been stated that 3 and 2 become partially symmetrical to one another. This may seem puzzling. If the values are combining through the operation, then the word symmetrical would imply that the natural numbers or groups, however they are abstractified, would have to become symmetrical to one another to an even greater degree, if not at least remain symmetrical to one another in order for the transfer of the value to take place.

However, all things measure through one (1). As 3 and 2 relate to one another, say, as sitting groups of objects of the given numbers 3 and 2, understood in the count is that one (1) object is worth one (1), and that each count increases by one (1). Each group of objects therein shares the same exact relationship to one (1), and it is from one (1) that is reflected all of symmetry. Therefore, we say that the two groups have values which are symmetrical to one (1) and also to each other. Now when the factor 3 becomes non-conserved when factoring in to the operation of multiplication, this exact process is performed by adopting the value of 2 for its effect, for the unit value upon which that factor 3 newly operates. The factoring says to add the group or unit of 2 twice more to itself, forming 3 groups of 2, which add up to 6. The fact that the factor 3 adopts a new unit value as found in its multiplicand 2 means that 3 symmetrizes 2 in an active sense, but since 3 as a factor is symmetrizing something other than one (1) itself as a raw value, this symmetrization is thought of as partial symmetrization. Even though the precise value of 3 still is understood as 3/1, the new disposition of the factor 3 becomes 3/1 times 2 per group; before that change in disposition of 3, 3 stood alone as 3 relates simply to one (1). Now, since the operation of multiplication dictates a new value to be related to 3, that of 2, the symmetry of 3 to one (1) is extended to the extent that the 3 becomes subsumed in the result of extension, or operation. Since (1) is all-pervasive, the quintessential symmetry of 3 to one (1) is the same. The change in the value which is the outcome of the multiplication implies that there was a change in the symmetrical relationship somewhere in the process of the operation. Logic tells that the change in symmetry would have been between the two terms of the multiplication, not that they symmetrize totally or only to one another, since the change would even imply a re-measuring, a non-conservation of value as found in the factor; these terms measure together as units before the operation as separate entities which are symmetrical to one another through one (1). If when they combine, one (1) does not change, which it does not, then it is more likely that the transfer of value should be regarded as a sign of asymmetry between the terms of the multiplication. How the values reflect to one (1) during the operation will tell the status of the symmetry, and when the factor 3 opens up its value so as to order the replication of another value, 2 in this example, then the factor 3 is relating now to another value besides one (1) in a very real and active sense which gives results. All of symmetry is born of one (1), so that opening up to another value during the operation lessens the one-to-one reflection of one (1) in the factor 3 for one (1), the value which is thus non-conserved accordingly. Thus, since one (1) remains as the unit quantifier even throughout the asymmetrically effected operation of multiplication, even though the two terms of the multiplication become asymmetrized one to the other, since partial symmetry is an asymmetrization of symmetry per se, this can only be partial in nature. Therefore, multiplication is a partially symmetrical operation. Since the operation of multiplication is commutative, meaning that a x b = b x a, then the factor and multiplicand are interchangeable so as to give equivalent products when commutated.

The key for understanding the concept of symmetry can be realized by understanding that the non-dual nature of one (1) subsists always as an absolute quantifier similar to a limit for all relative measurement and changes in values of those measurements.