Can irrational numbers exist on the numberline?

[Number Nine]

...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.

 

[Studiot]

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.

 

[mesa]

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.

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.

 

[webberfolds]

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.

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.

 

[HallsofIvy]

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.

 

[webberfolds]

Ya it doesn't matter what numeral system use. I wrote that because I didn't know at the time. I don't know if irrational numbers exist though but math can prove it does or doesn't.

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.

 

[funcalys]

A rational number a/b where b is nonzero can, however, be exactly represented on the real line, can't it?. If irrational numbers didn't exist, then the the number line would have all elements being rational, which can be disproved. Then they must somehow exist .

 

[Vorde]

Doesn't it totally depend on what you mean by number line?

If you are talking about a geometrical line with lengths and numbers, then of course rational numbers would exist.

I can't imagine how else you would imagine it, but if you decided to mean something else but a geometric line then the answer would lie in that field.

 

[Robert1986]

So, I don't want to beat a dead horse, but I think that the OP is, in some sense, on the exact right track. He talks about "moving" along the number line and "slowing" down at an irrational point. Of course, strictly speaking, this makes no sense; as many people have pointed out, numbers don't move. But his thought process is still valid. I mean, what he has essentially described is a sequence that converges to an irrational number, but this is one of the ways we define irrational numbers. We say "you know what, there are lots of rational sequences that converge to something that isn't rational so let's just 'create' numbers that fill these gaps."

Now, the actual question of whether irrational numbers are on the number line or "exist" in some sense aren't nearly as interesting as the intuition that the OP has. I haven't seen any posts mention this explicitly (that's not to say there aren't any, I might have missed some) and I think it is important to point out.

 

[Studiot]

First this seems to be an old thread (2010) revived.

Secondly I assume you were referring to the second quote, from the originator, Robert.

Robert 1986
So, I don't want to beat a dead horse, but I think that the OP is, in some sense, on the exact right track. He talks about "moving" along the number line and "slowing" down at an irrational point. Of course, strictly speaking, this makes no sense; as many people have pointed out, numbers don't move. But his thought process is still valid. I mean, what he has essentially described is a sequence that converges to an irrational number, but this is one of the ways we define irrational numbers. We say "you know what, there are lots of rational sequences that converge to something that isn't rational so let's just 'create' numbers that fill these gaps."

Now, the actual question of whether irrational numbers are on the number line or "exist" in some sense aren't nearly as interesting as the intuition that the OP has. I haven't seen any posts mention this explicitly (that's not to say there aren't any, I might have missed some) and I think it is important to point out.

Mu naught
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.

I do believe in my post#32 I alluded to what you are saying, as did others in other ways.

A point is "that which has no part"

A line is an assembly of an infinite number of such points.

We can prove that the cardinality of this infinity is greater than the cardinality of the (also infinite) set of rational numbers.
In other words the set of real numbers has more points than the rationals.

Since they are not rational, we call these 'extra' points non rational or irrational.

It does not actualy matter which model we use to assemble the points into the real number line, the crucial fact is that there are more of them.

 

[Natko]

This is like The arrow paradox; you can't reach the point on the line as you'd have to go through infinity points in fieri.

 

[Vorde]

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

But as has been said. You can easily construct a line segment of irrational length. One could claim that this is just an approximation done by the tools involved, but then you could make the same claim for rational numbers.