Why Do Irrational Numbers Exist?

[csprof2000]

"You would use the digits 0 and 1."

Hmmm... alright. Interesting. So you can easily get 0, sqrt(2), 2, 2+sqrt(2), 2sqrt(2), etc.

Strangely, though, I think that:

110 = 2 + sqrt(2) + 0 ~ 3.4
1000 = 2sqrt(2) + 0 + 0 + 0 ~ 2.8

110 > 1000 in this system.

Is this alright? This seems to go against intuition, but I don't know enough about place value systems to know whether this invalidates it or not. Clearly in any integer base this can never happen... thoughts? Maybe I'm missing something.

It seems to be I should be able to divide both sides of the inequality above by 10, leaving
11 > 100, which is oddly enough also true.

 

[csprof2000]

And, for the record, I have said I believe sqrt(2) exists even though its decimal representation is non-terminating and non-repeating. Perhaps you recall the simple algorithm I gave for finding its digits?

I think the more interesting question has to do with numbers for which no algorithm can give the digits. Again, I'd like to throw an example out there, but how could I?

Maybe somebody can come up with a good example of a way to specify an incomputable number, so we can have something to work with.

For instance, is Chaitin's constant a well-defined real number? It is certainly real. There is a formula which gives it. Thoughts?

 

[CRGreathouse]

Strangely, though, I think that:

110 = 2 + sqrt(2) + 0 ~ 3.4
1000 = 2sqrt(2) + 0 + 0 + 0 ~ 2.8

110 > 1000 in this system.

Is this alright? This seems to go against intuition, but I don't know enough about place value systems to know whether this invalidates it or not. Clearly in any integer base this can never happen... thoughts? Maybe I'm missing something.

This is different from positive integer bases. Also, simplifying the form of a number is different from positive integer bases. (11 + 10 = 21 = 101 in binary, but 11 + 10 = 21 = 1001 in base sqrt(2).) Also Google for "phinary", base phi.

 

[csprof2000]

Hmmm... alright, then. I guess I don't have to like it...

 

[CompuChip]

no, i don't think sqrt(2) exists.
[...]
In other words, sqrt(2) by definition is a number that you multiply by itself in order to get 2. However, we will never be able to get that number so it should be undefined for the same reason that infinity is undefined.

OK, so you are saying that when I draw a square which has sides of exactly 1 unit, then the length of the diagonal does not exist?

Your objection could be that it is impossible to draw a perfect square with sides of exactly one unit, and that would be right: in a sense all numbers are "idealized" mathematical constructs.

 

[CRGreathouse]

Hmmm... alright, then. I guess I don't have to like it...

For bases greater than phi they compare the way you want, since then
b^2 > b + 1

Neat, huh?

 

[Gerenuk]

Is it correct to say that rational number are just the outcome of algorithms that produce rational numbers that are supposed to satisfy an equation? For example x^2=2 can be approximated with increasing precision by rational numbers. However the algorithm for sqrt(2) never stops at a perfect result.

In fact all irrational numbers are outcomes of a limiting process in algorithms?!

 

[csprof2000]

I would argue that infinite-precision numbers don't "exist", in a colloquial sense at the least. The only sensible way to talk about real numbers (not the set, mind you... lol) in my opinion is to define the precision. So sqrt(2) = 1 to one significant digit, 1.4 to 2 significant digits, etc.

If you define numbers this way, then certain irrational numbers - and all rational numbers - exist.

So, to answer your question, no. I don't think that any numbers "exist" as a limiting process of algorithms. I believe numbers exist which are the output of some algorithm which computes them. Non-terminating algorithms don't produce any numbers.

 

[Hurkyl]

Is it correct to say that rational number are just the outcome of algorithms that produce rational numbers that are supposed to satisfy an equation?

I'm really confused by this; I can't figure out what you're thinking.

In fact all irrational numbers are outcomes of a limiting process in algorithms?!

It really depends on what exactly you mean by "outcome", "limiting process", and "algorithm".

For example, every real number is the limit of a (Cauchy) sequence of rational numbers. However, there are irrational, real numbers that cannot be printed by a Turing machine.

 

[Hurkyl]

I would argue that infinite-precision numbers don't "exist", in a colloquial sense at the least.

"Exist" isn't particularly well-defined as a colloquial word -- I assert that it's much better to simply define a new word that is meant to refer to whatever notion you're trying to discuss, rather than debating what 'really exists' and what-not.

I believe numbers exist which are the output of some algorithm which computes them.

e.g. why not just talk bout "computable real numbers"? (for some particular specification of what it means to be 'computable')

Actually, "computable decimal numberals" is probably better for what you describe, since you seem to focus on the decimal representation specifically; Wikipedia implies that a slightly different concept is more typical.

The only sensible way to talk about real numbers (not the set, mind you... lol) in my opinion is to define the precision. So sqrt(2) = 1 to one significant digit, 1.4 to 2 significant digits, etc.

There is no such thing as the "precision of a real number" -- precision is a quality of {approximations to real numbers}.

Non-terminating algorithms don't produce any numbers.

This is somewhat artificial, because you can generally switch back and forth between terminating and non-terminating versions of the same algorithm.

e.g. a non-terminating algorithm A that outputs the decimal digits of some decimal representation of a real number^* can easily be converted into a terminating algorithm that will take an integer n as input, run A long enough to compute the first n digits, then output those digits and stop.

*: remember, some reals have two decimal representations!

 

[John Creighto]

Irrational numbers exist either because:

A: We assume the Pythagoras theorem always has a solution
or
B: We accept the supremum axiom.

 

[Hurkyl]

Irrational numbers exist either because:

A: We assume the Pythagoras theorem always has a solution
or
B: We accept the supremum axiom.

Both of which are provably true. Remember that if a set of 'numbers' doesn't satisfy the supremum axiom, then it's not a model of the real numbers.

Irrational numbers in other number systems can follow from much more modest assumptions. For example, the "circle continuity principle" of Euclidean plane geometry implies that irrational numbers exist, as does the "intermediate value theorem for polynomials".

For reference, the circle continuity principle says that if you have:

* Circles C and D,
* D contains a point inside of C,
* D contains a point outside of C,​

then C and D intersect.

 

[csprof2000]

Alright, so my wording was a little sloppy. Let me rephrase everything.


I believe that one must keep the ideas of "approximation to a real number" and "real number" separate.

There are no infinite-precision "approximations of real numbers". It only makes sense to talk about these in terms of how much information we have about them (significant digits, for instance).

Measurement can only return approximations to real numbers. Computers can only deal with approximations to real numbers. The human mind possesses only a finite number of neurons, and therefore deals with real numbers - and all numbers, really - in an approximate (throw away information) or symbolic (ignore how much information something really contains) way. Therefore, when one talks about "real number" in a colloquial sense, I assume they mean "approximate real number" in my sense of the word.

If one would like to talk about numbers of potentially infinite (though not actually infinite) precision, algorithms in the most general sense of the word can produce arbitrary amounts of precision. All the numbers I'm talking about are therefore computable.

And I wish you would let go the notion that my arguments depend on using a base-10 representation. They don't. Base 10 is one way to express computable numbers, the most common way (arguably), and that's why I'm framing these algorithms in terms of them. We could talk about roman numerals, or tick marks, or whatever.

"e.g. a non-terminating algorithm A that outputs the decimal digits of some decimal representation of a real number* can easily be converted into a terminating algorithm that will take an integer n as input, run A long enough to compute the first n digits, then output those digits and stop."

Apparently you are missing the point. The algorithms I'm talking about are exactly what you describe - they find the first n digits, and nothing after that. They work for all positive n, of course, and they don't have to be executable on an actually existing computer, but they should in principle be executable. Therefore the algorithm

number FindThreePointTwo(int n)

result = ""

for i = 1... n
if i = 1 then append(result, "3.").
else
if i = 2 then append(result, "2").
else
append(result, "0")

return result

Is what I've been saying is enough to define a number, for me. The vast majority of real numbers have no such algorithmic representation. All integers, rationals, roots, logarithms, exponentials, sines and cosines, etc. do. Most real numbers don't.

 

[Hurkyl]

I believe that one must keep the ideas of "approximation to a real number" and "real number" separate.

Agreed.

There are no infinite-precision "approximations of real numbers".

This is either a meaningless or a false statement. By any reasonable definition of the word 'precision', each of the following is going to be an infinitely precise representation of a real number:

(a) 1
(b) $\pi$
(c) 31.59918374
(d) $61.4\overline{09}$
(e) The real number whose decimal representation is computed by a particular Turing machine
(f) The real number whose decimal representation consists of 0's to the left of the decimal point, and whose n-th digit to the right of the decimal point is 0 if the binary representation of n denotes a Turing machine that halts, 1 if the binary representation of n denotes a Turing machine that does not halt, and 2 if the binary representation of n does not denote a Turing machine. (For some chosen way of encoding Turing machines as bits)
(g) a (where a is chosen to be a specific real number)
(h) x (where x is an indeterminate variable of type "real number")
​

so the question boils down to whether or not you are defining "approximation" to mean something that isn't infinitely precise.

Computers can only deal with approximations to real numbers.

Therefore, when one talks about "real number" in a colloquial sense, I assume they mean "approximate real number" in my sense of the word.

(moderator hat on) This is unacceptable. You'll note that this is a math subforum. Also, one of the primary goals of physicsforums.com is to promote education in science and math -- this cannot happen if you fill readers' heads with errors and misinformation. To be sure, the theory of computation is a very interesting subject, but you do the reader a great disservice to masquerade it as if you were actually talking about the real numbers. Hijacking threads is similarly problematic.

Maybe I should have taken some action earlier to split the computability stuff into a separate topic. *shrug* Nobody's complained, though; I think unless someone does, I'll let things continue. (moderator hat off)

And I wish you would let go the notion that my arguments depend on using a base-10 representation. They don't. Base 10 is one way to express computable numbers, the most common way (arguably), and that's why I'm framing these algorithms in terms of them. We could talk about roman numerals, or tick marks, or whatever.

I mean to distinguish it from a scheme such as the one at wikipedia -- there is not a computable transformation for converting a computable number (as defined by that scheme) into an algorithm that enumerates its decimal digits.

"e.g. a non-terminating algorithm A that outputs the decimal digits of some decimal representation of a real number* can easily be converted into a terminating algorithm that will take an integer n as input, run A long enough to compute the first n digits, then output those digits and stop."

Apparently you are missing the point. The algorithms I'm talking about are exactly what you describe...

I'm having difficulty imagining anything that could reasonably be described as a "outcome of a limiting process in algorithms" that doesn't involve an algorithm of the type I describe...

 

[Gerenuk]

I'm really confused by this; I can't figure out what you're thinking.

However, there are irrational, real numbers that cannot be printed by a Turing machine.

Can you write down these numbers for me please? :)
Well, I associated every irrational number with a little program that would give me the number digit by digit. This is not sufficient?

 

[CRGreathouse]

Well, I associated every irrational number with a little program that would give me the number digit by digit. This is not sufficient?

There are only countably many Turing machines -- not enough to have one for each of the uncountably many irrational numbers.

 

[Hurkyl]

Can you write down these numbers for me please? :)

I already wrote one down: the real number described in point (f) at the top of my previous post.

 

[HallsofIvy]

Can you write down these numbers for me please? :)
Well, I associated every irrational number with a little program that would give me the number digit by digit. This is not sufficient?

Hurkyl did in the post you refer to. "$\pi$" is an infinite precision way of writing down a particular irrational number. You are still confusing "a real number" with a particular representation of that number.

 

[johansebastia]

Im really curious about what an Irrational Number is, In the case of Pi it represents a way to calculate a physical object, a circle, how can a circle possably be irrational, I suppose a circle could ,in some context, be considered infinate.

 

[PeterJ]

I can't follow most of this, but is there not a sense in which all numbers are ill-defined in the sense that they represent a region on the number line that can never be reduced to a point? In this way could the OP's question be something to do with the relationship between a continuum and a series of points?

 

[Hurkyl]

I can't follow most of this, but is there not a sense in which all numbers are ill-defined in the sense that they represent a region on the number line that can never be reduced to a point? In this way could the OP's question be something to do with the relationship between a continuum and a series of points?

I don't think the OP is paying attention anymore.

And while I'm sure there are number systems with that property that individual numbers represent a region on the line,

• Such numbers would not be ill-defined (unless they were still conjectural)
• Such a number system would not be the the real number system with its usual correspondence to the line

 

[PeterJ]

Forgetting the OP then, I'd like to ask something about this.

I see that a region may be well-defined, so that being a region would not necessarily entail that a number is ill-defined. (Is this what you meant?) But... couldn't we say it is ill-defined when we forget that it's is a region and treat it like a point, as people often do? And, if the end points of a region cannot be well-defined, (because they are points), then couldn't we say that the region is ill-defined?

Just exploring, not promoting a view. I suppose I'm thinking about this mechanically. If I put a number on the number line then I have to let it cover more than one point - which seems to make it's position ill-defined, or less well defined than it is when we're counting apples.

 

[abcd1019]

If sqrt(2) does not exist, does this mean that the number 2 does not exist? Or in general, 2^n does not exist?

 

[PeterJ]

For me the question is not whether these numbers exist but what they actually are. Whether they exist would seem to depend on how we define them.

 

[HallsofIvy]

In what sense do you mean "what they are"? They are numbers as defined in many equivalent ways. Do you know, for example, the Dedekind cut definition of real numbers?

 

[PeterJ]

Yes. Is it not a surreal fantasy about infinitely thin knives that produces a useable definition for infinitely 'thin' numbers? Whether such numbers exist (or whether numbers can be coherently defined in this way), I was suggesting, can be determined from examining the definition.

I suppose my thought is that there is no way to define a continuous series of numbers such that the numbers are determinate. Or something like that. Counting apples is easy enough, but dividing a continuum into finite parts requires a leap of imagination that leaves reality behind.

 

[disregardthat]

My theory is that people who disagree about these things are the ones who would answer the question "does god know all decimals of pi?" differently.

 

[PeterJ]

Differently to what? Damn silly question is the answer most people would give.

 

[disregardthat]

Differently to what? Damn silly question is the answer most people would give.

Different from each other, of course.

 

[PeterJ]

Oh yes. I see. Thanks. In that case it's a good point. But I wouldn't agree.

 

[Hurkyl]

Ah here it is! I lost track of this thread.

Forgetting the OP then, I'd like to ask something about this.

I see that a region may be well-defined, so that being a region would not necessarily entail that a number is ill-defined. (Is this what you meant?)

Right.

But... couldn't we say it is ill-defined when we forget that it's is a region and treat it like a point, as people often do?

People don't often work with number systems whose numbers are regions. If you are using a number system incorrectly, then you are simply using it correctly.

And, if the end points of a region cannot be well-defined, (because they are points), then couldn't we say that the region is ill-defined?

I'm not sure what sort of geometry you're envisioning where regions have endpoints but points are ill-defined. I suspect you have forgotten that points are points!

If I put a number on the number line then I have to let it cover more than one point

No! The line you drew on a sheet of paper, and the point you marked on it, is not a number line and it is not a geometric point. If you are using such a physical object to help you visualize the mathematical ideas, then you need to understand what parts of the object really do correspond to the math and what parts are simply errors of approximation.

Conversely, if it is the physical object we are trying to study, then the mark on the paper is not a geometric point. For many purposes it is useful to use a geometric point as an approximation, but you would be similarly in error if you think the two are the same.

 

[Hurkyl]

Incidentally, there's a subject jokingly called "pointless topology" where one defines things called locales without reference to the notion of point -- they are just made out of "regions", and you can take finite intersections and arbitrary unions of regions.

But even locales (usually) do have points, and many (most? all?) can be completely and perfectly described as a topological space -- the usual notion of a set of points together with a set of regions that define a topology.

 

[PeterJ]

Hurkyl - I think I can accept what you say (and I do) without it altering my general point, which comes exactly from trying to understand what parts of the object really do correspond to the math and what parts are simply errors of approximation. For most objects there may be no problem being precise since when we define the object we define it as being one object. An abacus raises no problems of precision. But an infinitely divisible or continuous number line is a unique and idealised object. Or so it seems to me at the moment. Danzig proposes that foundational (and thus metaphysical) issues arise from trying to match the staccato of the numbers to the legato of the number line (or of the world itself), and it's this issue that I find interesting.

I see your point about the dangers of using a physical object to visualise the maths, but I was only using a physical object as a metaphor for the number line. Not so sure I see the importance in this context of the difference between a geometric and a mathematical point. Are they not both man-made objects?

Pointless topology sounds like my kind of thing! That we can describe a locale as a set of points and set of regions may be irrelevant to my concerns, however, since a definition need not be coherent outside of the theory it's designed to support. I was suggesting (if we use the everyday meaning of these words) that a point is a region is a locale, depending on where we are standing, such that that the universe is a point if we stand in the right place. There would be no points on the number line, only arbitrarily defined regions seen from a distance. Or, looking at it the other way, no regions only ill-defined points under magnification.

If this sort of woolly talk is innapropriate here just let me know. I learn in this way, and it's not that I've got some half-baked uber theory of numbers that I'm peddling.

 

[Hurkyl]

I was suggesting (if we use the everyday meaning of these words) that a point is a region is a locale,

Locale theory doesn't equate point with region -- it just takes region as the fundamental primitive all by itself, and tries to describe the relationships between regions without reference to the notion of point.

The contrast with topology is that it takes both "point" and "region" as primitives. (but by invoking set theory, we can identify regions with sets of points)


The net trick is that, using only the notion of "region", we can still develop the notion of a point. One way of looking at it (I believe) boils down to identifying points by specifying a sequence of regions whose "limit" would define the point.

For example, in the locale version of the number line, we can identify the point 3 via the infinite sequence of intervals

(2,4), (2.9,3.1), (2.99,3.11), (2.999,3.111), ...​

(I'm using the normal naming scheme for the open intervals of the number line, because we are all familiar with that naming scheme)

Then, all properties of the point "3" are simply certain kinds of properties of the above sequence of regions. This, of course, is very similar to the classical notion of limits and completeness.



As for "woolly talk", if you're just lightly throwing out any idea that comes to mind, it is inappropriate here. However, if you're serious about trying to pin down actual meanings to the things you say and see how they might be arranged into coherent ideas and how they might relate to things that people have already developed, it might be appropriate in one of the forums here depending on the direction you're going.

Over the years, I have become rather convinced that most people who have some idea of "ill-defined regions" are really struggling to develop the various concepts of topology on their own -- but they have crippled themselves by developing a serious allergy to the notion of a point, so they never even have a chance to learn whether or not their is already a mathematical approach to working with their ideas.

 

[PeterJ]

Locale theory doesn't equate point with region -- it just takes region as the fundamental primitive all by itself, and tries to describe the relationships between regions without reference to the notion of point.

The contrast with topology is that it takes both "point" and "region" as primitives. (but by invoking set theory, we can identify regions with sets of points)

Thanks. (I was careful to add 'in the everyday sense' when I used these words.)

The net trick is that, using only the notion of "region", we can still develop the notion of a point. One way of looking at it (I believe) boils down to identifying points by specifying a sequence of regions whose "limit" would define the point.

For example, in the locale version of the number line, we can identify the point 3 via the infinite sequence of intervals

(2,4), (2.9,3.1), (2.99,3.11), (2.999,3.111), ...​

(I'm using the normal naming scheme for the open intervals of the number line, because we are all familiar with that naming scheme)

That's how I imagine points are usually defined, as the end point of a never ending process. I assume that they simply have to be defined in this sort of way. It's the issues this raises that interest me.

Over the years, I have become rather convinced that most people who have some idea of "ill-defined regions" are really struggling to develop the various concepts of topology on their own -- but they have crippled themselves by developing a serious allergy to the notion of a point, so they never even have a chance to learn whether or not their is already a mathematical approach to working with their ideas.

Very excellent comment. I'd never thought of this. Must be annoying. But there can be some reasoning behind a dislike of points, as you'll be well aware, and it's not necessarily just an allergy. I see no possibility of constructing a coherent metaphysic for which points are not a convenient fiction, and I think this is a quite widespread view. This raises some important mathematical issues, and it makes the relationship between mathematics and reality an important issue for everybody, not just mathematicians.

Now you come to mention it I realize that it's true, I've never enquired whether there's a mathematical approach that would encompass my ideas about the numbers. I expect the answer would take me well out of my depths, and anyway, it seems to me all the maths is already done. Peirce's arithmetic of circles and Spencer-Brown's calculus of indications would get my vote as a place to start, as they're conceptually simple and I share their view of points/numbers'regions etc as far as I can tell, but I don't know whether they'd be relevant here. I don't think they'd have any bearing on the definition of points, for example, for this would be a matter of convenience.

Is it safe to say that a continuum, whether it is the number line or spacetime, and regardless of whether it is conceptual or real, cannot be made of points according to reason. Or is even this debatable?