Can irrational numbers exist on the numberline?

In summary, the conversation discusses the representation of irrational numbers on a number line and whether they can be represented by a definite point. Some argue that they can be represented by a point that is continuously moving towards the value but never reaching it, while others argue that they can be represented by a definite point on the line. It is also mentioned that the concept of irrational numbers is often defined in terms of rational cauchy-sequences and that the idea of constructibility is not relevant when it comes to the number line as it is merely an abstraction of an ordered set. The conversation ends with a question about whether rational numbers can exist on the number line.
  • #1
Mu naught
208
2
This may be an elementary question, but I've been thinking about it a little bit and wondering what other people thought.

First, let me say that I'm talking about a number line not as a set but in the more literal sense, like a partitioned line that might exist as the axis of a graph.

So, if you want to represent a number as a point on this line, you can do so by starting at zero, and moving a certain distance to a point which corresponds to n units, the direction depending on if it's positive or negative. So what about an irrational number? It seems to me that you'd have to approach some point on the line, but continuously move toward it at a slower and slower rate, moving a thousandth of a unit, then a millionth, then a billionth and so on, constantly moving but constantly slowing down and never actually reaching any fixed point on the line. Is this a sound conception?

This also raises questions about real life objects having lengths which we can calculate to be irrational, but that's probably another thread.
 
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  • #2
The "real line" is an abstract idea. We draw a line on a sheet of paper to help us visualize its properties. But in the sense of a geometry construction, you can certainly plot a point that represents an irrational number. Just draw an x-axis and a y-axis perpendicular to it. Construct the 45 degree line y = x and mark 1 unit from the origin on it in the first quadrant. Drop a perpendicular from that point to the x axis. It will hit the x-axis at sqrt(2) / 2, which is irrational. But, of course, your pencil representation won't be exact. :frown:
 
  • #3
A much better question is can rational numbers exist on the numberline? After all, almost all of the numbers on the number line are irrational. A point picked at random on that number is almost surely irrational. There are lots of (an infinite number of) irrationals that are constructible on that numberline -- and yet the set constructible numbers is still of measure zero. Just because you cannot construct a number doesn't mean the number doesn't exist.
 
  • #4
So rational numbers don't exist on a numberline?
 
  • #5
Irrational numbers are often defined in terms of rational cauchy-sequences. Any real number is identified with an equivalence class of rational cauchy-sequences. The equivalence is as such: two rational cauchy-sequences are identical if their difference converges to 0. In a way an irrational number can be seen as an algorithm of rational numbers growing closer together.

So you could actually identify sqrt(2) with an algorithm which produces a rational sequence q_n such that |sqrt(2)-q_n| --> 0 as n --> infinity. The 'infinity' of information is not more a problem. Here, sqrt(2) have been identified in the (arguably) finitistic terms of a sequence, which can be considered a recursive rule.

However, the debate of whether irrational numbers exists more or less than rational numbers is actually irrelevant when it comes to the number line. The number line is merely an abstraction from an ordered set. A set is ordered if; given any two elements (a,b), then either a=b, a>b or b>a. If we identify two points a and b on the number line such that a < b, then the line segment in between them represent the set { x | a<x<b }. The number line serves as nothing but a consistent picture of an ordered field such as the real numbers.
 
  • #6
D H said:
A much better question is can rational numbers exist on the numberline? After all, almost all of the numbers on the number line are irrational. A point picked at random on that number is almost surely irrational. There are lots of (an infinite number of) irrationals that are constructible on that numberline -- and yet the set constructible numbers is still of measure zero. Just because you cannot construct a number doesn't mean the number doesn't exist.

I still don't see how an irrational number can be represented by a definite point on a number line. Any point you choose, you can always move to the right by n x 10-j for whatever decimal you want to bring the number out to. This is what leads me to understand that an irrational number can only be represented by a point which is moving infinitesimally slowly towards some value but can't ever reach it, and if that is a valid way of thinking of things, then I don't believe you can ever represent an irrational number as a point on a number line.

Char. Limit said:
So rational numbers don't exist on a numberline?

Is this directed at me? If so, please explain how anything I've said would imply that.
 
  • #7
Mu naught said:
I still don't see how an irrational number can be represented by a definite point on a number line. Any point you choose, you can always move to the right by n x 10-j for whatever decimal you want to bring the number out to.
I don't quite understand your problem. Why do you think that an irrational number cannot be represented by a definite point on a number line, while a(ny) rational number can? If you worry about constructibility, LCKurtz has given you a prescription to construct sqrt(2)/2.
 
  • #8
Landau said:
I don't quite understand your problem. Why do you think that an irrational number cannot be represented by a definite point on a number line, while a(ny) rational number can? If you worry about constructibility, LCKurtz has given you a prescription to construct sqrt(2)/2.

I think I explained why I think this pretty clearly:

Any point you choose, you can always move to the right by n x 10-j for whatever decimal you want to bring the number out to. This is what leads me to understand that an irrational number can only be represented by a point which is moving infinitesimally slowly towards some value but can't ever reach it
 
  • #9
Mu naught said:
Any point you choose, you can always move to the right by n x 10-j for whatever decimal you want to bring the number out to. This is what leads me to understand that an irrational number can only be represented by a point which is moving infinitesimally slowly towards some value but can't ever reach it

I don't understand. How can I move to the right of a point that I'm not actually at? And if you move by some power of ten to the side you're on another irrational number, which also doesn't exist apparently
 
  • #10
Mu naught said:
This is what leads me to understand that an irrational number can only be represented by a point which is moving infinitesimally slowly towards some value but can't ever reach it, and if that is a valid way of thinking of things,
Let's get away from the idea of a "moving point" and an "infinitessimal" and just say the following: you are thinking about the rational number line, and you are representing a number by a Cauchy sequence. (e.g. by a sequence of decimal approximations)


We can go further. We can define the sum, difference, or product of two sequences. Then, we might define two Cauchy sequences to be equivalent if their difference converges to zero. We can then define an ordering, and we might even be inclined to make a "Cauchy sequence line". We can even work out how to do calculus with these things.

And our crowning achievement will be to show that we have constructed a complete ordered field!

But wait a moment -- all that says is we have just created a model of the real numbers. So why did we bother going through all of this trouble? :confused:
 
  • #11
Mu naught said:
I think I explained why I think this pretty clearly:

Any point you choose, you can always move to the right by n x 10-j for whatever decimal you want to bring the number out to. This is what leads me to understand that an irrational number can only be represented by a point which is moving infinitesimally slowly towards some value but can't ever reach it

Numbers don't move because they don't have legs :smile:

When you get a bit farther along in your mathematics training you may run across the idea of Dedekind cuts. That may help you understand the construction of the irrational numbers without thinking about moving approximations. Until then I wouldn't worry too much about it. You are safe in assuming that the pencil and paper representation of the real line is a pretty good physical representation of an abstract idea.
 
  • #12
A decimal approximation of an irrational number is an arrow pointing towards a limit.

That the limit exists can be proven, that the approximation can be extended to reach the limit can be proven, but you often can't directly calculate the entire string of digits.

Usually you can settle for being able to show that it must be smaller than n, and larger than m.

The rational line is full of holes, within those holes lie the real numbers.

The integers are a nice grassy lawn, big gaps, evenly spaced, the rational numbers are a forest beside it, much smaller gaps, but still spread out nicely. The real numbers are the dense jungle which is visible behind (and indeed all around) the rational forest, and the whole lawn. The complex numbers are a mountain range surrounding the valley of real numbers.
 
  • #13
Minor correction: a decimal approximation is simply a number. It doesn't have an extent, it doesn't move, it doesn't point.

A sequence of decimal approximations, I suppose, can be viewed in a variety of ways. I wouldn't begrudge thinking of a Cauchy sequence as "pointing" to something.

And just to be clear for the opening poster, the (possibly infinite) decimal expansion of a number is not an approximation. Again, it is simply a number.


(to be overly precise decimals are often used as notations for numbers, rather than literally being numbers themselves. But that's far more pedantic than most people will ever care about, and in some presentations, decimals really are numbers rather than being merely numerals)
 
  • #14
Well, I simply wanted to give an impression of what a decimal approximation is without having to go into the deep definition of limits and such.

You are right that it doesn't "point" at a spot on the line, but it isn't too far off to say it helps indicate where the real number is located, in a way which could perhaps help deal with the issue the OP had regarding the infinity of the rationals without delving into Cauchy and such.


Like pi for example, the string 3.14159265358979323846264338327950288419716939937510... is only a suggestion of what the actual value is, kinda like saying "were you able to continue this to an infinite number of places, you'd reach the true value of pi", but the string there is not pi exactly, and it could be far from it if I put a 3 or a 9 next instead of a 5. All of those possible variations fan out from the irrational value, which is the only one that follows the sequence exactly.
 
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  • #15
(Talking about a graph with a number line) If you put your pencil on a whole number for example 3 and move the pencil to 4 then you must have passed through many irrational numbers, therefore you can find an irrational number on a number line. Does that sound credible?
 
  • #16
Mu naught said:
I still don't see how an irrational number can be represented by a definite point on a number line.

Consider a unit square. Put one corner of the square at the origin of your number line. The lower-left corner, for definiteness.

Now rotate the square 45 degrees clockwise, bringing the upper right corner of the square to the line. Now what point on the line is the corner of the square sitting on? Exactly square root of 2, right?

There's your irrational number on the number line.
 
  • #17
CodyOwen said:
(Talking about a graph with a number line) If you put your pencil on a whole number for example 3 and move the pencil to 4 then you must have passed through many irrational numbers, therefore you can find an irrational number on a number line. Does that sound credible?

Not really, because that already assumes that irrational numbers exist. Their existence is a consequence of how the system of real numbers is built up from the system of rational numbers (which, as has already been mentioned, is done via Cauchy sequences or Dedekind cuts). Part of the reason we define the real number system like this is aesthetic. To understand this, first assume that we have only rational numbers. It is easy to find (Google it) a proof that there is no rational number whose square is 2. (This is usually phrased, "√2 is irrational," but we are not yet assuming that irrational numbers exist.) Define X to be the set of positive rational numbers x such that x2<2, and define Y to be the set of positive rational numbers y such that y2>2. Then, define two sequences {an} and {bn} as follows:

a1=1, b1=2

an+1=(an+bn)/2, bn+1=bn, if [(an+bn)/2]2<2

an+1=an, bn+1= (an+bn)/2, if [(an+bn)/2]2>2.

{an} and {bn} are increasing and decreasing sequences, respectively, whose difference converges to 0, but neither of these sequences converges to a rational number. In other words, assuming we only have rational numbers, these two sequences are getting arbitrarily close together for large enough indices n, but neither sequence is going anywhere in particular. If you think about this, I hope you'll agree that this is an undesirable property for our number system to have if we are to represent it by a continuous line (the number line). Loosely speaking, it means that there is an infinitely small hole located where √2 should be. Defining the real number system as the set of equivalence classes of Cauchy sequences fixes this. In particular the equivalence class of {an} (or {bn} -- it doesn't matter) has the property that its square is 2.
 
  • #18
You should get together with Pythagoras and commiserate with him. It is alleged that his school drowned one of its own members for proving that sqrt(2) is not rational.
 
  • #19
Mu naught said:
I still don't see how an irrational number can be represented by a definite point on a number line.

Let me ask you a question.

Say you have a number line in front of you. Now put your pen down and draw a line from 0 to 2. The square root of two is somewhere in between 0 and 2. Did your pen cross over the square root of 2?

I think what you are missing is that points on a number line have 0 width. That is how a line or line segment can have an infinite number of points on it.
 
  • #20
Of course!
 
  • #21
Draw two points on the number line as follows: Draw a point 1 unit along the number line. Label it point A. To draw the second point, construct a square which has the line segment extending from the origin to A as one of its sides. Draw the diagonal of this square, and then rotate the diagonal down to the number line (see attached image). Label this point B.

Now create two line segments: one from the origin of the number line to point A and one from the origin to point B. Call these segments SA and SB respectively.

SB corresponds to an irrational number. Here is why:

  • First, I need one definition: Two line segments S1, S2, are called commensurate if there exists a third line segment, E, that can be successively lined up with itself N times and fit perfectly across S1 and be successively lined up with itself M times and fit perfectly across S2. (N and M are integers).
  • SA and SB are not commensurate. For a proof of this (and you should thoroughly look at this proof because this is the most important step of my argument here), see the section "Geometric proof of the irrationality of the sqrt(2)" of the following paper: http://www.bsu.edu/libraries/virtualpress/mathexchange/04-01/Coleman.pdf.
  • Now, suppose the length of SB was rational. Then Length(SB) = n/m for some integers n and m. But consider the segment, E, that is formed by chopping SA up into m equal parts. E can be successively lined up with itself m times and fit perfectly across SA. Also, E can be successively lined up with itself n times and fit perfectly across SB (because Length(SB) = n/m = n*(1/m)). Thus, SA and SB are commensurate.
  • From this, we can conclude that if the length of SB is rational, then SA and SB are commensurate. However, the link I included proved that SA and SB are not commensurate. Hence, the length of SB is not rational.

So, B is a point that corresponds to an irrational number.

That there are irrationals on the number line basically means that, if you chop your unit length up into pieces of equal length, there will always be points on the number line that you cannot get to by successively lining up an integer number of these pieces, no matter how small you try to make the pieces. There are points on the number line that simply 'do not submit' to that type of process.
 

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  • #22
Mu naught said:
It seems to me that you'd have to approach some point on the line, but continuously move toward it at a slower and slower rate, moving a thousandth of a unit, then a millionth, then a billionth and so on, constantly moving but constantly slowing down and never actually reaching any fixed point on the line. Is this a sound conception?

I think this is a good question. If you will allow me to rephrase what you are saying:
(1) We claim that the numbers (both rational and irrational) correspond to points on the number line.
(2) The way to find out what point a given number corresponds to is to by looking at its decimal expansion. If x=0.d1d2d3d4... then, to find the point it corresponds to, you start at the origin and move to the right d1 10ths, then d2 100ths, then d3 1000ths, and so on.
(3) But for numbers whose decimal expansion does not terminate (which, by the way, does not include only irrationals but also many rationals such as 1/3), we will find ourselves stuck in an infinite process when trying to apply (2) and we will never happen upon any final point.

So, how can we say that an irrational number corresponds to a point if, when we attempt to figure out what point it corresponds to, we end up eternally jumping from point to point, never landing on a final one. Might it not be the case that every point on the number line corresponds to a rational number and that the irrationals just specify an infinite sequence of points just as the sequence {1,2,3,4,5,...} specifies a infinite sequence of points but does not correspond to a point on the number line?

But, what if you ask the question a different way? Given the points on the number line, do we find any whose decimal expansion happens to be non-terminating. The answer is yes. An example is the point B constructed in my previous post.

You may be interested to hear that the question you raise was an open question back in early Greek mathematics. It was not settled whether or not every length could be expressed rationally.
 
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  • #23
This thread was very helpful to me.

Max™ said:
A decimal approximation of an irrational number is an arrow pointing towards a limit.

That the limit exists can be proven, that the approximation can be extended to reach the limit can be proven, but you often can't directly calculate the entire string of digits.

Usually you can settle for being able to show that it must be smaller than n, and larger than m.

The rational line is full of holes, within those holes lie the real numbers.

The integers are a nice grassy lawn, big gaps, evenly spaced, the rational numbers are a forest beside it, much smaller gaps, but still spread out nicely. The real numbers are the dense jungle which is visible behind (and indeed all around) the rational forest, and the whole lawn. The complex numbers are a mountain range surrounding the valley of real numbers.

This was a great explanation.

Hurkyl said:
And just to be clear for the opening poster, the (possibly infinite) decimal expansion of a number is not an approximation. Again, it is simply a number.

This is the statement that allowed me to lock it into my memory. Well put, folks.
 
  • #24
shreyakmath said:
Of course!



If you don't quote someone else's answer/question, that "of course!" is almost impossible to guess what its meaning is.

DonAntonio
 
  • #25
DonAntonio said:
If you don't quote someone else's answer/question, that "of course!" is almost impossible to guess what its meaning is.
"Almost impossible"? Seems pretty obvious to me. Generally speaking, if you aren't quoting a previous post, then you are, by default, replying to the OP, which was a question to which the answer is "Of course!".
 
  • #26
oay said:
"Almost impossible"? Seems pretty obvious to me. Generally speaking, if you aren't quoting a previous post, then you are, by default, replying to the OP, which was a question to which the answer is "Of course!".



For you, but for me it is almost "pretty obvious" he's addressing the message immediately before his, so

I think it is safe enough to say that it is, at least, some people can be confused. That's the reason we

have the very nice "quote" option...

DonAntonio
 
  • #27
All I can add here is that some people were talking about always approaching a point on the number line but never actually getting there because it is an non-terminating decimal.

This seems a lot like Zeno's paradox and how Zeno 'proved' that motion is impossible (fun fact: everyone contemporary with Zeno thought he was crazy, it was only after he died that some people started believing him).

I think the point (as highlighted early on by a couple people) is that the numberline is wholly our own construction, and therefore it seems silly to argue about this thing without a stringent definition of what the numberline is, in which case you can answer this question mathematically.
 
  • #28
Vorde said:
This seems a lot like Zeno's paradox and how Zeno 'proved' that motion is impossible

It is said Zeno's paradox is solved by the help of limit later. But I still wander because in limit, we take the value of the limit to be something that is never reached.

So the paradox to me is still a paradox.


__________________________
I am a beginner in this field.
 
  • #29
rajeshmarndi said:
It is said Zeno's paradox is solved by the help of limit later. But I still wander because in limit, we take the value of the limit to be something that is never reached.

So the paradox to me is still a paradox.

The premise of Zeno's paradox is that in order to move any distance, a person must first move an infinite number of smaller distances, each requiring a positive amount of time to traverse. Calculus demonstrates conclusively that this infinite number of positive "times" sums to a finite number; therefore, the time taken to move a distance if finite, therefore, there is no paradox. There is nothing that is "never reached". Zeno's paradox asserts that an infinite series diverges when it does not; there is no paradox, only an error.
 
  • #30
Number Nine said:
The premise of Zeno's paradox is that in order to move any distance, a person must first move an infinite number of smaller distances, each requiring a positive amount of time to traverse. Calculus demonstrates conclusively that this infinite number of positive "times" sums to a finite number; therefore, the time taken to move a distance if finite, therefore, there is no paradox. There is nothing that is "never reached". Zeno's paradox asserts that an infinite series diverges when it does not; there is no paradox, only an error.

I'm always puzzled by this argument.

To be sure, an infinite series can converge in mathematics. But how can we be sure that same logic applies to physics? Do you believe in the physical existence of Dedekind cuts? If the real line has a direct analog in the physical world, does that mean that the Continuum hypothesis is subject to physical experiment? Or that it might someday be, given enough technology?

I don't personally believe these things; therefore, I don't believe that the mathematical theory of infinite series resolves Zeno's paradoxes.

After all, Zeno was making a point about physical motion. Does one have to believe that math = physics in order to accept that the theory of infinite series resolves Zeno? Or is there something else I'm misunderstanding?
 
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  • #31
...Does one have to believe that math = physics in order to accept that the theory of infinite series resolves Zeno?

No, because Zeno makes a mathematical argument. He attaches a positive real number (denoting time) to each infinite distance that must be traversed and argues that motion between two points is thus impossible, because the amount of time required must be infinite. He has explicitly set up an infinite series. There is no need to appeal to physics, because the is already very clear (things do, in fact, move), so all we're left with is the theoretical (mathematical) justification for his paradox.
 
  • #32
Lot's of posters here presenting arguments about a 'point' or 'points'.

Any care to offer a definition that can be used in the arguments?

A good definition provides the answer to the original question.
 
  • #33
LCKurtz said:
The "real line" is an abstract idea. We draw a line on a sheet of paper to help us visualize its properties. But in the sense of a geometry construction, you can certainly plot a point that represents an irrational number. Just draw an x-axis and a y-axis perpendicular to it. Construct the 45 degree line y = x and mark 1 unit from the origin on it in the first quadrant. Drop a perpendicular from that point to the x axis. It will hit the x-axis at sqrt(2) / 2, which is irrational. But, of course, your pencil representation won't be exact. :frown:

Abstract idea or not, that is a clever solution. You have quite a knack for the cartesian coordiante system LCKurtz.

I liked the question too.
 
  • #34
I'm pretty sure I understand what mean and yes the concept that nothing can reach infinity is true. Now if irrational numbers exist is another question and that may depend on the numerical system.
Mu naught said:
So, if you want to represent a number as a point on this line, you can do so by starting at zero, and moving a certain distance to a point which corresponds to n units, the direction depending on if it's positive or negative. So what about an irrational number? It seems to me that you'd have to approach some point on the line, but continuously move toward it at a slower and slower rate, moving a thousandth of a unit, then a millionth, then a billionth and so on, constantly moving but constantly slowing down and never actually reaching any fixed point on the line.
 
  • #35
webberfolds said:
I'm pretty sure I understand what mean and yes the concept that nothing can reach infinity is true. Now if irrational numbers exist is another question and that may depend on the numerical system.
No, irrational numbers do exist no matter what numerical system you use. The existence of any numbers is independent of how you wish to write them.
 
<h2>1. What are irrational numbers?</h2><p>Irrational numbers are numbers that cannot be expressed as a ratio of two integers. They are non-repeating and non-terminating decimals, meaning they have an infinite number of decimal places.</p><h2>2. How do irrational numbers differ from rational numbers?</h2><p>Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot. Additionally, rational numbers have either a finite or repeating decimal representation, while irrational numbers have an infinite non-repeating decimal representation.</p><h2>3. Can irrational numbers exist on the numberline?</h2><p>Yes, irrational numbers can exist on the numberline. They are located between rational numbers and can be represented as points on the numberline, although their exact value cannot be determined.</p><h2>4. How are irrational numbers used in mathematics?</h2><p>Irrational numbers are used to represent quantities that cannot be expressed as whole numbers or fractions, such as the circumference of a circle or the diagonal of a square. They are also used in various mathematical proofs and calculations.</p><h2>5. Are there any real-life applications of irrational numbers?</h2><p>Yes, irrational numbers have many real-life applications in fields such as science, engineering, and finance. They are used to model and predict natural phenomena, design structures and machines, and calculate interest rates and financial growth.</p>

1. What are irrational numbers?

Irrational numbers are numbers that cannot be expressed as a ratio of two integers. They are non-repeating and non-terminating decimals, meaning they have an infinite number of decimal places.

2. How do irrational numbers differ from rational numbers?

Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot. Additionally, rational numbers have either a finite or repeating decimal representation, while irrational numbers have an infinite non-repeating decimal representation.

3. Can irrational numbers exist on the numberline?

Yes, irrational numbers can exist on the numberline. They are located between rational numbers and can be represented as points on the numberline, although their exact value cannot be determined.

4. How are irrational numbers used in mathematics?

Irrational numbers are used to represent quantities that cannot be expressed as whole numbers or fractions, such as the circumference of a circle or the diagonal of a square. They are also used in various mathematical proofs and calculations.

5. Are there any real-life applications of irrational numbers?

Yes, irrational numbers have many real-life applications in fields such as science, engineering, and finance. They are used to model and predict natural phenomena, design structures and machines, and calculate interest rates and financial growth.

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