What Makes Irrational Numbers So Intriguing Between Rational Ones?

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SUMMARY

The discussion centers on the intriguing nature of irrational numbers in relation to rational numbers. It is established that between any two distinct rational numbers, there exists an infinite number of irrational numbers. This concept highlights the density of both rational and irrational numbers on the number line, emphasizing that no matter how close two rational numbers are, an infinite set of irrational numbers can always be found in between them.

PREREQUISITES
  • Understanding of number lines and their properties
  • Familiarity with rational and irrational numbers
  • Basic knowledge of mathematical infinity
  • Concept of density in real numbers
NEXT STEPS
  • Explore the concept of Cantor's diagonal argument
  • Learn about the properties of real numbers and their subsets
  • Investigate the implications of density in mathematical analysis
  • Study the historical context of irrational numbers in mathematics
USEFUL FOR

Mathematicians, educators, students studying real analysis, and anyone interested in the foundational concepts of number theory.

icystrike
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They can fit into number lines but not marked on a sewing thread ?

I love to think of between 2 infinity small rational numbers there is a infinity deep hole that you can always pick a different irrational number out of it. (Is it a safe idea? )
 
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I'm not sure what you mean by a sewing thread, but yes, it is true that between any two distinct rational numbers there are always an infinite number of irrational numbers, no matter how close together you pick the rational numbers. It's also true that there are an infinite number of rational numbers between any two different rational numbers.
 

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