Are all irrational numbers equally irrational, or are there degrees or subsets of irrationality?
The rational numbers can all be expressed as a ratio of integers. The irrational numbers, like pi or sqrt2, cannot be expressed as ratios of integers. But some of them at least can be defined, like pi being the ratio of a circle’s circumference to its diameter, or the square root of an integer, or a sequence like 0.100111000011111 which concatenates n 1’s (if n is odd) with n 0’s (if n is even) for all n. On the other hand, I would guess that there are irrationals that can’t be expressed with a formula or algorithm or such, but I can’t think of an example ;).
Are such formulaic/defitional/algorithmic numbers a legitimate subset of the irrationals? Would they be ‘less irrational’ than an irrational that can’t be so described, like some infinities are smaller than others? Do the irrationals have some analogues of the primes, or perfect squares, or that sort of thing?
There are the algebraic numbers which are irrational but not transcendental like pi. Algebraic numbers are the roots of polynomials with integer coefficients.
Numbers can be rational or irrational, but almost all numbers are irrational.
Numbers can be constructable or unconstructable, and all rational numbers are constructable, bu almost all numbers are unconstructable. The square root of 2 is an example of a number that is constructable but not rational.
Numbers can be algebraic or transcendental (AKA non-algebraic), and all constructable numbers are algebraic, but almost all numbers are transcendental. The cube root of 2 is an example of a number that is algebraic but not constructable.
Numbers can be computable or incomputable, and all algebraic numbers are computable, but almost all numbers are incomputable. Pi is an example of a number that is computable but not algebraic.
Numbers can be definable or undefinable, and all computable numbers are definable, but almost all numbers are undefinable. The halting probability for any given programming language is an example of a number which is definable but not computable.
And even though almost all numbers are undefinable, by their very nature I cannot tell you any specific examples.
You raise a very interesting question, i.e. that of Kolmogorov-Chaitin complexity. This asks the question, what is the size of the smallest computer program that can be used to specify a number. This can be applied to numbers with a finite decimal expansion or an infinite one. You can write a small program to calculate pi or the square root of two, so these irrationals have low complexity. Such numbers are a small subset of all irrationals. From Wikipedia, it seems that for almost all irrationals, it is impossible to compute even a lower bound on the Kolmogorov-Chaitin complexity.
It is not the case that all constructible numbers are algebraic. Both pi and e are perfectly constructible but not algebraic. But the constructible numbers are rare. Here is an odd fact. Following Errett Bishop’s Constructive Analysis book, a number is constructive if it can be given as the limit of a sequence a_n/b_n where both sequences a_n and b_n are sequences of integers that can be generated by a computer program (technical name: recursively enumerable) and such that |a_n/b_n - a_m/b_m| < 1/n +1/m for all m and n ranging over the positive integers. A function is constructible if there is a computer program that computably turns one such constructible number into another. Then there is a constructible function that whose value at 0 is -1 and at 1 is +1, but is not 0 at any constructible number in between. This failure of the intermediate value theorem makes constructive analysis a very weak theory.
Thank you very much. By looking up this video which has one argument for how to measure the “irrationality” of numbers I learned of the existence of irrational number deniers.
It’s been a while since I learned this stuff, but I’m pretty sure that (a) all constructible numbers are the roots of some polynomial, and therefore algebraic; and (b) if you could construct pi, then you could square the circle.
ETA: I suspect that we’re looking at different definitions of “constructible”; the one I’m more familiar with involves constructing the numbers with a compass and straightedge, not a computer. I think that your definition is what I’ve heard called a “computable” number instead.
Note that there is an infinite extension of the computable number hierarchy.
Suppose you lucked into an “oracle” that answered all halting problem questions. The numbers computable with this added oracle attached to your Turing Machine is a larger set than the ones you could compute without it.
But those extra TMs still have their own halting problem. Suppose you were given a better oracle for those. Then you could define a larger still set of numbers computable using those numbers, and on and on.
What if you gave up doing this one level at a time and just “jumped” ahead thru all the levels using a super-oracle? Still larger set. And even more on and on.
Going the other direction, in the completely computable way, you can define the numbers generated in linear time, quadratic time, exponential time, linear space, non-deterministic double exponential time, etc. So a ton more sets (but not clear in all cases which are different).
Note: Generated in linear time for an infinite output would mean theres a c so that the nth digit in generated in at most cn (plus lower order terms) time.
“Constructible” means that it is possible, using a compass and straightedge, to construct two line segments such that the ratio of their lengths is exactly that number. So, for instance, to construct sqrt(2), I can draw one line segment, then make a perpendicular line to that segment, then mark off an equal distance on that perpendicular segment, and then connect the endpoints of the two segments (making a right isosceles triangle).
It is, of course, trivial to construct a curve whose length is equal to pi times the length of a given segment, but not a straight line.
Here’s a bonus bit of mathematical weirdness: The integers, the rationals, the constructable numbers, the algebraic numbers, the computable numbers, and the definable numbers all have the same cardinality (that is to say, all of those sets have the same size). The real numbers, the irrational numbers, the unconstructable numbers, the transcendental numbers, the incomputable numbers, and the undefinable numbers also all have the same cardinality as each other, which is larger than the cardinality of the integers. But is there some set of numbers that has a cardinality in between, strictly greater than the integers but strictly less than the real numbers?
And the answer is not only “nobody knows”, but “nobody can know”. It’s been rigorously proven that both possibilities are perfectly valid. Or, to put it another way, there are (at least) two different ways of defining “sets” and “numbers” and so on, that both look completely identical for all finite cases, but which differ in the answer to that question.
I don’t think “nobody can know” is particularly valid, although I might see what you’re getting at slightly. What’s known is that either conclusion is consistent with the rest of set theory, in the same way that there are multiple ways to construct geometries based on varying a parallel postulate. It’s just that no one has any more intuition for one than the other, while people have a great deal more intuition for Euclidean geometry compared to the alternatives. Some parallel postulate can only be postulated, not proven, and the same is true for the continuum hypothesis. People definitely know which one is true, because they decided on it themselves.
That’s sort of like saying “people know how game pieces are allowed to move on the board”. Sure, there’s one right answer for how chess pieces move, because people have chosen those rules, but that doesn’t mean that checkers (or Monopoly) is any less valid a game. Just as people have chosen one set of rules for “the standard set theory”, but that doesn’t stop anyone from doing mathematics in non-standard set theories, or in other fields of mathematics that don’t even resemble set theory at all.
As for geometry, one could even argue that Euclidean isn’t the standard. The origin of the word is “measurement of the Earth”, and when you’re restricting measurements to the surface of the Earth, you’re doing non-Euclidean (specifically, spherical) geometry.