The Missing Operation
The true successor to multiplication?
I’ll be covering ‘‘The Missing Operation’’ by Epic Math Time1, and extrapolating on it with help from the comments section and other sources. The intention of this post is to bring more awareness to this operation and hopefully catalyze discussion on it.
Addition is commutative,
multiplication is commutative,
exponentiation is NOT commutative.
Can we fix this?
Let us define a new operator called the Powerlog, denoted by a ᴖ b.
For the sake of generalization, the logarithm is defined in base B. For the rest of the post, I shall take B = e . Please let me know if there is a more appropriate base.
We can see that a ᴖ b is commutative:
Multiplication distributes over addition and powerlog distributes over multiplication:
Multiplication is associative and so is powerlog:
The identity element of the operator is equal to e (the chosen base):
The inverse of every element under this operator can be found:
In order to determine if powerlog is to multiplication as multiplication is to addition, it should be sufficient to find a field of some real numbers with multiplication and powerlog as the operators which is isomorphic to the field of real numbers with addition and multiplication as operators. In other words:
To find a ring isomorphism, we need to determine a function that maps to both sets while preserving both the identities and both operations.
This function must also be bijective in nature. As it turns out, the standard exponential function is a perfect fit for all the conditions!
So there we have it, the powerlog seems to be the true successor in all but obvious intuition!
In addition to above, by using the alternate definition to the powerlog proposed in the beginning we can define different tier operations as well.
Multiplication (1-Tier operation) :
Powerlog (2-Tier operation):
In this manner, we can define higher and higher tier operation, and a n-tier operation as n→ ∞ would resemble the following:
It isn’t clear as to what this function signifies.
We can see that addition is itself, ie. a+b = a+b.
Continuing this pattern for lower tier operations,
For a n-tier operation as n→ - ∞, it would resemble the following:
Such a operator would completely neglect the small of the two inputs and return the greater of the two values. Thus:
On analysis of the domains of ‘a’ and ‘b’ at both extremes of the tiers: (To be continued)
( I have decided to publish the article in a somewhat unfinished state, as this article has been on the rails for an uncomfortable amount of time and I can’t seem to find time to continue my inquiries. I hope the sources will guide those who want to continue the journey.)
Some food for thought:
Can powerlog be defined without using exponentiation or power series?
Where does it “pop” up in mathematics?
Where is this operator utilised outside maths?
Does the operator have an intuitive definition?
Sources and additional reading:
His video on the topic. Highly recommended.