Why Do Irrational Numbers Exist?

[timjones007]

why do irrational numbers exist? I am well familiar with the proof that irrational numbers exist, but why do they?

 

[Hurkyl]

why do irrational numbers exist? I am well familiar with the proof that irrational numbers exist, but why do they?

Your question doesn't really make sense. If you know the proof, then what's your problem?

Are you using the word "why" in some unusual way? If so, you really should have said that up front...

 

[Unknot]

Well, your question does seem odd, but my guess is that you want to ask a philosophical question.

Let me ask you a question. Do rational numbers exist? how do you know this?

 

[CRGreathouse]

For me, the intuitive answer is "because there aren't nearly enough rationals to 'fill in' all the gaps".

 

[Unknot]

Well, I don't understand why people think rational numbers exist and some numbers don't. It's just easier to think that all numbers are mathematical constructs and real numbers are simply, yes, way to fill in gaps between rational numbers.

I wonder what the op thinks of complex numbers.

 

[csprof2000]

Oh jeez, I'll try to tread softly in this thread.

It's actually a very interesting question the OP is getting at, and one I've often thought about myself. How much information do we need to have about a number before we can consider the number to be well-defined? Are all numbers which provably exist well-defined under our definitions of well-definedness of a number? Is there a definition of the well-definedness of numbers?

One can also talk about whether numbers are computable or not. It's interesting that the real numbers are most incomputable... what does this mean? What can even be meant by incomputable number?

I think it's an interesting discussion. To the OP: do you think that sqrt(2) exists, and in what sense do you think this? I mean, we both know that there is a proof that it is not rational. It's a relatively tame irrational number. Why do you feel the way you do? I could enjoy this conversation.

 

[timjones007]

no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely.

So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself.

It's the same logic that goes with the fact that we can't multiply any number by infinity. For example, 0 x infinity is an ideterminate form, because although we know logically that you will get 0 if you keep multiplying 0s, we will never finish multiplying 0 infinitely many times so we say that it is undefined.

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.

 

[CRGreathouse]

no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely.

So you don't accept 1/9 = 0.111111... or 1/10 = 0.10000... either?

What about this: I define "foo" as an ordered pair (a, b) where (a, b) = (c, d) iff (a - c)(b - d) = 0 and the operations plus and times are defined by (a, b) + (c, d) = (a + c, b + d) and (a, b) * (c, d) = (ac + 2bd, bc + ad). Do "foo"s exist?

How about "bar"s, where "bar" is an ordered pair (a, b) where (a, b) = (c, d) iff ad = bc and the operations plus and times are defined by (a, b) + (c, d) = (ad + bc, bd) and (a, b) * (c, d) = (ac, bd)? Do "bar"s exist?\

Maybe "baz", where a "baz" is (a) where (a) = (b) iff a - b = 0 and the operations plus and times are defined by (a) + (b) = (a + b) and (a) * (b) = (ab). Do "baz"s exist?

 

[Unknot]

no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely.

So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself.

It's the same logic that goes with the fact that we can't multiply any number by infinity. For example, 0 x infinity is an ideterminate form, because although we know logically that you will get 0 if you keep multiplying 0s, we will never finish multiplying 0 infinitely many times so we say that it is undefined.

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.

So tell me, what does "exist" mean? Do you think 2 exists? How so? Is there a realm of numbers where 2 exists but sqrt(2) doesn't?

what about sqrt(2) km? Does that exist?

 

[HallsofIvy]

It might be good to point out that while asking if a number is "well-defined" it makes no sense to focus entirely on the decimal representation of the number, as timjones007 does in #7. The decimal representation of a number is just that- a representation- and has little to do with the properties of the number itself.

 

[Hurkyl]

It's actually a very interesting question the OP is getting at ...

I would be very surprised if he was actually asking interesting questions about formal language, computability theory, or anything like that. I think he simply doesn't have a clear understanding of what others (and he) means by 'number', and lacking such clarity, is flailing about with his intuition.

 

[csprof2000]

"no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely. "

So you take the definition of a number as its decimal representation? This would take a little elaboration to take into account the (very valid) objection raised by CRGreathouse. For instance, you could say that a number is well defined if its decimal representation repeats with a string of digits of finite length L for all places N at least N_0 to the right of the decimal. This covers repeating decimals (1/9 = 0.1111... letting N_0 = 1 and the string of digits being "1", and 1/10 = 0.1000... is covered letting N_0 = 2 and the string of digits being "0", etc.) Obviously, the choice of 10 as the base doesn't make any difference... you could allow this to vary as well.

But the real problem with that is that you're taking the properties of rational numbers and saying that's what makes a number well-defined. Does that make sense? I mean, if we are trying to show that irrational numbers are not well defined, it's a little self-serving to equate a property of rational numbers with well-definedness. Savvy?


"So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself. "

Well, the problem with this is that, as CRG said, 1/10 = 0.100... and this technically also goes on forever... perhaps a better way of saying what you're thinking is that you have a finite set of rules with which you can always generate the next digit in the decimal (or some other sensible) representation. For instance, 1/10 is well-defined because I can say "tenths' place 1, all other places 0" and you can use the two rules to write out the number to any desired number of digits. Does this sound alright, tim?

The only snag with that, of course, is that sqrt(2) is also well defined by this definition of well-definedness. Consider this: sqrt(2) can be found as follows:

sqrt(n)::
x := 0
p := n // could be made more efficient, but who cares?

for p = n to p_min
begin

while x <= n
begin
x = x + 10^p
end
x = x - 10^p

end

Let's see this operating on n = 2.

x = 0.
x = 100, p = 2.
x = 0, p = 2.
x = 10, p = 1.
x = 0, p = 1.
x = 1, p = 0.
x = 2, p = 0.
x = 1, p = 0.
x = 1.1, p = -1
x = 1.2, p = -1
x = 1.3, p = -1
x = 1.4, p = -1
x = 1.5, p = -1
x = 1.4, p = -1
x = 1.41, p = -2
etc.

As you can see, this will always allow you to find the nth decimal digit in a finite number of steps... so you would need a stricter definition than the one I provided to exclude sqrt(2).

 

[csprof2000]

What does everybody else think about what it takes to define a number? Do numbers have to have a value? If so, and you know a number exists for which you cannot possibly find its value... does this mean anything?

 

[Hurkyl]

What does everybody else think about what it takes to define a number?

A type of number system is defined by a list of properties. If a particular set^* has those properties, it's a model of that number system, and we would call its elements numbers (of the appropriate type).

(The properties don't have to be complete -- though the definitions for common number systems like the integers or the reals are complete in the appropriate sense)

*: Or type or class or language or whatever foundational gadget you want to use.

Once you actually have an actual, concrete list of properties to work with, you can usually answer simple questions relatively easily. e.g. it's fairly straightforward to show that

* in the rational numbers, 2 doesn't have a square root. (what would its factorization be?)

* in the real numbers, 2 does have a square root. (construct it as the least upper bound)

* for fields the question is undecidable -- some fields do and some fields don't have a square root of 2. (as shown by the previous two examples)

 

[Take_it_Easy]

no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely.

So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself.

It's the same logic that goes with the fact that we can't multiply any number by infinity. For example, 0 x infinity is an ideterminate form, because although we know logically that you will get 0 if you keep multiplying 0s, we will never finish multiplying 0 infinitely many times so we say that it is undefined.

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.

You are right! sqrt(2) does not exist.
And in fact also 1 does not exist.
1 is the multiplication of 13/7 and 7/13 now
13/7 and 7/13 are just symbols for their decimal representations which are
13/7 = 1,85714285...
7/13 = 0,53846153...
and the decimal places continue on infinitely.

So, if we will never reach the last digit in the decimal places for these two numbers, how can we multiply them together?

In other words, 1 is a number that you get multiplying 13/7 by 7/13.
However, we will never be able to get that number so it should be undefined for the same reason that infinity is undefined.

 

[Dadface]

This is an interesting question and I think it is one that has been discussed since ancient times.According to David Wells (the penguin dictionary of curious and interesting numbers) pi is the only irrational and transcedental number that occurs naturally.People here have been using root 2 as an example and I have been trying to think of an example where this number can be given a unit.Suppose we were told that a square had an area of root 2 metres squared.Does this mean anything when such a square cannot be consructed or have I picked on a dopey example?

 

[csprof2000]

Dadface:
That could be a dopey example.
What about the distance between opposite corners of a square of area 1?

Hurkyl:
I see what you're saying, but I think the problem we're all having is in communicating. I agree that you're absolutely right about numbers... a very clear and thoughtful exposition.

However, I think that the OP means to talk about the value of numbers, not their properties... to know what the number is, not whether it is there or not. I mean, 2 *is* an integer, but how big is 2? We can get to 2 using a finite number of logical steps. Is sqrt(2) a real number? The OP didn't think so, but perhaps after my last post he will agree that sqrt(2) must exist as well... since we can get as close as we like to it on a whim. But in what sense do the numbers which we cannot find values for have these values - even if we know the number must exist?

I apologize that the discussion is a little vague. I'd love to give you an example of such a number, but obviously I can't... I don't know, maybe the reason this topic isn't more mainstream is that it's a rabbit hole, makes no sense, and has no good answer.

 

[Dadface]

Yes csprof2000,yours is a good example,so numbers like root 2 come up but we can't measure them.It is a rabbit hole.

 

[Dadface]

You can construct a square of side 1 by drawing the sides and then you can draw in the two lines from opposite corners.You can't do it the other way round though by drawing the two lines first.Sorry but I am not exactly sure what I am getting at here,just chucking a few thoughts in as they come.This rabbit hole can make the brain ache-time for a nice cup of tea.

 

[csprof2000]

No, you're right, it is sort of strange. Although, I challenge you to draw a line that's exactly 1 unit long, and prove that you got that right...

Basically, how do you get anything exactly right? How close is close enough to be exactly right?

 

[Hurkyl]

No, you're right, it is sort of strange. Although, I challenge you to draw a line that's exactly 1 unit long, and prove that you got that right...

Basically, how do you get anything exactly right? How close is close enough to be exactly right?

But now, you're not doing mathematics anymore -- you've crossed over into physics, or possibly epistemology.

 

[HallsofIvy]

Yes csprof2000,yours is a good example,so numbers like root 2 come up but we can't measure them.It is a rabbit hole.

What do you mean we can't measure them? We can construct and measure a length \sqrt{2} as well as we can a length 1. It is true that we cannot write that out in terms of decimal numerals, but that is a problem with the numeration system, not the number.

 

[HallsofIvy]

No, you're right, it is sort of strange. Although, I challenge you to draw a line that's exactly 1 unit long, and prove that you got that right...

Basically, how do you get anything exactly right? How close is close enough to be exactly right?

What exactly do you mean by "draw a line that's exactly 1 unit long"? In Euclidean geometry, we simply declare a segment to have length 1 and base everything else on that. I can then construct a segment that has length exactly \sqrt{2}. (The physical "compasses" and "straight edge" represent the mathematics that is going on. Physical measurement is "approximate". Mathematical construction is not.) If you want to continue in this line, you should discuss \sqrt[3]{2} which is not a "constructible" number!

In any case, \sqrt{2}, and even \sqrt[3]{2} are as well defined as "1", "2", or "1/2".

 

[CRGreathouse]

If you want to continue in this line, you should discuss \sqrt[3]{2} which is not a "constructible" number!

Not with compass and straightedge, anyway. But it's possible with a marked straightedge, a http://www.museo.unimo.it/labmat/trisetin.htm , or origami...

 

[csprof2000]

"What exactly do you mean by "draw a line that's exactly 1 unit long"?"

Exactly what you said in your first post after me, if you think about what the meaning of that is. The point is both 1 and sqrt(2) are equally hard to nail down to infinite precision. I was just using the same language as Dad to say it, and you were using other language.

And what about the discussion of the value of numbers? Do numbers have to have value? And do you have to be able to find the value, at least in principle?

 

[HallsofIvy]

"What exactly do you mean by "draw a line that's exactly 1 unit long"?"

Exactly what you said in your first post after me, if you think about what the meaning of that is. The point is both 1 and sqrt(2) are equally hard to nail down to infinite precision. I was just using the same language as Dad to say it, and you were using other language.

And what about the discussion of the value of numbers? Do numbers have to have value? And do you have to be able to find the value, at least in principle?

Do you understand the difference between the value of a number and the value as written in a particular number system? Certainly, in our standard decimal number system, "1" is already written to infinite precision, \sqrt{2} is not. But, as I said before, that is an artifact of the numeration system, not the numbers themselves. If I were to use a place-value system, base \sqrt{2}, I can write \sqrt{2} to "infinite precision": 10.

 

[CRGreathouse]

If I were to use a place-value system, base \sqrt{2}, I can write \sqrt{2} to "infinite precision": 10.

Yep. That's similar to my suggestion in post #8 (work in \mathbb{Z}[\sqrt2]).

 

[timjones007]

Let me just say this in response to post number 17 by csproof2000.

Suppose there are two pieces of string forming the legs of an isosceles right triangle and each string measures 1 meter in length.

Now suppose that I ask you to cut a string of sqrt(2) meters to complete the triangle. Even if you use the most accurately calibrated meter stick that can be created, you wouldn't be able to do it because you would know where to stop cutting.

You might want to stop at 1.41 m, but that's to0 small. Then you'd try 1.414213562 m, but that is also too small.
(by the way, not that it matters, but I, the op, am a female...just thought I might clear that up since everyone keeps saying he)

 

[Unknot]

Let me just say this in response to post number 17 by csproof2000.

Suppose there are two pieces of string forming the legs of an isosceles right triangle and each string measures 1 meter in length.

Now suppose that I ask you to cut a string of sqrt(2) meters to complete the triangle. Even if you use the most accurately calibrated meter stick that can be created, you wouldn't be able to do it because you would know where to stop cutting.

You might want to stop at 1.41 m, but that's to0 small. Then you'd try 1.414213562 m, but that is also too small.
(by the way, not that it matters, but I, the op, am a female...just thought I might clear that up since everyone keeps saying he)

This is very strange. I don't believe this.

Regardless, I just want to say, I find it very strange that you think you can somehow cut something down to a rational number but not an irrational number.

You should surely know that there are infinitely more irrational number than rational number? If you cut a stick, for example, the probability of getting a rational length is 0.

 

[Werg22]

Rational/irrational are abstract concepts and the length of a string is a property of abstract line segments. To say that a physical string has exactly rational length or irrational length is absurd. Only within an accepted error range does it make sense.

 

[Dadface]

I think there has been a misunderstanding of timjones007 post here in that she was describing a principle and you can,in principle, cut a string so that it has a length of say one metre.The practical difficulties of doing this and any experimental errors/uncertainties are not relevant to the point being made.Here is another example,we can and do take a certain platinum iridium bar and define this to have a mass of 1 Kg.Having made this definition can we construct,in principle,something that has a mass of root 2 kg?We cannot.
What am I talking about?Time ,I think,for another nice cup of tea.Lovely.

 

[csprof2000]

"Do you understand the difference between the value of a number and the value as written in a particular number system? Certainly, in our standard decimal number system, "1" is already written to infinite precision, is not. But, as I said before, that is an artifact of the numeration system, not the numbers themselves. If I were to use a place-value system, base , I can write to "infinite precision": 10."

Of course I understand the difference between value and its representation in a particular base. I apologize if I made it seem that I did not.

What I'm not sure I understand is how one can work in a place-value system where the base is not an integer, or perhaps in some exotic sense a rational number. Unless you're talking about something more interesting than I've ever seen, it doesn't make sense to say you have a number in base sqrt(2)... I mean, what are the finite set of symbols one uses in such a notation to denote place value? In binary, they are {0, 1}, in decimal {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, etc. For any integer base, it's easy to come up with the set of symbols.

But for sqrt(2)? How many should there be? 2? 1? 3? And what on Earth would you call them, and what does it mean? How would you right 2.5 in base sqrt(2)... I understand it's irrational in this base, if anything, but still...


However, you are committing a fallacy, Halls, and I think it's time to call you out on it. You keep saying that it's not valid to rely on the decimal representation of a number in order to talk about its value. My problem with this is, quite simply, that the decimal notation is one representation of a number, and I see no reason to exclude it as a way of talking about a number's value. Of course, if you have other notations which make the number's value easy to arrive at, fine, but it's all too easy to avoid talking about value by hand waving arguments related to their representation. Yes, the representation isn't the number. But we need a representation to talk about its value in a clear way. If you have another way, please, let me know. Otherwise, let's drop the whole "the representation isn't the number" thing. Ink isn't the same as the Declaration of Independence, but it makes it a lot easier to read.

 

[Unknot]

However, you are committing a fallacy, Halls, and I think it's time to call you out on it. You keep saying that it's not valid to rely on the decimal representation of a number in order to talk about its value. My problem with this is, quite simply, that the decimal notation is one representation of a number, and I see no reason to exclude it as a way of talking about a number's value. Of course, if you have other notations which make the number's value easy to arrive at, fine, but it's all too easy to avoid talking about value by hand waving arguments related to their representation. Yes, the representation isn't the number. But we need a representation to talk about its value in a clear way. If you have another way, please, let me know. Otherwise, let's drop the whole "the representation isn't the number" thing. Ink isn't the same as the Declaration of Independence, but it makes it a lot easier to read.

I'm lost. We can have sqrt(2) to infinite precision by, writing, sqrt(2). You don't want to exclude decimal notation, but then why exclude the possibility of simply writing down sqrt(2) in a precise way? Aren't you making a self-contradiction? The representation is NOT a number! Everyone knows sqrt(2) does not have a "nice" representation in decimal notation.

I will tell you a number you cannot rely on a decimal system - what about an ordered pair? What about sqrt(-1)? Decimal numbers cannot represent every number, so the converse should not be considered at all.

 

[HallsofIvy]

However, you are committing a fallacy, Halls, and I think it's time to call you out on it. You keep saying that it's not valid to rely on the decimal representation of a number in order to talk about its value.

I did not say any such thing!

My problem with this is, quite simply, that the decimal notation is one representation of a number, and I see no reason to exclude it as a way of talking about a number's value. Of course, if you have other notations which make the number's value easy to arrive at, fine, but it's all too easy to avoid talking about value by hand waving arguments related to their representation. Yes, the representation isn't the number. But we need a representation to talk about its value in a clear way. If you have another way, please, let me know. Otherwise, let's drop the whole "the representation isn't the number" thing. Ink isn't the same as the Declaration of Independence, but it makes it a lot easier to read.

If someone were to say that the Declaration of Independence is meaningless because the ink is too faded to be read, don't you think that talking about difference between the content and the representation in ink is relevant?

 

[CRGreathouse]

But for sqrt(2)? How many should there be? 2? 1? 3? And what on Earth would you call them, and what does it mean? How would you right 2.5 in base sqrt(2)... I understand it's irrational in this base, if anything, but still...

You would use the digits 0 and 1.

Numbers are rational or irrational regardless of how you display them -- I think you meant "non-terminating".

2.5 terminates in base sqrt(2): it is exactly 100.01. Pi is 1000.00010001000000000000010010000000000100001... 1/3 is the repeating 'decimal' 0.00010001000100010001...

 

[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![/size]

 

[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)[/color]

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?