Are all irrational numbers rational?

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Discussion Overview

The discussion revolves around the nature of irrational numbers, specifically questioning whether all irrational numbers could be considered rational. Participants explore the definitions and implications of irrationality in the context of mathematical concepts such as pi and the properties of circles.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants propose that since pi is defined as the ratio of the circumference of a circle to its diameter, it raises the question of whether a fraction could exist for all non-repeating decimal values.
  • Others argue against this notion, stating that it is incorrect to assume a circle with rational circumference and diameter exists, thus implying that pi cannot be expressed as a fraction of integers.
  • A participant humorously questions the logic by comparing irrational numbers to absurd statements about humans and mortals, suggesting a playful critique of the initial question.
  • Another participant reiterates the definition of a rational number, emphasizing that it must be expressible as a fraction of integers, which pi does not satisfy.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the nature of irrational numbers and their relationship to rational numbers.

Contextual Notes

The discussion highlights assumptions about the definitions of rational and irrational numbers, as well as the implications of mathematical properties related to circles. There are unresolved questions about the nature of these definitions and their applications.

Skhandelwal
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Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?
 
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Skhandelwal said:
Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?

Short answer: No.

Tongue-in-cheek answer: Yes, but the fraction would have at least one noninteger.

Longer answer: You're wrongly assuming that a circle with rational circumference and diameter exists.
 
? are all humans non human? are all m ortals immortal? are al...
 
mathwonk said:
? are all humans non human? are all m ortals immortal? are al...

lol :smile:
 
Skhandelwal said:
Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?
The definition of "rational number" is that it can be written as a fraction with numerator and denominator integers. The "ratio of the circumference of the circle to its diameter" is not a ratio of integers.
 

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