Where are the irrational numbers?

  • Thread starter smolloy
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  • #1
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Rational numbers are those that can be represented as a/b.

It is simple (I think) to demonstrate that the series of rationals is continuous, since, for any two rational numbers, X=a/b, and Y=c/d, you can always find at least one rational number between them.
[tex]\frac{X+Y}{2} = \frac{ad+bc}{2bd}[/tex]
This derivation works for *any* pair of rational numbers, no matter how close, so the series of rationals is continuous.

But, doesn't this mean that there is nowhere -- no gaps -- in which we can fit irrational numbers?

Does this mean that, logically speaking, pi doesn't really exist between 3.14 and 3.15 on rational number line?

Do the irrationals exist in a completely separate series to rationals (similarly to the way in which reals and imaginary numbers are on different series)?

Perhaps I should put my copy of Russell's "Principles of Mathematics" away, and stop pretending I'm a mathematician? ;)
 

Answers and Replies

  • #2
mathman
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Your "proof" is that between any 2 rationals is another rational. This still does not make a continuum.
 
  • #3
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Hmmm... I thought that that was the definition of a continuum. No?

Between *any* two rationals is another, third, rational. So between the third rational and one of the first two is yet another, fourth, rational. Etc. Ad infinitum.
 
  • #4
gb7nash
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Like mathman said, just because there exists a rational number between rational numbers a and b, this doesn't mean that there's no irrational numbers between a and b. In fact, there exists an infinite number of irrational numbers between a and b.
 
  • #5
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Thanks guys.

So, between the rationals a and b, there exists an infinite number of other rationals, as well as infinity of irrationals. Two series interlaced within the series of the reals. Yes?
 
  • #6
gb7nash
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Thanks guys.

So, between the rationals a and b, there exists an infinite number of other rationals, as well as infinity of irrationals. Two series interlaced within the series of the reals. Yes?
Yes, in fact the union of the set of rational numbers and irrational numbers is equivalent to the set of real numbers.
 
  • #7
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OK. This is becoming clearer. I was tricked by the seemingly illogical impossibility of interleaving two continuous series, but I guess "common sense" breaks down when you start to consider infinite numbers and infinitesimals.

Could an irrational be considered to be defined as the limit of some converging series of rationals? In fact, can't such a series be used to converge to pi?
 
  • #8
gb7nash
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OK. This is becoming clearer. I was tricked by the seemingly illogical impossibility of interleaving two continuous series, but I guess "common sense" breaks down when you start to consider infinite numbers and infinitesimals.

Could an irrational be considered to be defined as the limit of some converging series of rationals? In fact, can't such a series be used to converge to pi?
Well, I know some irrational numbers can. For instance:

[tex]\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi ^2}{2}[/tex]

All of them though? That I'm not sure about.
 
  • #9
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Oh, I wasn't suggesting all irrationals could be defined as a limit. I meant "an irrational" -- meaning that there are one or more irrationals that could be defined in this way. Then it struck me that there was an ancient method for finding an approximation to pi that can be extended to an infinitely long series converging to the irrational value of pi.

Something to do with finding the circumference of an n-sided regular polygon, and then letting n go to infinity and then dividing by the diameter.
 
  • #10
StatusX
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Actually the property of the rationals you are describing when you say that "between any two rationals (or indeed, any two real numbers) is another rational" is the property of the rationals being dense in the space of all real numbers. And this is equivalent to the existence, for any real number x, of an infinite sequence of rational numbers which converges to x. For example, any irrational number can be written as a non-repeating decimal, eg, 0.343512309..., and the following is a sequence of rationals converging to it: 0.3, 0.34, 0.343, 0.3435, ... .
 
  • #11
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Well, I know some irrational numbers can. For instance:

[tex]\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi ^2}{2}[/tex]

All of them though? That I'm not sure about.
.


I don't know if this is what you are referring to, but you can cut the first terms of the
decimal expansion of an irrational to get a sequence of rationals that converge to it,
e.g., for Pi:

a<sub>1</sub>=3

a<sub>2</sub>=3.1

............................

.............................

a<sub>n</sub>=3...... (first n digits in decimal expansion of Pi)


Same for any irrational --or rational.

I didn't notice that Static X had already said this.

Sorry, my "quote" button suddenly became disabled after my first quote.


Smolloy: AFAIK, there are no infinitesimals at play here; you seem to be doing standard
analysis. It is an issue of order properties of the reals and their subsets. By a different
perspective, the rationals are not really continuous: the rationals are a totally-disconnected subspace of the reals.
 
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  • #12
gb7nash
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I don't know if this is what you are referring to, but you can cut the first terms of the
decimal expansion of an irrational to get a sequence of rationals that converge to it,
e.g., for Pi:
Of course. I think the OP was looking for a closed formula to determine the nth digit of an irrational number (in this case pi) in the form of a series, but I could be mistaken.
 
  • #13
disregardthat
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Every real number can be written as the limit of a converging sequence of rationals, (e.g. their decimal expansion up to the n'th digit), and we often define the real numbers as the equivalence classes of converging sequences of rational numbers, where two sequences are equal if their difference converge to 0.
 
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  • #14
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Well, I know some irrational numbers can. For instance:

[tex]\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi ^2}{2}[/tex]

All of them though? That I'm not sure about.
yes you can construct a cauchy seqence of rational numbers by "truncation of digits" given an irrational number this is how you complete the rationals into the reals
 
  • #15
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prove that between any two rational numbers, there is a rational number; then prove that between any two rational numbers there is an irrational number ( you can take an irrational number between 0 and 1 to start with ) ; then you are all set to prove that for any x and y in the reals there is an irrational number in between them
 
  • #16
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Rational numbers are those that can be represented as a/b.

It is simple (I think) to demonstrate that the series of rationals is continuous, since, for any two rational numbers, X=a/b, and Y=c/d, you can always find at least one rational number between them.
[tex]\frac{X+Y}{2} = \frac{ad+bc}{2bd}[/tex]
This derivation works for *any* pair of rational numbers, no matter how close, so the series of rationals is continuous.

But, doesn't this mean that there is nowhere -- no gaps -- in which we can fit irrational numbers?

Does this mean that, logically speaking, pi doesn't really exist between 3.14 and 3.15 on rational number line?

Do the irrationals exist in a completely separate series to rationals (similarly to the way in which reals and imaginary numbers are on different series)?

Perhaps I should put my copy of Russell's "Principles of Mathematics" away, and stop pretending I'm a mathematician? ;)
The amount of irrational numbers is VERY MUCH MORE than rational numbers. Draw a line and put 2 notches, label the notch on the left 0 the notch on the right 1. Place the tip of your pencil somewhere on the line. Do this every second for the next BILLION YEARS! I would be very surprised if you could hit a rational number even once! In fact, i'm not even sure you could hit the line, but lets pretend you could hit the line every time. I still don't think a rational number could be hit even after a TRILLION YEARS etc. So according to this 'thought' experiment someone can say rationals DO NOT EXIST because you are pointing to an irrational ALL THE TIME! In probability, the probability of pointing to a rational is 0% the probability of pointing to an irrational is 100% The infinity of irrationals 'swallows' up the infinity of rationals and rationals become insignificant as compared to irrationals. Isn't that amazing? I recommend you read about Georg Cantor, father of elementary set theory and how he analyses infinity.
 
  • #17
gb7nash
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The amount of irrational numbers is VERY MUCH MORE than rational numbers. Draw a line and put 2 notches, label the notch on the left 0 the notch on the right 1. Place the tip of your pencil somewhere on the line. Do this every second for the next BILLION YEARS! I would be very surprised if you could hit a rational number even once! In fact, i'm not even sure you could hit the line, but lets pretend you could hit the line every time. I still don't think a rational number could be hit even after a TRILLION YEARS etc. So according to this 'thought' experiment someone can say rationals DO NOT EXIST because you are pointing to an irrational ALL THE TIME! In probability, the probability of pointing to a rational is 0% the probability of pointing to an irrational is 100% The infinity of irrationals 'swallows' up the infinity of rationals and rationals become insignificant as compared to irrationals. Isn't that amazing? I recommend you read about Georg Cantor, father of elementary set theory and how he analyses infinity.
The more technical reason for this is that the rational numbers between 0 and 1 are countable and the irrational numbers between 0 and 1 are uncountable. It's pretty cool that there's practically no chance of "picking" a rational number.
 
  • #18
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Also, consider the famous Dirichlet function in x-y plane. It goes like this...'let y= 0 for x rational, let y=1 for x irrational This 'piecewise' defined function has the amazing property of being DEFINED everywhere on the real x-axis but being continuous NOWHERE in the x-y plane! So the notion of continuity is 'tricky' and the notion of 'in-between' becomes meaningless as soon as you dive into the realm of fractions and beyond. One final 'nightmarish' thought...if you are fortunate enough to swim out to the realm of complex numbers...you lose the notion of ORDER so you cannot tell which one comes before or after another! 1 and 2 are integers but are also in the set of complex numbers. When you 'treat' them in the set of complex numbers, 1 does NOT come before 2, 1 does NOT come after 2, 1 does NOT EQUAL 2 WTF??? right? This only happens AFTER you extend the real numbers to include complex. In the set of real numbers ORDER still persists. Isn't that amazing?
 
  • #19
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The more technical reason for this is that the rational numbers between 0 and 1 are countable and the irrational numbers between 0 and 1 are uncountable. It's pretty cool that there's practically no chance of "picking" a rational number.
It makes me very happy that you (like myself) think it's pretty cool. The technical explanation completely disregards the 'cool' part. The way math is taught...they teach the technical reasons, they disregard the cool parts. I like to be FASCINATED (like Spock) If you teach me the rationals are countable, the irrationals are not, it doesn't mean very much to me. Show me Cantor's diagonal argument, still doesn't mean much more. (Although I agree it is a very clever argument) If you show me the consequences of the ideas formed by thousands of years of itellectual, independant, creative thought of humanity, then chances are i will be fascinated. Most mathematicians are not good story-tellers and as a consequence popularity of mathematics suffers. The way math is taught...they teach the technicalities and leave the 'cool consequences' to be discovered by the student. This is wrong approach, not everyone is genius enough to discover cool consequences for themselves, consequences that took humanity thousands of years to discover. However, EVERYONE, whether genius or not, has the RIGHT to know about the 'cool' MATHEMATICS. Also, is it any wonder why most mathematicians SEEM arrogant. In this system of teaching they have had to figure out almost everything by themselves and now feel 'privelaged' for having this knowledge. You ask a question and they throw some technical theorem at you which really doesn't explain much, unless you know, but if you know then you wouldn't ASK the question. They appear more like MAGICIANS guarding their secrets instead of professors who's JOB is to spread the knowledge. Am i wrong?
 
  • #20
mathwonk
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take an interval of length 1/1000, and cut it in half. Plop the first half down on top of the rational number 1/2. Then cut the rest in half again plop half down on top of the rational number 1/3. Do it again and plop halkf down on 2/3. Keep on covering the rational numbers 1/4, 3/4, 1/5, 2/5,...... by increasingly smaller intervals. Continue forever.

Or work faster and faster so that you finish the job in 1 second if you prefer.

Then you will covered all the positive rational numbers by an interval of length 1/1000.

so there is not a very big set of rational numbers. i.e. they have length zero.
 
  • #21
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take an interval of length 1/1000, and cut it in half. Plop the first half down on top of the rational number 1/2. Then cut the rest in half again plop half down on top of the rational number 1/3. Do it again and plop halkf down on 2/3. Keep on covering the rational numbers 1/4, 3/4, 1/5, 2/5,...... by increasingly smaller intervals. Continue forever.

Or work faster and faster so that you finish the job in 1 second if you prefer.

Then you will covered all the positive rational numbers by an interval of length 1/1000.

so there is not a very big set of rational numbers. i.e. they have length zero.
I loooooooooove this explanation, you know why? Cause it's FASCINATING! Let me tell a personal story. Years ago I was reading a book on elementary number theory. I didn't know much about math then (don't know much about math now lol) There was a graph in there sort of like a 45 degree rotated bell curve not too complicated, i think it was cubic in x-y plane, and the author claimed the graph did NOT pass through any rational point with BOTH x,y co-ordinates rational. I found that fascinating even though i understood the explanation why. The equation was fairly simple. I showed it to a professor friend of mine who had a Phd in Autumorphic Forms and knew a lot about number theory. 'This is amazing' i said to him ' you have this graph with an infinite number of points extending in opposite directions and you mean to tell me it's never going to hit a rational point?' He said 'y = pi never hits a rational point' 'but it's not a straight line, it's curvy' i said
'doesn't matter' he said 'given the fact that the set of irrational points is SO MUCH BIGGER than the set of rational points, the AMAZING thing is that ANY function hits ANY rational points at all'

I was fascinated. We never spoke of it again because i knew he didn't like to answer questions too much. I was not angry, i was grateful because most didn't like to answer ANY questions at all so i appreciated what little time he spent with me. A few weeks later and intense study on my part, i agreed with him.
 
  • #22
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I’ve been noting the discussion about irrational numbers on the number line with great interest and have two questions: Is anyone aware of attempts to correlate points on the traditional number line with the idea that real world space and time are probably discrete at the scale called the Plank distance or Plank volume? My understanding is that in the our world where nothing exists in lengths, areas and volumes smaller than the Plank size, the experience of an infinite number of points or the experience of irrational numbers altogether would not exist (except of course, in our minds). If this is true, in efforts to understand the so called “real world” using mathematics, might it be more productive to modify the number line to reflect a finite number of points of Plank size and investigate descriptions of the world using mathematics based on that kind of fundamental structure?
 
  • #23
gb7nash
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I’ve been noting the discussion about irrational numbers on the number line with great interest and have two questions: Is anyone aware of attempts to correlate points on the traditional number line with the idea that real world space and time are probably discrete at the scale called the Plank distance or Plank volume?
I've heard of this, though my knowledge on it is very limited. This is more of a physics question than mathematical question though. In the sense of the real numbers, time and length are not discrete.

If this is true, in efforts to understand the so called “real world” using mathematics, might it be more productive to modify the number line to reflect a finite number of points of Plank size and investigate descriptions of the world using mathematics based on that kind of fundamental structure?
In terms of preciseness, this would probably be better. However, you would need some kind of mechanism/scheme to determine the actually length of a planck. This would require insane detail, since if you're off by just a tiny bit, the error between the actual length and the observed length will amplify when measuring visible objects.
 
  • #24
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This was in the news recently ...

http://io9.com/5818008/the-universe-probably-isnt-a-giant-hologram-after-all

Evidently some clever physicists measured something down to 1 ten-trillionth of the Planck length. I don't know enough physics to make sense of the article but it came to mind as I was reading this discussion.

In any event, math and physics are not the same thing. Certainly it's fair to say that the mathematical notion of the real numbers as the continuum does not correspond to what modern physicists think about physical space.

That doesn't mean math isn't useful. Even if physical space is quantized, the mathematical continuum is still a handy model for doing calculations. But we have no evidence for infinite sets in the physical universe, let alone uncountable ones.
 
  • #25
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In words, the Plank length is defined as the square root of (the reduced Plank constant times the Gravitational constant divided by the speed of light cubed). The reduced Plank constant component is the Plank constant divided by 2 pi. The yield from this function is an approximation of 1.616252(81)*10-35 meters. This number has a fairly long history of being tested experimentally and indicates a scale below which not only does nothing exist, but nothing can exist even philosophically. Nothing, that is, but what a conscious mind might consider – and a conscious mind can consider anything, including a universe constructed of green elephants. Green elephants don’t count without evidence. Nor apparently, as SteveL27 mentioned, do infinite sets.

As for the debate about whether the number line represents a continuous or discrete reality, doesn’t the existence (?) of irrational numbers themselves hint at discontinuities on the number line within the current framework of traditional mathematics?

A Plank distance is extraordinarily small, but as gb7nash mentioned, it is still an approximation, but one being refined as experimental evidence continues to come along. His additional comment that any size error would cause problems is accurate also. Size error would be due to ultimately and irreconcilably to field fluctuations at that scale, but I wouldn’t think of that as a reason not to pursue the idea. Averages and approximations can be standardized and updated in a fair way in order to illuminate the terrain. And although it may be difficult at present to stamp an exact size, the concept exists and is accepted because of the evidence and that in itself should have some meaning or significance that could inform the subject of number theory even as mathematics informs physics. As an approximation it would seem a useful tool and if nothing else, would remove irrational numbers and infinities from the number line. That would be just a beginning.

If one were to accept this and think it out, other possibilities might derive from the use of a change in what we’ve known as the traditional number line. What would it mean to construct lines, areas and volumes from Plank size “points”? Formal mathematical systems would probably have to be deveoped to make things coherent. Examples that might be derived: descriptions what of the exact beginning of time and space may have looked like and a more certain proposition that the universe must be bounded if it contains only a finite number of Plank “points” to elimination or confirmation of the possibility of time travel. Now, how one gets to there from here, I don’t think I have all the knowledge or horsepower needed to make that trip which is why I asked the question I did in my previous post. But, maybe some young soul somewhere….thinking his or her thinks…about mathematics….could help the physicists along.

For myself, I would choose the found knowledge about the world through physics and other disciplines and try to bend the traditional philosophies of mathematics to conform to that knowledge in hope of discovering more about what the world is and how it works. It has been done before and it was mathematicians who did it.
 

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