Are all irrational numbers rational?

AI Thread Summary
Irrational numbers, such as pi, cannot be expressed as fractions of integers, which defines rational numbers. The discussion highlights a misunderstanding regarding the nature of pi as a ratio of circumference to diameter, emphasizing that this ratio does not yield a fraction of integers. It is clarified that the existence of a rational circle with both rational circumference and diameter is not possible. The conversation also touches on the broader implications of definitions in mathematics. Ultimately, all irrational numbers remain non-rational by their very nature.
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.
 
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mathwonk said:
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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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