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.
  • #36
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.
HallsofIvy said:
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.
 
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  • #37
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 :smile:.
 
  • #38
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.
 
  • #39
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.
 
  • #40
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.
 
  • #41
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.
 
  • #42
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

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.
 
Last edited:
<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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