Where are the irrational numbers?

[agentredlum]

The arctan function is the bijection. The arctan function maps the reals bijectively to a bounded open interval.

Any non-vertical line through the origin has slope y/x, where (x,y) is any point on the line. In particular if you choose a point on the unit circle, then the line intersects the unit circle at the point (cos(t), sin(t)) where t is the angle the line makes with the positive x-axis.

What's the slope of the line passing through the origin and the point (cos(t), sin(t))? It's sin(t)/cos(t) = tan(t).

We are interested in the restriction of the tangent function to the open interval ]-pi/2, pi/2[. That restriction maps an angle in the open interval ]-pi/2, pi/2[ to a slope in the reals. And the map is bijective.

Since the (restricted) tan is bijective, it has an inverse. What's its inverse? It's the arctan. So the arctan function maps all the reals to the interval ]-pi/2, pi/2[.

It's helpful to look at the graphs of the tan and arctan to see how we're selecting one of the many connected components of the graph of the tan; and using that as a bijection.
Not sure exactly what you mean. The arctan is the function that maps the real numbers to the angles between -pi/2 and pi/2. Nothing "intersects arctan." And the line only goes halfway around the circle, if that's your concern. We don't care about angles you get when you go past the y-axis. Was that your concern? That's the restriction idea above.
No, that's not true. The tangent function is not defined at +/- pi/2. We are only concerned about tan on the open interval ]-pi/2, pi/2[. It's not correct to say that it's "plus or minus infinity."

There are some situations in general where it's useful to define the values of a function in the extended real numbers; but this is not one of those situations! If we restrict our attention to the open interval where tan does not blow up, we avoid exactly the problem you mentioned.

Not sure what the concern is. These are just visualizations to show that a bounded line segment is bijectively equivalent to an unbounded one. In fact they're topologically equivalent: you can choose a bijection that's continuous in both directions. This example shows that a continuous function can transform a bounded set into an unbounded one and vice versa.

Credit where credit's due. Micromass already gave the function that maps the reals to the open interval ]-a, a[ using the arctan function. Earlier you mentioned you can't see the TeX, here's the ASCII:

R -> ]-a, a[ : x -> (2a/pi) * arctan(x)

This entire discussion is already implicit in that symbology. I'm just providing the visualization.

Oh i get it now, is x any real number? The domain of arctanx is -infinity, +infinity the range is -pi/2, pi/2 this shows a fit of all real numbers in that interval ]-pi/2,pi/2[ why couldn't you guys say so to begin with?

Concerning my comment about hitting arctanx twice...if you rotate a line on the x-axis counterclockwise using origin as pivot then the left part of the line hits arctanx as well as the part on the right. You can fix this if you use half a line not the whole x-axis. but that does not mean using half a line won't cause other difficulties, i can think of a few.

You talked about rotating a line sitting on the x-axis this will hit arctanx twice, once on the right once on the left except when the line makes angle 90 degrees, then it hits arctanx only once. Have i misunderstood your original post?

about my use of infinity, didn't you use it first?

Steve, if you approach zero angle from above on the x-axis the right part of your line aproaches x=+infinity in arctanx and y approaches pi/2. However the left part of your line aproaches x=-infinity in arctanx and y approaches -pi/2 so your observation that tan(pi/2) is undefined is a bit misleading

 

[SteveL27]

Oh i get it now, is x any real number? The domain of arctanx is -infinity, +infinity the range is -pi/2, pi/2 this shows a fit of all real numbers in that interval ]-pi/2,pi/2[ why couldn't you guys say so to begin with?

The domain is the open interval ]-infinity, +infinity[. That notation is a shorthand for "the domain is all of the real numbers." That's a legitimate use of infinity. The open brackets mean that +/- infinity are NOT part of the domain; nor are they in the range of the tangent function. Using infinity that way is just a shorthand. And it's essential to understand that +/- infinity are not elements of the domain of the arctan.

Concerning my comment about hitting arctanx twice...if you rotate a line on the x-axis counterclockwise using origin as pivot then the left part of the line hits arctanx as well as the part on the right. You can fix this if you use half a line not the whole x-axis. but that does not mean using half a line won't cause other difficulties, i can think of a few.

If you prefer to think of the directed ray emanating from the origin, that's fine. But you don't actually need to.

Consider the line y = 2x. It passes through the point (1,2) so its slope is 2. But if we instead take the point in the third quadrant (-1, -2), the slope is still -2/-1 = 2. The tangent is the slope, period. And the angle is the angle made with the positive x-axis in the counterclockwise direction. That's the standard convention.

Why you keep saying it "hits arctan" is a complete mystery to me. It shows that you are misunderstanding something. The tangent is the slope as a function of the angle. The arctangent is the angle as a function of the slope.

You talked about rotating a line sitting on the x-axis this will hit arctanx twice, once on the right once on the left except when the line makes angle 90 degrees, then it hits arctanx only once. Have i misunderstood your original post?

In your latest post you seem to have some misunderstandings. I never said any such thing as "hitting arctan." You keep saying that, and I keep trying to correct that misunderstanding.

The slope of a vertical line is undefined.

about my use of infinity, didn't you use it first?

I used the notation ]-infinity, +infinity[ as a shorthand for "all the real numbers. That's a legitimate usage. The slope of a vertical line is undefined. The tangent of pi/2 is undefined.

Steve, if you approach zero angle from above on the x-axis the right part of your line aproaches x=+infinity in arctanx and y approaches pi/2. However the left part of your line aproaches x=-infinity in arctanx and y approaches -pi/2 so your observation that tan(pi/2) is undefined is a bit misleading

Do you understand the slope of a line? What is the slope of the line y = 2x? Does it matter whether you compute the slope using a point in the first quadrant or in the third quadrant?

The angle a line makes with the positive x-axis in the counterclockwise direction is unambiguous.

 

[agentredlum]

The domain is the open interval ]-infinity, +infinity[. That notation is a shorthand for "the domain is all of the real numbers." That's a legitimate use of infinity. The open brackets mean that +/- infinity are NOT part of the domain; nor are they in the range of the tangent function. Using infinity that way is just a shorthand. And it's essential to understand that +/- infinity are not elements of the domain of the arctan. If you prefer to think of the directed ray emanating from the origin, that's fine. But you don't actually need to.

Consider the line y = 2x. It passes through the point (1,2) so its slope is 2. But if we instead take the point in the third quadrant (-1, -2), the slope is still -2/-1 = 2. The tangent is the slope, period. And the angle is the angle made with the positive x-axis in the counterclockwise direction. That's the standard convention.

Why you keep saying it "hits arctan" is a complete mystery to me. It shows that you are misunderstanding something. The tangent is the slope as a function of the angle. The arctangent is the angle as a function of the slope.

In your latest post you seem to have some misunderstandings. I never said any such thing as "hitting arctan." You keep saying that, and I keep trying to correct that misunderstanding.

The slope of a vertical line is undefined. I used the notation ]-infinity, +infinity[ as a shorthand for "all the real numbers. That's a legitimate usage. The slope of a vertical line is undefined. The tangent of pi/2 is undefined.

Do you understand the slope of a line? What is the slope of the line y = 2x? Does it matter whether you compute the slope using a point in the first quadrant or in the third quadrant?

The angle a line makes with the positive x-axis in the counterclockwise direction is unambiguous.

Are you not using a line and arctanx to establish a one-to-one correspondence between points on the line and points on the graph of arctanx?

Saying tan(pi/2) is undefined only helps up to the level of precalculus, it does not help after that when limits are explored. The tan(pi/2) depends on which way you approach pi/2 on the x-axis, if you approach pi/2 from the left, with positive dx, tan(pi/2-dx) increases without bound, if you approach pi/2 from the right, with positive dx, tan(pi/2+dx) decreases without bound so these answers are not meaningless because they explain the behavior of tanx. To say tan(pi/2) is undefined doesn't help anyone beyond precalculus.

Like i said i can see it your way, but asking me what the slope of y=2x is hurts my feelings a little bit.


agentredlum at rest.

 

[SteveL27]

Are you not using a line and arctanx to establish a one-to-one correspondence between points on the line and points on the graph of arctanx?

Not in the slightest. I can't imagine where you got that idea.

Not only aren't we doing that; but it wouldn't even be interesting to try! If you look at the graph of arctan you see it's a curvy line in very obvious 1-1 correspondence with the points on the x-axis. Each vertical line in the plane passes through exactly one point of the x-axis and one corresponding point on the graph of arctan(x). This is the least interesting thing anyone could say about the arctan.

Saying tan(pi/2) is undefined only helps up to the level of precalculus, it does not help after that when limits are explored.

That remark is irrelevant to the discussion. What on Earth do limits have to do with this discussion?

The tan(pi/2) depends on which way you approach pi/2 on the x-axis, if you approach pi/2 from the left, with positive dx, tan(pi/2-dx) increases without bound, if you approach pi/2 from the right, with positive dx, tan(pi/2+dx) decreases without bound so these answers are not meaningless because they explain the behavior of tanx. To say tan(pi/2) is undefined doesn't help anyone beyond precalculus.

It would help you to understand what micromass was saying when he gave the arctan as a specific function that maps the reals to a bounded open interval.

I honestly cannot tell if you are just a little confused, or deliberately obfuscating the discussion.

Like i said i can see it your way, but asking me what the slope of y=2x is hurts my feelings a little bit.

If you understood that slope of a line through the origin is the same as the tangent of the angle the line makes with the positive x-axis, there would be no more confusion. I'm not trying to hurt your feelings, I'm just trying to explain the arctan function.

In any event, it's often the case that we may have studied math to a particular level, yet be totally confused about much more elementary things. We are using the high-school math idea of slope to visualize a bijection between the reals and a bounded open interval. So we're taking a more sophisticated look at something elementary here, and there's no harm in trying to review the basics.In any event ... micromass already gave a bijection between the reals and a bounded open interval. I mentioned a visualization that helps me to understand that bijection. However if it's not helpful to you, it's not worth further flogging this deceased equine.

 

[Robert1986]

Admittedly, I haven't read every single post. But it seems as though agentredlum has gotten (at least) two bijections confused.

Given that most people on this forum are better at explaining stuff that I am, this might be a futile attempt on my part, but here goes:

First, forget about the semi-circle thing for now. This doesn't have anything to do with the arctan bijection. You asked for a bijection and micromass (or someone) gave you one. It is just a bijective function from the entire real line to to the interval (-a,a). (Graph it in WolframAlpha.) However, I am not quite sure how Steve has come up with his visulisation. The slope of arctanx is not 0 at x=0, it is 1. And, as x -> infinity, the slope goes to 0. Now, it is 100% possible that I, too, have misunderstood (or not read) something, but SteveL seems to have gotten arctan confused with tan. Yet, this doesn't really matter. Even in the visualisation that SteveL gave (which I think is of the tanx function on he interval (-a,a)) there is still a bijection between the reals and this interval. As someone (I think Steve) mentioned, graph (I would use wolframalhpa) tanx and arctanx and you will see that there is a bijection.

 

[SteveL27]

However, I am not quite sure how Steve has come up with his visulisation. The slope of arctanx is not 0 at x=0, it is 1. And, as x -> infinity, the slope goes to 0. Now, it is 100% possible that I, too, have misunderstood (or not read) something, but SteveL seems to have gotten arctan confused with tan.

I'm tempted to just let this go. I'd invite you to reread my posts. The slope of arctan, by which I imagine you mean the derivative of the arctan function, has nothing to do with this.

Briefly, as a line through the origin goes between -pi/2 and pi/2, its slope -- the tan of the angle -- goes from -infinity to +infinity. The inverse function is the arctan, mapping the reals to the bounded open interval ]-pi/2, pi/2[. That's all I've ever said. It's how I visualize the arctan.

I'm quite surprised that two people have now read what I've written and decided that I'm trying to say something about the slope of the graph of the arctan function. I'm literally baffled by that interpretation of what I wrote. I can't say anything more in this thread that I haven't already said.

 

[agentredlum]

Not in the slightest. I can't imagine where you got that idea.

From the first paragraph of post #46 which i quote below.

'It's just the arctan function. The way I like to think of this bijection is by imagining a horizontal line in the plane through the origin. It makes an angle of zero with the x-axis and its slope is zero.'

I thought you were trying to create a bijection between a line and arctanx subsequently my objections followed.

 

[I_am_learning]

Although, I haven't followed whole of the thread, I came across the thought experiment, where you randomly place the tip of your pencil on a line marked ----------> 0--------1.
Why do you say that the pencil always lands at irrational number? Because there are just as many rationals as irrationals (both infinite), the chances must be equal.
I know I am wrong (because you appear to be great mathmatician :) ), but I would like to learn. :]

 

[gb7nash]

No. There are more irrational numbers than rational numbers. As I said earlier in the thread, the more technical reason for this is that the rational numbers are countable and the irrational numbers are uncountable.

 

[I_am_learning]

No. There are more irrational numbers than rational numbers. As I said earlier in the thread, the more technical reason for this is that the rational numbers are countable and the irrational numbers are uncountable.

Is there a simple logical explanation for that? (Countable/uncountable don't appear to have enough logic)
Sorry if I am having you to repeat.

 

[HallsofIvy]

Yes, every number, rational or irrational, can be represented as a limit of a sequence of rational numbers. In fact, you make use of that when you represent an irrational number in "decimal form". Saying that \pi= 3.1415926... means precisely that the sequence of rational numbers (any terminating decimal is a rational number) 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, ... has \pi as limit.

One way of defining the real numbers, in terms of the rational numbers, is to use equivalence classes of sequences of rational numbers:

Let S be the set of all increasing, unbounded (equivalently "Cauchy") sequences of rational numbers. (Such sequences do not necessarily converge in the rational numbers.) We say that two such sequences, {a_n} and {b_n} are "equivalent" if and only if the sequence {a_n- b_n} converges to 0.

A "real number" is an equivalence class of such sequences.

 

[HallsofIvy]

Is there a simple logical explanation for that? (Countable/uncountable don't appear to have enough logic)
Sorry if I am having you to repeat.

They do if you understand the technical meaning of "countable" and "uncountable".
An infinite set is said to be "countable" if and only if there is a one to one mapping of the set onto the natural numbers, 1, 2, 3, ... An infinite set is said to be "uncountable" if and only if it is not countable.

A simple illustration that the set of all rational numbers is countable is given here:
http://www.homeschoolmath.net/teaching/rational-numbers-countable.php

A discussion of Cantor's proof that the set of all real numbers (and hence the set of all irrational numbers) is uncountable is given here:
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

 

[agentredlum]

Although, I haven't followed whole of the thread, I came across the thought experiment, where you randomly place the tip of your pencil on a line marked ----------> 0--------1.
Why do you say that the pencil always lands at irrational number? Because there are just as many rationals as irrationals (both infinite), the chances must be equal.
I know I am wrong (because you appear to be great mathmatician :) ), but I would like to learn. :]

The thought experiment is designed to be simple enough that any non-expert can understand and to boggle the mind. If it has done this for you then it is a success. It is designed to bring into question the pedestrian understanding of infinity and invite your curiosity to learn more.

Georg Cantor was the first to show that some infinities are equal, some infinities are larger than others. Then he showed that if you consider any infinity, there is always another infinity larger. That is MIND-BOGGLING!

Cantor did not do this by counting numbers. He did it by establishing a one-to-one correspondence between members of sets.

Here is a simplified example of his technique. Suppose there is a classroom with chairs and students outside in the hallway. Someone asks, 'which set is larger, the set of chairs, or the set of students? Somebody else says, 'I know what to do, count the number of chairs and count the number of students, then we will know which set is greater.' 'WAIT!' says Cantor, 'ask the students to sit down, if there are chairs left over then the set of chairs is larger, if there are students left over then the set of students is larger, if all students are seated and all chairs have 1 student then the sets are equal.'

This simplified version shows it is possible to determine which set is larger WITHOUT COUNTING the members of any two sets and without knowledge of the magnitude of each set by using a one-to-one correspondence, to every student is assigned 1 chair, and to every chair is assigned 1 student.

This is very helpfull when considering infinite sets because no one can actually count up to infinity but after Cantor things you can count and things you can't count were forced to have a different meaning. That is where the terms 'countable infinity' and 'uncountable infinity' originated.

 

[Robert1986]

It's just the arctan function. The way I like to think of this bijection is by imagining a horizontal line in the plane through the origin. It makes an angle of zero with the x-axis and its slope is zero.


As you rotate the line counterclockwise, as the line goes from horizontal to vertical the angle goes from 0 to pi/2; and the slope goes from 0 to +infinity.

Likewise as you rotate a horizontal line clockwise, the angle goes from from 0 to -pi/2 (= 3*pi/2), and the slope goes from 0 to -infinity.

Ahhhh. Now I see what you are saying. My apologies; my interpretation of your post was, in fact wrong. When you started off by saying "It's just the arctan function." I thought that what you described next was how you constructed the graph of the arctan function in your mind.



Now that I see what you mean, I see that it is actually a pretty interesting way of looking at it.

My apologies.


EDIT: If this post sounds sarchastic, I don't meant it to be; it is sincere.

 

[I_am_learning]

They do if you understand the technical meaning of "countable" and "uncountable".
An infinite set is said to be "countable" if and only if there is a one to one mapping of the set onto the natural numbers, 1, 2, 3, ... An infinite set is said to be "uncountable" if and only if it is not countable.

A simple illustration that the set of all rational numbers is countable is given here:
http://www.homeschoolmath.net/teaching/rational-numbers-countable.php

A discussion of Cantor's proof that the set of all real numbers (and hence the set of all irrational numbers) is uncountable is given here:
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

Nice.
So, rational numbers is countable because it can be precisely located in the infinity X infinity (2-dimensional) matrix of that homeschoolmath site.

Natural number is countable because it can be precisely located in 1x inifinity matrix (1-dimensional) (1 2 3 4 5 ... inf)

Irrational numbers are uncountable because no such matrix exists. Infact, we would have to develop a infinite dimensional matrix (infinity X infinity X infinity X infinity ...) so as to make a matrix that would contain all the Irrational numbers. So, that makes it un-countable and infinitely many than the rational number.
Pretty satisfied now.
However, I am sure you folks have much better grasp of the concept than this.
Thanks.

 

[spamiam]

First, forget about the semi-circle thing for now. This doesn't have anything to do with the arctan bijection. You asked for a bijection and micromass (or someone) gave you one. It is just a bijective function from the entire real line to to the interval (-a,a). (Graph it in WolframAlpha.)

Actually, if you write out the function for my semicircle bijection, you'll see it's more or less the inverse to the one given previously. If you work it out, the semicircle method gives g : (-c, c) \to \mathbb{R} by
g(x) = \frac{2c}{\pi}\tan\left(\frac{\pi}{2c}x\right)<br />
and the one given before was f : \mathbb{R}\rightarrow (-a,a) by f(x) = \frac{2a}{\pi} \arctan (x) \, .

But I like describing the bijection I gave pictorially because I think it's much more intuitive.

 

[Robert1986]

Actually, if you write out the function for my semicircle bijection, you'll see it's more or less the inverse to the one given previously. If you work it out, the semicircle method gives g : (-c, c) \to \mathbb{R} by
g(x) = \frac{2c}{\pi}\tan\left(\frac{\pi}{2c}x\right)<br />
and the one given before was f : \mathbb{R}\rightarrow (-a,a) by f(x) = \frac{2a}{\pi} \arctan (x) \, .

But I like describing the bijection I gave pictorially because I think it's much more intuitive.

Correct. I meant to mention that he should also consider your example.

Your example is certainly more intuitive and should be what he is looking for given he didn't like the more technical stuff. I also like your description, as well, for the same reason that you like it.

 

[cant_count]

Not a mathematician. I've tried to do some reading on this. My intuitive thought is this: between any two rational numbers there's an infinite number of rational numbers because you can invent any fraction (1.1, 1.11, 1.111, ...). I guess everybody would agree on that. However, when you "hit" infinity in your inventing, then you get an irrational number. Physically impossible, but in theory, if there really is such thing as infinity, then you could. Is that a paradox? Does it mean rational numbers tend towards continuity? - Does it mean anything at all? (I think I've blown a few brain-cells.)

It's a bit like: what's the definition of random? If the digits of Pi are random (and I haven't yet understood whether they are or not) - then does that not mean this: that any sequence of digits you can invent will occur somewhere in it ... including Pi "eventually" repeating itself .. ?

I guess it's like saying you're trying to reach infinity.

(This is the trouble with letting anybody into your forums ! )

 

[pessimist]

your intuitive thought is absolutely not the way to go. The notion of infinity in itself is a counter intuitive notion + your thought about hitting infinity doesn't really make much sense.

 

[agentredlum]

Not a mathematician. I've tried to do some reading on this. My intuitive thought is this: between any two rational numbers there's an infinite number of rational numbers because you can invent any fraction (1.1, 1.11, 1.111, ...). I guess everybody would agree on that. However, when you "hit" infinity in your inventing, then you get an irrational number. Physically impossible, but in theory, if there really is such thing as infinity, then you could. Is that a paradox? Does it mean rational numbers tend towards continuity? - Does it mean anything at all? (I think I've blown a few brain-cells.)

It's a bit like: what's the definition of random? If the digits of Pi are random (and I haven't yet understood whether they are or not) - then does that not mean this: that any sequence of digits you can invent will occur somewhere in it ... including Pi "eventually" repeating itself .. ?

I guess it's like saying you're trying to reach infinity.

(This is the trouble with letting anybody into your forums ! )

I like your idea about rational numbers 1/n as n goes to infinity. Yes, i believe some of them tend toward becoming irrational. Never thought of it that way until you mentioned it so thank you.

Heres what i mean, for a prime p, 1/p can have at most p-1 digits after the decimal before it starts to repeat.

Some primes do not exhaust the possibilities, some primes do.

Example for the prime 3, 1/3 can have at most 2 digits after the decimal before it starts to repeat but
1/3 = .3333... repetition begins after 1 digit so 3 does not exhaust the possibilities

Example for the prime 7, 1/7 can have at most 6 digits after the decimal before it starts to repeat
1/7 = .142857142857142857... repetition begins after the 6th decimal position so 7 exhausts the possibilities.

It's interesting to look at primes in this way.

1/11 = .090909... repetition begins after the 2nd position so 11 does not exhaust the possibilities.

1/13 = .076923 076923 076923... repetition begins after the 6th decimal position so 13 does not exhaust the possibilities.

1/17 = .0588235294117647 0588235294117647 0588235294117647 ... repetition begins after the 16th decimal position so 17 exhausts all the possibilities.

Now, we know that the primes are infinite in number, this means they get bigger and bigger. There may be many primes out there larger than 10^1000000 that have somewhere in the neighborhood of 10^1000000 - 1 non repeating digits, and then repetition begins. As far as calculation is concerned this 1/p would be as difficult to calculate as sqrt(2) to the 10^1000000 - 1 decimal position. So in some sense, there are rational numbers that tend to be irrational.

This idea is going to generate a lot of controversy here so pretend that i am joking.

There is a method similar to long division for finding square roots without using a calculator, you can find it toward the bottom of the following page

http://en.wikipedia.org/wiki/Methods_of_computing_square_roots

 

[Robert1986]

A few things.

First, the number 1/n does not tend to go anywhere: it just is. Only variables can "tend" to do anything. So, saying that 1/n (with n some fixed constant), 1.1, or 1.11 "tends" to go anywhere is just flat out wrong.

Cant Count is certainly correct in stating that there is an infinite number of rational numbers between any two rational numbers. Let a and b be two rationals. Now, since the rational numbers are countable, we can make a list of the rational numbers between a and b (though that fact isn't of critical importance in what follows). So, make the list. Each number on this list is a number that stays exactly as it is. It is NOT, in anyway, tending to be irrational. And it doesn't matter how difficult it is to compute a rational number, either.

If this is the first time you started dealing with this stuff, then (and I mean no offence) you don't really have intuition: this takes time and experience to build. Instead, you are using your "common sense." When I was first learning this stuff (and I still don't really have any intuition) it blew my mind. It gets really strange. Leave your common sense at the door in mathematics and trust proofs.

Cant Count, I would suggest googling the Cantor Set (this will really blow you away) and the countability and uncountability of the rationals and irrationals.

 

[agentredlum]

If you want to do something fun, pick up pencil and paper and compute 1/97.

It's very easy to show 1/2 is rational by computation. Not so easy to show 1/17 is rational. you would have to compute at least 16 decimal positions. Then you would see the pattern repeat and you can conclude that it is rational

 

[pessimist]

A few things.

First, the number 1/n does not tend to go anywhere: it just is. Only variables can "tend" to do anything. So, saying that 1/n (with n some fixed constant), 1.1, or 1.11 "tends" to go anywhere is just flat out wrong.

Cant Count is certainly correct in stating that there is an infinite number of rational numbers between any two rational numbers. Let a and b be two rationals. Now, since the rational numbers are countable, we can make a list of the rational numbers between a and b (though that fact isn't of critical importance in what follows). So, make the list. Each number on this list is a number that stays exactly as it is. It is NOT, in anyway, tending to be irrational. And it doesn't matter how difficult it is to compute a rational number, either.

If this is the first time you started dealing with this stuff, then (and I mean no offence) you don't really have intuition: this takes time and experience to build. Instead, you are using your "common sense." When I was first learning this stuff (and I still don't really have any intuition) it blew my mind. It gets really strange. Leave your common sense at the door in mathematics and trust proofs.

Cant Count, I would suggest googling the Cantor Set (this will really blow you away) and the countability and uncountability of the rationals and irrationals.

The cantor set is indeed one of the most remarkable things one encounters in mathematics. In my opinion it is the height of non intuitiveness.

 

[HallsofIvy]

If you want to do something fun, pick up pencil and paper and compute 1/97.

It's very easy to show 1/2 is rational by computation. Not so easy to show 1/17 is rational. you would have to compute at least 16 decimal positions. Then you would see the pattern repeat and you can conclude that it is rational

You understand, don't you that most people learn the definition of "rational number" as "can be written as a ratio (fraction) of integers". That is, 1/2 and 1/17 are rational (by that definition) just be looking at them, not by any computation. One can then show that every rational number can be written as a repeating decimal (thinking of 0.5 as "0.5000000" where "0" is the repeating part). You may well have learned to define a rational number as "a repeating decimal" and then learned to show that every fraction is a "rational number" and vice versa. If so, you are the second person in about twenty years that I have met who learned it that way! Where and when did you learn this definition of "rational number"?

 

[agentredlum]

You understand, don't you that most people learn the definition of "rational number" as "can be written as a ratio (fraction) of integers". That is, 1/2 and 1/17 are rational (by that definition) just be looking at them, not by any computation. One can then show that every rational number can be written as a repeating decimal (thinking of 0.5 as "0.5000000" where "0" is the repeating part). You may well have learned to define a rational number as "a repeating decimal" and then learned to show that every fraction is a "rational number" and vice versa. If so, you are the second person in about twenty years that I have met who learned it that way! Where and when did you learn this definition of "rational number"?

It was a learning process, of negotiation between both notions. At some point after a few years I decided I liked the repeating decimal explanation better than the ratio explanation. I think Niven and Zuckerman Elementary Number Theory influenced me but I can't be sure cause i don't remember. Also i read a few history of mathematics books and the impression i got was great emphasis was placed on computation for thousands of years. It is only within the past 100 years that mathematics has moved toward abstractness more and more.

Personally I like and trust the computing part but I don't throw away the abstract part because it has many uses.

The examples i used above with 1/p i saw for the first time in 'Recreations in the Theory of Numbers, The Queen of Mathematics Entertains' by Bieler. Highly recommend this one is full of treasure.

I found this idea fascinating but i only remember a sketch, Beiler goes into it in much greater detail and the results are fascinating. Beiler definitely loves computation cause he does a lot in that book!

Also i worked in a math center for quite some time and professors would donate books so i pounced on everything. I came across a book on numerical analysis and some amazing things were done in there, that i had never seen or even imagined so my fondness for numerical computations grew.

I also saw formulas developed by Ramanujan that converge rapidly to higher precision. These formulas fascinated me and felt like having a glimpse at glorious possibilities that would take me a lifetime to understand. It's easy for others to say how he did what he did...AFTER HE DID IT! That man was self taught.

Very important to me increasing my respect for numerical calculations was how Euler noticed an agreement to some decimal approximation between 2 relations that led him to discover his famous e^(ix)=cos(x)+ isin(x)
I learned about that in 'The sqrt(-1), An Imaginary Tale' by Paul J, Nahin. Highly recommend this, there are many treasures, tricks, lots of Algebra and great historical account.

 

[agentredlum]

The book by Bieler 'Recreations in the Theory of Numbers, the Queen of Mathematics Entertains'

ALMOST EVERY SINGLE PAGE CONTAINS A TREASURE!

 

[agentredlum]

If you were to ask 'prove 1/17 is rational' most people would say 'it's the ratio of two integers, therefore it's rational' and that's fine by me. I would prefer to say 'it has a repeating decimal expansion' and then i would calculate it and show you.

Bielers example which i used above for the decimal expansion of 1/p, somebody at some time in the past did the calculations and noticed a pattern. Namely, some primes exhaust the possibilities, others do not. Years later others proved the results abstractly and fit them into a more general theory full of symbols and few numbers but they woudn't have been able to do that unless many before them picked up pencil and paper and did ARITHMETIC.

 

[Robert1986]

Hmmmm. Here's an easy proof that 1/17 is rational: it can be written as the ratio of two integers.

 

[Robert1986]

Despite the fact that you are using a (IMHO) a non-standard definition of rational, this doesn't change the fact that numbers are not tending anywhere.

 

[agentredlum]

Hmmmm. Here's an easy proof that 1/17 is rational: it can be written as the ratio of two integers.

Yes that's true but you miss all the fun of discovery about decimal expansions. Let's pretend repeating 0's don't count, after all, repeating 0's don't add anything to the VALUE of a number. An interesting question then would be 'why do some rational numbers have terminating decimal expansions and others do not?' Is there a way to tell which ones terminate, which ones repeat?

I ask... If the sequence of non repetition has googolplex^(googolplex) digits before it starts to repeat, does it tend toward irrationality.

Of course its rational. I know that, you know that, I'm just thinking here.

 

[agentredlum]

Despite the fact that you are using a (IMHO) a non-standard definition of rational, this doesn't change the fact that numbers are not tending anywhere.

The idea of a repeating decimal is not worthless...am i challenging your 'standard' notion that numbers must be well defined? That's not what I'm after.

 

[Hurkyl]

Also i read a few history of mathematics books and the impression i got was great emphasis was placed on computation for thousands of years.

"Computation" is not a synonym for "express in decimal form". In fact, I'm under the impression their pervasive use is a fairly recent phenomenon. (and possibly even that widespread knowledge of the existence of such a system is relatively recent)

Historically, people have liked expressing answers using ratios or radicals or geometric constructions; all are perfectly good ways of "computing" an answer. For a variable precision system, continued fractions were once a popular method. (and, I believe, even preferred to decimals)

 

[Robert1986]

The idea of a repeating decimal is not worthless...am i challenging your 'standard' notion that numbers must be well defined? That's not what I'm after.

Agreed. But the fact that 1/17 might be hard to compute by hand doesn't, in any way, mean that it is "tending" to be irrational. Numbers don't tend anywhere, they just are. And the fact that 1/17 is hard to compute by hand doesn't in any way, make it any less rational than 1/2.


So, I don't really know what your point was other than some rational numbers are difficult to compute by hand.

 

[agentredlum]

Agreed. But the fact that 1/17 might be hard to compute by hand doesn't, in any way, mean that it is "tending" to be irrational. Numbers don't tend anywhere, they just are. And the fact that 1/17 is hard to compute by hand doesn't in any way, make it any less rational than 1/2. So, I don't really know what your point was other than some rational numbers are difficult to compute by hand.

My point? (IMHO) The posters intuition about irrationals and infinity is not 100% incorrect. I knew there would probably be many others here who would say that it was in order to discourage that way of thinking, thus my attempt with the Examples using 1/p. No one has commented about that so maybe people don't find it interesting?

The 2 ideas of irrational, infinity, are connected at least by way of calculation with positive integers, and they are connected in many more ways. It appears to me the poster understands they are connected but is unsure why.

 

[agentredlum]

The fractions,

3/2,

17/12,

99/70,

577/408,

665857/470832

etc.

Are all approximations that give better and better precision to sqrt(2). Just consider the concept of a fraction as an idea. What is happening here? The numerator is going to infinity, the denominator is going to infinity and because of this you get what you want, sqrt(2), an irrational number.

Take a look at this. Please notice in particular the PRODUCT FORMULA for sqrt(2). It gives a fraction whose numerator tends to infinity and whose denominator tends to infinity and it does this by using positive integers.

http://en.wikipedia.org/wiki/Square_root_of_2

I really don't understand why this idea is so offensive?

 

[micromass]

The fractions,

3/2,

17/12,

99/70,

577/408,

665857/470832

etc.

Are all approximations that give better and better precision to sqrt(2). Just consider the concept of a fraction as an idea. What is happening here? The numerator is going to infinity, the denominator is going to infinity and because of this you get what you want, sqrt(2), an irrational number.

Take a look at this. Please notice in particular the PRODUCT FORMULA for sqrt(2). It gives a fraction whose numerator tends to infinity and whose denominator tends to infinity.

http://en.wikipedia.org/wiki/Square_root_of_2

I really don't understand why this idea is so offensive?

Take

9/10

99/100

999/1000

9999/10000

This sequence has numerators and denominators that grow to infinity, but still the sequence does not converge to an irrational. It converges to 1.

However, the idea to use fractions to approximate numbers is an interesting one. I'd suggest looking up "Cauchy sequences" in http://en.wikipedia.org/wiki/Construction_of_the_real_numbers
However, checking rationality is in general a very hard problem!

 

[agentredlum]

Take

9/10

99/100

999/1000

9999/10000

This sequence has numerators and denominators that grow to infinity, but still the sequence does not converge to an irrational. It converges to 1.

However, the idea to use fractions to approximate numbers is an interesting one. I'd suggest looking up "Cauchy sequences" in http://en.wikipedia.org/wiki/Construction_of_the_real_numbers
However, checking rationality is in general a very hard problem!

Good point.

Yes, i know that (infinity)/(infinity) is an indeterminate form and can represent any real number. And i certainly don't want to give the impression that all fractions are irrational! I merely wanted to put forth the idea that it is possible to construct fractions, in a simple way, that kinda, sort of 'blur' the distinction between rational and irrational. I say this with 'tongue in cheek' cause I know it's controversial and not standard.

I agree with you about using fractions to approximate numbers is interesting, but i would like to take it one step further, nobody knows what 1/17 represents until the decimal expansion is given. Of course algebra can be done with 1/17 and you can get correct results, but IMHO 1/17 is similar to a variable x whose value is yet to be determined in a linear equation.

I am not saying replace 1/17 with a decimal in all calculations. Not at all! This would be a computational nightmare!

I am putting forth the idea that dividing 1 by 17 can sometimes provide insight.

 

[agentredlum]

However, the idea to use fractions to approximate numbers is an interesting one. I'd suggest looking up "Cauchy sequences" in http://en.wikipedia.org/wiki/Construction_of_the_real_numbers
However, checking rationality is in general a very hard problem!

The link you provided is very helpful and interesting so thanx!

In particular, i never heard of surreal numbers, hyperreal nymbers before today.

 

[I_am_learning]

Good point.

Yes, i know that (infinity)/(infinity) is an indeterminate form and can represent any real number. And i certainly don't want to give the impression that all fractions are irrational! I merely wanted to put forth the idea that it is possible to construct fractions, in a simple way, that kinda, sort of 'blur' the distinction between rational and irrational. I say this with 'tongue in cheek' cause I know it's controversial and not standard.

I agree with you about using fractions to approximate numbers is interesting, but i would like to take it one step further, nobody knows what 1/17 represents until the decimal expansion is given. Of course algebra can be done with 1/17 and you can get correct results, but IMHO 1/17 is similar to a variable x whose value is yet to be determined in a linear equation.

I am not saying replace 1/17 with a decimal in all calculations. Not at all! This would be a computational nightmare!

I am putting forth the idea that dividing 1 by 17 can sometimes provide insight.

I like your explanation that rational fraction tends to irrational fraction when we increase the numerator and denomarator towards infinity, but following some rule.
In your increasingly more accurate apporximation to sqrt(2) series, suppose that the series can be written as
sqrt(2) =~ x(n)/y(n)
=~ means approximately equal to
x(1) = 3, y(1) = 2
x(2) = 17, y(2) = 12
and so on.
Don't we all agree that, when n tends to infinity
sqrt(2) = x(n)/y(n) (n tends to infinity)

For every finitie n, x(n)/y(n) is rational, however for n tends to infinity, x(n)/y(n) is irrational.
Nice thing. Don't this prove your point that rational faction can tend to irrational.


And for your 1/17 thing, I feel just the opposite way. 1/17 clearly speaks to me of 1 part in 17 however, 0.05882... don't much make sense to me. :)

 

[agentredlum]

I like your explanation that rational fraction tends to irrational fraction when we increase the numerator and denomarator towards infinity, but following some rule.
In your increasingly more accurate apporximation to sqrt(2) series, suppose that the series can be written as
sqrt(2) =~ x(n)/y(n)
=~ means approximately equal to
x(1) = 3, y(1) = 2
x(2) = 17, y(2) = 12
and so on.
Don't we all agree that, when n tends to infinity
sqrt(2) = x(n)/y(n) (n tends to infinity)

For every finitie n, x(n)/y(n) is rational, however for n tends to infinity, x(n)/y(n) is irrational.
Nice thing. Don't this prove your point that rational faction can tend to irrational.And for your 1/17 thing, I feel just the opposite way. 1/17 clearly speaks to me of 1 part in 17 however, 0.05882... don't much make sense to me. :)

Thanx for giving me hope that i am not a total fool.

well for me, i can more easily understand 6 parts in 100 or 59 parts in 1000 or 588 parts in 10000 than i can understand 1 part in 17 because of the use of powers of ten which are easy to visualize for me.

IMHO there is nothing wrong with your visualization but if 0.05882... doesn't make much sense, i hope my way of visualizing it helps.

Would you have any trouble visualizing

(1/17)^2,

3/(17)^3,

7/37

19/53

23/127 ?



My point is that as numerators and denominators become larger, the decimal representation becomes more useful in understanding the actual value of the fraction.

 

[I_am_learning]

Thanx for giving me hope that i am not a total fool.

well for me, i can more easily understand 6 parts in 100 or 59 parts in 1000 or 588 parts in 10000 than i can understand 1 part in 17 because of the use of powers of ten which are easy to visualize for me.

IMHO there is nothing wrong with your visualization but if 0.05882... doesn't make much sense, i hope my way of visualizing it helps.

Well that was really silly of me to not realize that 0.05822 is just around 5.822/100 ( 5.8 part in 100) or even better, 5822 part in 100000 (don't count the 0's because, I didn't .

However, after all its just a crude guessing. I can't really find any difference in my visualization of either 5 part in 237 or 37 part in 806 or 41 part in 1000. In all case I visualize the part being a tiny fraction of the whole.
Do you have any such sharp visualization?

 

[Hurkyl]

Are all approximations that give better and better precision to sqrt(2). Just consider the concept of a fraction as an idea. What is happening here? The numerator is going to infinity, the denominator is going to infinity and because of this you get what you want, sqrt(2), an irrational number.

Ah, you conceive wrongly! It's not the concept of a fraction you're thinking of, it's the concept of a sequence of fractions. (And specifically, of a sequence that converges to something)

 

[agentredlum]

Ah, you conceive wrongly! It's not the concept of a fraction you're thinking of, it's the concept of a sequence of fractions. (And specifically, of a sequence that converges to something)

Not at all. I talked about a single fraction, you mentioned sequence. Can you think of a fraction as an abstract idea or do you have to quantify it with numbers?

In the product formula for sqrt(2) how many fractions do you see?

 

[agentredlum]

Look, you can think of it as a limit if you want. Take the limit as x and y go to infinity of x/y. This is an indeterminate form (infinity)/(infinity) which means it could be any real number, including an irrational number.

 

[agentredlum]

Well that was really silly of me to not realize that 0.05822 is just around 5.822/100 ( 5.8 part in 100) or even better, 5822 part in 100000 (don't count the 0's because, I didn't .

However, after all its just a crude guessing. I can't really find any difference in my visualization of either 5 part in 237 or 37 part in 806 or 41 part in 1000. In all case I visualize the part being a tiny fraction of the whole.
Do you have any such sharp visualization?

Not beyond the first few decimal points. I can get a rough estimation at a glance using notion of distance but then i zoom in, in my minds eye and start comparing to other objects that i am familiar with, such as cells bacteria, DNA strands, molecules, atoms, protons, electrons. Then I start thinking in terms of wavelength of light. Ultraviolet, x-ray, etc. Particularly useful is an ANGSTROM because it's 10^(-10). This is not easy and takes great effort but you get better as you practice.

Something that helped me get an understanding about decreasing quantities was a video by Arthur C. Clarke, may he rest in peace.



I'm going to watch it again now that your question reminded me of it.

Here is a great example of 'zooming in'. As far as fractals are concerned you can do this forever because they have infinite complexity.

http://www.youtube.com/watch?v=0jGaio87u3A&NR=1

 

[Hurkyl]

Not at all. I talked about a single fraction, you mentioned sequence.

Eh? You were not talking about a single fraction -- you fairly explicitly listed a family of fractions.

The inability to tell the difference between the idea of "a number" and "a family of numbers" is a rather serious liability in calculus. It is, for example, one of the major causes that lead people to insist that "0.999\ldots \neq 1" -- e.g. "0.999... is a number that tends to 1, but isn't actually 1".

I honestly can't tell if you are just in love with using "fraction" to describe "family of fractions", or if you are starting down a path that will have you coming back here in three months insisting that \sqrt{2} is a rational number.

Yes, I know you might think that last comment silly -- but there have been at least two people who have visited this forum who have insisted exactly that, using arguments that resemble what you are arguing.

In the product formula for sqrt(2) how many fractions do you see?

Explicitly, I see contained in the notation one fraction-valued expression in the variable n. Implicitly there are two related sequences of fractions: the infinite sequence of terms, and the infinite sequence of partial products.

And the product formula itself is, of course, not a fraction at all, e.g. because the outermost verb is "The infinite product of..." and not "The quotient of..."

 

[agentredlum]

Eh? You were not talking about a single fraction -- you fairly explicitly listed a family of fractions.

The inability to tell the difference between the idea of "a number" and "a family of numbers" is a rather serious liability in calculus. It is, for example, one of the major causes that lead people to insist that "0.999\ldots \neq 1" -- e.g. "0.999... is a number that tends to 1, but isn't actually 1".

I honestly can't tell if you are just in love with using "fraction" to describe "family of fractions", or if you are starting down a path that will have you coming back here in three months insisting that \sqrt{2} is a rational number.

Yes, I know you might think that last comment silly -- but there have been at least two people who have visited this forum who have insisted exactly that, using arguments that resemble what you are arguing.

Explicitly, I see contained in the notation one fraction-valued expression in the variable n. Implicitly there are two related sequences of fractions: the infinite sequence of terms, and the infinite sequence of partial products.

And the product formula itself is, of course, not a fraction at all, e.g. because the outermost verb is "The infinite product of..." and not "The quotient of..."

Well, i guess people see what they want to see depending on the point they want to make. Yes, i listed fractions that converge to sqrt(2), but i am not interested in ANY intermediate fractions. I am only interested in the fraction whose numerator and denominator have gone to infinity by some rule, as I_AM_LEARNING pointed out in post # 89. This is a fraction I CANNOT list in the normal sense so I am asking for a little lattitude here, and for people to use the power of their imagination. This is not a rigorous approach, i understand that, but my original observation was not intended to be unquestionable truth. I used the words 'in some sense'

When I look at the product formula for sqrt(2), I see all those things you mentioned, no question, but i also see a single fraction whose numerator and denominator have gone to infinity by some rule.

I know what a sequence is. I know what partial products are. I'm not sure if you know what 'taking a step back' and' looking at the big picture' means. That's what I'm trying to do here, in a way that makes a little sense, not perfect sense.

I think sqrt(2) is irrational, no question, but I am open to the possibility that sqrt(2) can be thought about as being rational 'in some sense'

I think .999... = 1 no question, but i am open to the possibility that this can create other problems 'in some sense'

 

[agentredlum]

How many fractions do you see when you look at 2/3 ? 'In some sense' I see an infinite product of fractions.

...(-3/-3)(-2/-2)(-1/-1)(2/3)(1/1)(2/2)(3/3)...

You can get creative.

...(-pi/-e)(-e/-pi)(2/3)(e/pi)(pi/e)...

you can fill in the dots however you like, just make sure it works.

The possibilities are only limited by a persons imagination.

 

[micromass]

How many fractions do you see when you look at 2/3 ? 'In some sense' I see an infinite product of fractions.

...(-3/-3)(-2/-2)(-1/-1)(2/3)(1/1)(2/2)(3/3)...

You can get creative.

...(-pi/-e)(-e/-pi)(2/3)(e/pi)(pi/e)...

you can fill in the dots however you like, just make sure it works.

The possibilities are only limited by a persons imagination.

This is very dangerous to do. There are many traps when dealing with infinite products, so you better say exactly what you mean with it. How do you evaluate such an infinite fraction??

 

[agentredlum]

This is very dangerous to do. There are many traps when dealing with infinite products, so you better say exactly what you mean with it. How do you evaluate such an infinite fraction??

I disregard everything and pick the number in the 'middle'.



Yeeess, I know it's dangerous but IMHO so are the subtle points of arithmetic.