Rotational numbers

In this post I’d like to talk about the mathematics section. In particular I would like to address some common objections.

Rotational numbers are not an advanced concept at the cutting edge of mathematical research, but a basic number set that, in my view, was passed over in the relentless drive towards abstraction. In fact it is telling that the professional mathematicians I discussed this with often seemed to prefer to derive this as a special case of a higher level concept, like a module over a ring, rather than accept the elegant derivation from first principles.

Other common objections are (1) concern about the absence of multiplication, and (2) discomfort about using the output of an open function to define new sets because it is contrary to the requirement that functions must have a well defined domain and range.

Multiplication is essential to many areas of mathematics and we learn it at primary school so it maybe helpful to take a step back and review what it is in essence.

We use unary inc open functions to generate the natural numbers. We can create a new binary function that repeats the application of inc. Clearly the second argument, the loop counter that determines the number of repetitions, must be a positive integer. We arrive at the binary addition function. We find that the function is commutative, meaning that we can swap the loop counter and the first argument and get the same result. The inverse function to addition is subtraction. This is not closed in the natural numbers. We extend the range set for subtraction and arrive at the set of integers. We can retroactively extend the domain for addition to include negative numbers. We don’t want to break commutativity, so we invent a rule to allow negative numbers as the second argument (the loop counter).

Now we repeat the process and repeat adding the same number to itself and arrive at multiplication. We find that this is also commutative. The inverse function is division which again is not closed under the integers, so we need to extend the range set to the fractions. Again we extend the domains for addition and multiplication to include fractions. Because we want to keep commutativity we allow fractions as the loop counter argument.

We repeat the process again and arrive at potentiation. This is no longer commutative, so we get two inverse functions: roots and logarithms. The inverse functions again require a range extension to real numbers. Retrofitting fractions and real numbers to addition, multiplication and potentiation results in some rules that don’t appear intuitive at first sight, but that work to maintain the expected laws.

The next level is tetration or the power tower. Again, there are two inverses. Finding sensible domain extensions becomes quite difficult from this point and is an active area of research.

The point of all this is to demonstrate that defining new sets as the range of functions is perfectly natural and that multiplication is merely the hyper2 function that has many applications and regularities.

The conventional insistence of all unity roots being complex numbers rather than defining them to have independent dimensions (where the complex numbers are the order 4 special case) stems from the desire to maintain general multiplication.

In the mathematics section we don’t strictly generalise complex numbers, but Gaussian numbers, because we only require integer coordinates. The same methods can be used to derive real numbers and some other interesting sets. I’ll do that in the next post.


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