New Number System Allows Pi to be Exact

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SUMMARY

The discussion centers on the concept of creating a new number system where Pi can be represented exactly. Participants argue that while Pi can be denoted as "1.0" in a Pi-based numeration system, both Pi and e cannot be represented as finite or repeating decimals simultaneously due to their algebraic independence. The conversation highlights the inherent contradictions in rational and irrational number classifications across different bases, particularly emphasizing that rational numbers in an integer-based system may appear irrational in a Pi-based system.

PREREQUISITES
  • Understanding of algebraic independence, particularly regarding Pi and e.
  • Familiarity with rational and irrational numbers in various numeral systems.
  • Knowledge of numeral systems and their bases, specifically integer-based systems.
  • Basic concepts of number representation in mathematics.
NEXT STEPS
  • Research the properties of algebraically independent numbers.
  • Explore the implications of numeral systems based on non-integer bases.
  • Study the representation of rational and irrational numbers in different bases.
  • Investigate the mathematical foundations of number theory related to Pi and e.
USEFUL FOR

Mathematicians, educators, and students interested in advanced number theory, particularly those exploring the implications of numeral systems and the nature of irrational numbers.

Zuryn
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Pi can be exact if we have a new number system. :mad:
 
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Wanna expand on that?
 
We already have an exact representation of pi: "pi".
 
So what is your exact idea about this ?
What do you think if we do create a new number system to make Pi and e rational, but 1/2 is irrantional number?
....
 
Obviously, we could choose pi itself as a base for our numeration system so that pi would be "1.0". Since pi and e are algebraically independent, I don't believe it is possible to choose a base so that pi and e are both represented by finite or repeating decimals. It is fairly easy to show that a number is "rational" in an INTEGER based numeration system if and only if it is "rational" in any INTEGER based numeration system but I'm pretty sure that all the rational numbers in an INTEGER based numeration system (e.g. base 10) would be "irrational" in a pi based numeration system.
 

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