An accurate representation of Irrational and rational numbers

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Discussion Overview

The discussion revolves around the representation of irrational numbers, their exact placement on the real line, and the implications of different numerical bases. Participants explore whether irrational numbers can be accurately represented and how this relates to their existence on the real line.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether there is an accurate way to write the value of an irrational number.
  • Others argue that while irrational numbers require infinite digits for representation, their points on the real line are fixed.
  • There is a discussion about the use of nested intervals to prove the existence of irrational numbers on the real line.
  • Participants propose that different bases can represent rational numbers like 2/3 accurately in finite digits, with base 3 being mentioned specifically.
  • Some participants express confusion about the distinction between an irrational number and its representation, questioning if they are fundamentally different concepts.
  • There is a debate about whether representations of numbers can convey new mathematical information beyond cardinality or ordinality.
  • Participants discuss the nature of pi and its representation, with some suggesting that it cannot be represented as a ratio of integers.
  • Concerns are raised regarding the definition of a "proper" representation of numbers, with some arguing that traditional arithmetic operations do not suffice for irrational numbers.
  • Some participants assert that the construction of the real line does not depend on digital representations, emphasizing the abstract nature of real numbers.

Areas of Agreement / Disagreement

Participants express differing views on the nature of representations of irrational numbers, with no consensus on whether a proper representation exists or what it entails. The discussion remains unresolved regarding the implications of these representations and the nature of irrational numbers themselves.

Contextual Notes

Participants highlight limitations in definitions and representations, noting that the discussion involves assumptions about what constitutes an accurate representation and the nature of mathematical operations.

  • #31
pi can be any 'place' on the real line? of course not, if we must think of it as some physical object, going left to right, positive to negative, then pi is always to the right of all of the following numbers:

3, 3.1, 3.14, 3.141,...

and is in fact the sup of that set, which would be one definition of pi as a real number (dedekind cut)

why do you think pi can be any place on the real line? (ie be any element of the real numbers?)
 
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  • #32
Because if the diameter can be any number, then pi (as you wrote) is the ratio of a circle's diameter to its circumference.

Therefore pi can be found in any "place" on the real line, which means, we only care for the ratio and not for any particular value.
 
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  • #33
The diameter can be any number. The circumference can be any number. The ratio of the two is a constant; it can only be one number.
 
  • #34
Dialog said:
Because if the diameter can be any number, then pi (as you wrote) is the ratio of a circle's diameter to its circumference.

Therefore pi can be found in any "place" on the real line, which means, we only care for the ratio and not for any particular value.

usually in a 'therefore' the consequent is implied by the antecedent. I don't see anything that can remotely justify that assertion mathematically. pi is fixed, the ratio between a circle's circumference and diameter is independent of the circle, in the sense that it is the same for any two circles.
 
  • #35
Hurkyl said:
The diameter can be any number. The circumference can be any number. The ratio of the two is a constant; it can only be one number.
So, is this constant is the "place" of pi on the real line?

Matt Grime said:
the ratio between a circle's circumference and diameter is independent of the circle, in the sense that it is the same for any two circles.
This is exactly what i mean, the pair(circumference,diameter) can be in any "place" on the real line as long as they stick together by their constant ratio.

In more general view, from this point of view there is no such a thing like a one real-line made up of fixed numbers, but infinitely many real-lines on top of each other where each real-line has its own 1 and we get a fractal, which its depth is infinitely many real-lines.
 
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  • #36
\pi is just a constant, like 1. On a circle if you have a diameter of 10 then you will have a circumference of 10\pi, if you have a diameter of 1 / \pi then you will have a diameter of 1.

The circumference of a circle divided by its diameter will always equal pi, no matter what the size of the circle is. So \pi is just a number.
 
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  • #37
So, is this constant is the "place" of pi on the real line?

Yes; this constant IS pi.


This is exactly what i mean, the pair(circumference,diameter) can be in any "place" on the real line as long as they stick together by their constant ratio.

I can't figure out what you mean in any part of this sentence.


but infinitely many real-lines

There are indeed infinitely many things called the real line, but they are all the same in roughly the same sense that 1, 1, and 1 are all the same.

But I don't see how this relates to anything you're saying.
 
  • #38
Dialog said:
This is exactly what i mean, the pair(circumference,diameter) can be in any "place" on the real line as long as they stick together by their constant ratio.

the pair (circumference,diameter) is a pair of real numbers, therefore it isn't a real number. But I think we can translate that sentence to mean: given any real number d (positive) there is another real number, C, which is the circumference of a circle with diameter d, for any such pair the ratio C/d is constant. This constant is what we call pi.

In more general view, from this point of view there is no such a thing like a one real-line made up of fixed numbers, but infinitely many real-lines on top of each other where each real-line has its own 1 and we get a fractal, which its depth is infinitely many real-lines.

Nonsense that sounds dangerously like Organic/WWW/Doron Shadmi.
 
  • #39
A Fractal is an invariant self-similarity over infinitely many scales, so what is the reason that we cannot use the fractal model instead of the real-line model?

The advantage of the fractal model is that we can use both relative and/or constant definitions, and by this model the real number system is richer then the constant-only real-line model.
 
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  • #40
thank you for offering conclusive proof of your identity. the usual rebuttals are valid:
1. you've no clue what the mathematical meaning of any of the words you use is
2 you're about to redefine a definition which isn't how things are done
3. you're not actually offering a definition in any mathematical sense just offering subjective opinions
 
  • #41
Matt,

Please response to all what I wrote in the previous post (after refreshing screen)
 
  • #42
that sounds dangerously like Organic/WWW/Doron Shadmi- Matt

Those little hairs on the back of my neck too were starting to rise. :smile:
 
  • #43
This does not belong here, Organic or not.
 

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