Irrational numbers are uncomputable with probability one. In that sense, numerical, they do not belong to nature. Animals cannot calculate it, nor humans, nor machines.
But algebra can deal with irrational numbers. Algebra deals with unknowns and indeterminates, exactly.
This would mean that a simple bee or fish can do algebra? No, this means, given the simple expression of their brains, that a higher entity is able to command them to do algebra. The same for humans and machines. We must be able also to do quantum computing, and beyond, also that way.
Thus, no one (animals, humans, extraterrestrials in the NASA search, and machines) is limited by their expressions, and all obey a higher entity, commanding through a network from the top down -- which entity we call God, and Jesus called Father.
This means that God holds all the dice. That also means that we can learn by mimicking nature. Even a wasp can teach us the medicinal properties of a passion fruit flower to lower aggression. Animals, no surprise, can self-medicate, knowing no biology or chemistry.
There is, then, no “personal” sense of algebra. It just is a combination of arithmetic operations.There is no “algebra in my sense” -- there is only one sense, the one mathematical sense that has made sense physically, for ages. I do not feel free to change it, and did not.
But we can reveal new facets of it. In that, we have already revealed several exact algebraic expressions for irrational numbers. Of course, the task is not even enumerable, but it is worth compiling, for the weary traveler. Any suggestions are welcome.
Intellectual work by humans is very inspiring, but Physics knows that nature is at least 13.8 billion years ahead of us, and there might be other universes still older.
There just was not a single Big-Bang -- if one can happen, more can happen ... We might be late-comers on a feeric Tohu-wa-bohu. Evolution is a very long road.
Infinity is lame to algebra, not unknown and not indeterminate. But it is not a number, so there are no arithmetic rules to multiply/divide or add/subtract it to 1 or any number. We know that 0.x = 0, where x is a number.
We need to be optimistic, because that is the lesson from nature. An animal can self-medicate, obeying natural laws in chemistry that are unknown to animals. A tree grows when pruned, so we can see this pandemic as an opportunity. Let's grow, nature is not a zero-sum game!
Irrational numbers and mathematical real-numbers are uncomputable, with probability 1.
But irrational numbers can be calculated exactly in algebra and that is how animals are able to calculate-- in a network of thoughts, which does not have to hierarchical!
It can exist as an ontology, like the Internet. We, then, have to he mindful who we connect to. We can always connect to "DNS" to get the correct directions, or to a rogue DNS, of an attacker...
Since the 17/19th centuries, “the differential and integral calculus is based upon two concepts of outstanding importance, apart from the concept of number, namely, the concept of function and the concept of limit.” according to the well-known textbook by R. Courant in Germany and worldwide.
Without including any philosophical opinions, the system of rational and irrational numbers build a continuum of numbers, such that each point on an axis corresponds to just one number, and each number corresponds to just one point on the axis, called the system of mathematical real numbers(MRN).
Therefore, all MRN in the number line can be separated into one of two sets — the rationals (expressible as the ratio of two integers), and the irrationals (all the remaining MRN, which are, thus, not rationals). Every MRN is one of either. This construction is often credited to Richard Dedekind in ETH Zurich, in 1888, writing, “Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things.”
Because uncountably infinite sets are always larger than countably infinite sets, one can conclude that there are more irrational numbers than rational ones. It is worth mentioning that the rationals and the irrationals are both dense, but unlike the rationals, the irrationals are not “sparse” and do not have zero measure mathematically.
This fact emphasizes that rationals and irrationals are really quite different -- even though one can find a rational between any two irrationals, actually an infinite number of them, and an irrational between anytwo rationals.
See more at "Algorithms for quantum computation: the derivatives of discontinuous functions", available at:
https://www.researchgate.net/publication/366124503/
The comments above show how following the RG ToS is necessary for a scientific conversation.
But the legal responsibility of enforcing the RG ToS falls only on RG, per EU law.
Meawhile, it is suggested to read the question text, with the latest references. Also, private messaging is available.
NO. All you calculated, SPK, are rational number approximations! No irrational can ever be calculated, because one cannot go to infinity. There are no irrational numbers, just irrational people. This is further explained in:
https://www.researchgate.net/publication/370755777/
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