Operation Table
From natural numbers to real numbers.
Last updated
From natural numbers to real numbers.
Last updated
There is also an alternative approach to analyzing trajectories which we refer to as the operational approach. Recall the functions and from the Introduction chapter.
These were defined for the natural number domain and odd number range. Now, we will introduce their brute versions, which take a real number input instead and output a real number. The parameter that was well-defined in the prior functions, is now given explicitly; and it does not respect whether the resulting number is an integer or not; hence the name "brute".
Brute Reduced Collatz function is defined as:
Brute Reduced Collatz Initializer function is defined as:
Since is given explicitly, we can give "wrong" parameters and end up with non-integer results. That is what makes these functions imprecise.
For an ECF , consider the imprecise trajectory starting from some number :
The notation is similar to what we have done in the Trajectory chapter:
ICF is defined exactly the same way for imprecise trajectories too! This makes it a really powerful tool, which we will soon make use of.
There are many ways we can analyze the table, see the following links:
Define as follows:
Consider an ECF , and its bijective map that is .
Consider the imprecise trajectory that is defined by this ECF and the starting number is :
As per our definition, the terminating number here is denoted as .
The function is then defined as .
Here is how the first 10 numbers look like in the operation table for :
If the cell contains 1, then the corresponding column actually represents the ECF that is of the Collatz sequence of . That is awesome, because you can immediately learn the sequence just by looking at the table.