Difference of two irrational numbers

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SUMMARY

The discussion centers on the mathematical concept of the difference between two irrational numbers, specifically whether x - y can be rational when both x and y are irrational. It is established that if y is defined as y = x + r, where r is any rational number, then x - y results in a rational number. This conclusion is crucial in the context of measure theory, particularly in constructing non-measurable sets using equivalence relations on the interval [0,1]. The equivalence relation x ~ y is defined such that x - y is rational, leading to the formation of equivalence classes for both rational and irrational numbers.

PREREQUISITES
  • Understanding of irrational numbers and their properties
  • Familiarity with rational numbers and their operations
  • Basic knowledge of equivalence relations in mathematics
  • Introduction to measure theory concepts
NEXT STEPS
  • Study the properties of equivalence relations in detail
  • Explore the Axiom of Choice and its implications in set theory
  • Learn about non-measurable sets and their construction in measure theory
  • Investigate the implications of rational and irrational number operations
USEFUL FOR

Mathematicians, students of measure theory, and anyone interested in the properties of rational and irrational numbers will benefit from this discussion.

ak416
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Im wondering if its possible given x,y irrational, that x-y is rational (other than the case x=y). The reason I am asking this is that I am reading a book on measure theory and they try to construct a non measurable set and they start with an equivalence relation on [0,1} x~y if x-y is rational. Then they construct a set using the axiom of choice which contains exactly 1 element from each equivalence class. I know that the set of all rational numbers in [0,1) is an equivalence class, also each irrational number forms an equivalence class because for each irrational number x, x-x=0 (rational). Is there any other possibilities?
 
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Let x be any irrational, and let y=x+r. (with r any rational number). Then y and x are irrational, and y-x is rational.
 
that was simple :)
 

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