Where Addition and Multiplication Meet

I have stressed repeatedly in these blog entries that - properly understood - there are two aspects of the number system (Type 1 and Type 2 respectively) in dynamic interaction with each other.

In defining a number fully, we must define it with respect to both a base and dimensional aspect (where the dimensional aspect represents the power to which the number is raised).

Therefore with respect to n^x, n represents the base and x the dimensional aspects of the number respectively.

The Type 1 aspect is then always defined with respect to a fixed default dimensional value of 1.

So  n¹ thereby (where a can take on any base value)  represents a number expression defined with respect to its Type 1 aspect. This aspect is directly suited to the cardinal treatment of number.

The Type 2 aspect, in an inverse manner, is always defined with respect to a fixed base value of 1.

So 1ⁿ (where n can now take on any dimensional value) represents a number expression defined with respect to its Type 2 aspect. This aspect is directly suited to the ordinal treatment of number.

The considerable issue then arises as how to convert a number, initially defined with respect to the Type 1 aspect as n¹, indirectly in Type 2 terms.

So in general terms we set n¹  = 1^x(where x represents the Type 2 expression of the number).

Then taking natural logs on both sides 1(log n) = x(log 1).

Therefore x = log n/log 1 = log n/(2iπ) = – {log n/(2π)}i

So therefore for the simple example of 2 (i.e. where 2 is expressed in Type 1 terms as 2¹), its corresponding Type 2 expression is given as – {log 2/(2π)}i = – .1103178 i (correct to six decimal places.

So fully expressed in terms of the Type 1 and  Type 2 aspects of the number system,

2¹ = 1 ^{– .1103178 i}

Now we can use these two aspects of the number system to illustrate precisely the relationship as between multiplication and addition respectively.

Put simply, multiplication in Type 1 format  is expressed as addition in terms of the corresponding Type 2 aspect.

So for example, in Type 1 format, 2 * 3 = 6. 

More precisely, 2¹ * 3¹ = 6¹.

However in terms of the Type 2 aspect,

2¹ = 1 ^{– {log 2/2π)} i}  = 1 ^{– .1103178 i}

3¹  = 1 ^{– {log 3/2π)} i}  = 1 ^{– .174850 i}

Then, 2¹ * 3¹  (in Type 1 terms) = 1 ^{– (.1103178 i + 174850 i)}

So when we multiply the two numbers (as base) in Type 1 terms, we add the two numbers (as dimensional powers) in the corresponding Type 2 manner.

Therefore  6¹ = 1 ^{– .285167 i} 

Of course, we can equally apply this in reverse, so what is addition (from the Type 2 perspective) is represented trough multiplication (in corresponding Type 1 terms).

Now, to convert from the Type 2 aspect to its corresponding Type 1 expression we simply replace the base number 1 (in the Type 2) with e^2iπ (as dictated by the famous Euler identity).

Therefore 1 ^{– .285167 i}   =   e^{2iπ  * ( – .285167 i)}   =    e ^1.791757   

= 6 (or 6¹ in a precise Type 1 manner).

So we have used this reverse means of conversion (from the Type 2 to the the Type 1 aspect) to verify that

2¹ * 3¹  = 6¹  can be consistently expressed in a Type 2 manner.


However it is important to stress once again that the Type 1 and Type 2 aspects of the number system are quite distinct. This also implies directly that the operations of addition and multiplication respectively are also quite distinct.

So once again whereas the Type 1 aspect is directly associated with the quantitative aspect of number (as independent) the Type 2 aspect is directly associated with its qualitative aspect (as interdependent with other numbers).

This also implies that whereas addition directly relates to the quantitative aspect of number, that in relative terms, multiplication relates to its corresponding qualitative aspect.

Of course in dynamic interaction with each other, these aspects overlap, so that both addition and multiplication entail (depending on the precise context) both quantitative (analytic) and qualitative (holistic) aspects.

However the key conclusion of all this is that the very paradigm which defines present mathematical interpretation is quite unsuited to the task.

Because of its underlying abstract assumptions regarding the nature of mathematical symbols such as numbers, this entails that qualitative holistic considerations are inevitably reduced (with respect to every context) in an absolute quantitative manner.

However, as I have sought to demonstrate through the number system, appropriately understood, the nature of mathematical reality is inherently dynamic, with both quantitative (analytic) and qualitative (holistic) aspects of interpretation.

And this fundamentally applies to the very nature of addition and multiplication, which can only be properly appreciated therefore in a dynamic relative manner.