Aspies For Freedom

B"H

Hello.  Thank you for reading my post.  I have a question for those more educated in number theory than I.  My question is as such; Is the term "irrational number" really valid?

Let me explain.  Let us look at all of the Transcendental numbers right now.  Let us focus on the most "irrational" of the irrational numbers.  Now, these numbers are not the result of finite ratios.  However, there is a misunderstanding afoot that they are not ratios at all.  Yet, Pi is defined as 4-(4/3)+(4/5)-(4/7)+(4/9)....a sequence that is *KNOWABLE* as it goes on forever.  Now, Pi is irrational, of course, because it is not the result of a *finite* numerator and denominator.  However, that does not mean that it can not be represented as an infinite series of fractions, each predictable, added on to each other.

OK, so we have our Taylor series.  But, what of those irrational numbers that are never detected in geometry, or ever "known"?  Well, what if we had the "infinite" computer that could somehow add an infinite number of Taylor series' and ratios together, each in a random way.  In other words, we might have the Taylor Series for e, but with random patterns added or subtracted from it, regular rational fractions as well as other Taylor series.  Would we not then get all possible real numbers, if the computer works to the point of infinity (or, say it could "do" infinity in some finite amount of time)?  If so, then we would describe all real numbers as either finite ratios, or as infinite series of some type (irrational).  Would we then truly have "irrational" numbers, in other words, numbers that are not ratios at all?  

There are arguments that my (mathematically less than fully educated) mind could bring to bear on this notion of all numbers being ratios of some type, even if we had to invoke Calculus and its "limit at the point of infinity" precedents.  Obviously, finite minds could not generate all real numbers, but could an "infinity computer"?  If so, then in some sense all numbers are "rational" at the limit of infinity, otherwise we could not represent them in any kind of equation dealing with ratios ([2*e]/e=2, etc.)

If such a computer could NOT generate all real numbers, then we would have a whole list of numbers that would not only be transcendental...they would not even be cardinal or ordinal in the sense that we understand them.  Rudy Rucker seems to hint at this being possible in Finite and Infinite Mind when he discusses infinitesmals, although he does not quite come out and say it.  If there is a truly "irrational" number, then we would not be able to represent it as any kind of ratio, or, frankly, use it in ratios.  Only by a slight of hand could we do so.

Now, there is a greater level of infinity for real numbers than there are for rational numbers.  This is a point of objection to my reasoning.  However, in this scenario we include replacement.  Replacement allows us to get out of this limitation.  I can use "1/2" over and over again, infinitely, in any series that I want.  Thus, we have no need to invoke Cantor and the diagonal set paradox, since we can use and re-use fractions or Taylor series to our heart's content.

So, I am truly lost here.  And, I have used up my good will with Ask Dr. Math, since I ask Dr. Math too many questions.  Therefore, I ask you folks.  Are "irrational numbers" truly irrational?  Or, ARE there potentially real numbers that are what we might term "truly transcendental", meaning that they have no ordinal properties in any sense?  I am not asking rhetorically...I honestly do not know.  It might depend on what model we use for an infinite Taylor series.  Most mathematicians seem to accept that 2*pi/pi=2, even though Pi is an infinite series.  Yet...are there doubters on the fringes of academia?  Are there, you know, "troublemakers"?

Personally, I go with the notion that there are no truly irrational numbers.  I am kind of Greek in that respect.  What about you?

Your call.  

All the best.

Actually I think humanity will always discover some new way to express numbers through integer numbers but there will always remain infintely more numbers which cannot be expressed.
See, first humans knew only integer numbers. Then somebody came up with the idea to divide them, rational numbers were born. So, a fraction can be expressed as a division of two integers. The rest was called irrational.
Some people started to study irrational numbers and found out that some of them are expressable as a root of an integer polynomial, they were called algebraic. All others are transcendent.
No, there is probably a way to express transcendent numbers. Now, some of them are expressible as infinte series of previously described numbers. But is there some finite way? Just like fractions were found for rational and polynomials for algebraic numbers? I think, there is. I even thing, one can go infinitely "deep" find some new methods which enable to express even more numbers, so that transcendent numbers wil also have to be divided in two groups - "known" and "unknown", then for  some of unknown numbers there will be found some method again etc. More and more numbers will be finitely expressible. But it will always be only a tiny bit of real numbers. Other numbers will remain inexpressible finitely - always less of them, but always uncountably infinitely many. Not even as series or series of series'.

This is a big IF you state there. What could be an infinitely deep process of digging up methods that bring certain constructs of the mind losing their infinity?

There may exist a finite way to describe a solution, like a program for the proposed infinite computer (whatever that may be). Actually, this finitiy of the description of the algorithm is one part of the definition of an algorithm (a finite, effective and definite description of a calculus).

But the computation takes an infinite amount of time, so this not a really satisfying solution. One should not try to realize the concept of infinity in mathematics by doing something for a really long time.

And even the abstract concept of infinity in mathematics is a process as well. There is no such thing as an infinite large number, for example. The static nature of a single, outstanding number is different from the process of converging to infinity.

'Irrational numbers' are defined as numbers being non-rational, where a rational number can be expressed as a ratio of two integer numbers. There it is. Nothing is said about Taylor here, no further taxonomy. I think the term is not as bad as 'atom' was for something that was supposed to resist any attempt of splitting.

Well, Yigal, how about irrational numbers which are algebraic? I mean, one can finitely express them , as a root of a polynomial. So, one can describe them. Of course, "numbers" like 4x^6+3x^5+2x^4+5x^3+4x^2+x+12=0 are hard to imagine in the sense one cannot construct a stick of this length, but they are not completely inaccessible to the mind. Each real solution of this equation is easy to imagine - through the equation itself. Even if one cannot even estimate the solution. by digging deeper I meant ways to express numbers. A number 5 or 1/2 doesnt need additional references. But square root of two cannot be expressed so easily. Still, you have the equation x^2-2=0, so, if you are able to imagine multiplication/squaring, you can say what this number is - namely a number, of which subtracted two you get zero (there are two solutions, but one can say f.x. the bigger solution).

Now, pi cannot be expreesed similar way. But is there ANY (finite) way to express pi or any other transcendent number which is not discovered?I think there is. That's what I meant.

the square of which subtracted two* (in my previous post).

Actually the easiest way to dig deeper in my post is to use hyper-operators... x^^^^^^3-2=0 - the solution is surely transcendent, but this equation expresses it. Still, even this way doesn't cover all real numbers, so one has to search for other methods.

The quest for a finite expression for Pi could turn out to be related to the halting problem.

I do not know what is out there exactly (in the world of mathematics), but there could be some spherical, hyper-dimensional Moebius-space in which Pi has a rather simple expression -- which could be a comforting thought.

I am curious which numbers sustain such a new description, still being interesting and hard to crack. But honestly, I think a more complex system of symbols (and thus concepts) only shifts the problem.

At least, I can express Pi in a very finite way, but it needs a huge system of symbols and concepts to deflate it. Here it comes: Pi.

π

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