New Number System Allows Pi to be Exact

  • Thread starter Thread starter Zuryn
  • Start date Start date
  • Tags Tags
    Pi
Click For Summary

Discussion Overview

The discussion revolves around the concept of a new number system that could allow for an exact representation of pi, exploring theoretical implications and the nature of rationality within different bases. The scope includes theoretical mathematics and conceptual exploration of number systems.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant suggests that pi can be exact if a new number system is developed.
  • Another participant questions the initial claim and asks for further elaboration on the idea.
  • A third participant asserts that pi already has an exact representation as "pi".
  • Further inquiry is made about the implications of creating a new number system where pi and e are rational, while 1/2 remains irrational.
  • One participant proposes using pi itself as a base for the numeration system, suggesting that pi could be represented as "1.0". They express skepticism about the possibility of representing both pi and e as finite or repeating decimals due to their algebraic independence.
  • The same participant notes that rational numbers in an integer-based system would be irrational in a pi-based system, indicating a potential conflict in representations across different bases.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of a new number system that allows for an exact representation of pi, with no consensus reached on the implications of such a system.

Contextual Notes

There are unresolved assumptions regarding the definitions of rationality and the properties of numbers in different bases, as well as the implications of algebraic independence.

Zuryn
Messages
9
Reaction score
0
Pi can be exact if we have a new number system. :mad:
 
Mathematics news on Phys.org
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.
 

Similar threads

High School Is PI (##\pi##) really a number?
  • · Replies 28 ·
Replies
28
Views
3K
High School How circular does it need to be?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Is e^pi a unique transcendental, or perhaps not transcendental?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Comparing ##a^b## and ##b^a##
  • · Replies 19 ·
Replies
19
Views
2K
Undergrad Is Riemann's Zeta at 2 Related to Pi through Prime Numbers?
  • · Replies 58 ·
2
Replies
58
Views
10K
Graduate My PI questions - I hope they are valid?
  • · Replies 10 ·
Replies
10
Views
2K
High School Dividing by infinity, exactly, finally!
  • · Replies 40 ·
2
Replies
40
Views
7K
Undergrad Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
4K
MHB Is $p\equiv 3\pmod{4}$ a condition for $\pi$ to be an even permutation?
  • · Replies 1 ·
Replies
1
Views
1K
MHB Why is Pi an irrational number?
  • · Replies 13 ·
Replies
13
Views
6K