Are all numbers properly classified in the real number system?

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SUMMARY

The discussion centers on the classification of numbers within the real number system, specifically addressing the relationships between natural numbers, integers, rational numbers, and irrational numbers. Participants confirm that natural numbers are a subset of integers, which in turn are a subset of rational numbers, all contained within the real numbers. The conversation also touches on the inclusion of irrational numbers in the real number system, with consensus on their classification as part of the reals. The mention of Gaussian integers highlights the complexity of number classification beyond the standard definitions.

PREREQUISITES
  • Understanding of number sets: natural numbers, integers, rational numbers, and irrational numbers.
  • Familiarity with the real number system and its hierarchy.
  • Basic knowledge of algebraic and transcendental numbers.
  • Awareness of advanced concepts like Gaussian integers and their field of fractions.
NEXT STEPS
  • Research the properties of algebraic vs. transcendental numbers.
  • Explore the concept of Gaussian integers and their applications.
  • Study the implications of irrational numbers within the real number system.
  • Investigate visual representations of number classifications and their relationships.
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Mathematicians, educators, students studying number theory, and anyone interested in the classification of numbers within the real number system.

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Have I correctly classified these sets of numbers? I am trying to diagram algebraic, transcendental, irrational...etc, numbers. Please see the attached picture.
Thanks
 

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First correction: put the natural numbers ellipse completely inside the integers. The attachment has been updated.
 
Last edited:
...comments?

Surely this isn't a difficult question.
 
naturals, integers, and rationals are contained in the reals by every definition I've seen (though there is the notion of the gaussian integers, it's field of fractions, etc). Irrationals can go either way, but the most common definition has them contained in the reals as well.
 

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