- #1
Aditya89
- 23
- 0
Is e^pi rational? I seem to have heard from one of my tacher that research was going. How far we have gone?
Is e^pi Rational? - Research Progress
- Context: Graduate
- Thread starter Aditya89
- Start date
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- Rational
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Discussion Overview
The discussion centers around the question of whether e raised to the power of pi (e^pi) is rational. Participants explore related mathematical concepts and the implications of transcendental numbers, referencing ongoing research and historical commentary.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant questions the rationality of e^pi and inquires about the current state of research on the topic.
- Another participant asserts that e^pi is transcendental, referencing Gelfond's theorem as a basis for this claim.
- A different participant introduces Euler's formula, suggesting it may be relevant to the discussion of e and pi.
- A subsequent reply quotes a historical comment attributed to Gauss regarding the clarity of Euler's formula, implying a standard of mathematical understanding.
Areas of Agreement / Disagreement
Participants express differing views on the rationality of e^pi, with some asserting its transcendental nature while others seem to seek further clarification or exploration of the topic. The discussion remains unresolved.
Contextual Notes
The discussion includes references to mathematical theorems and historical perspectives, but lacks detailed proofs or definitions that could clarify the claims made.
- #2
devious_
- 312
- 3
It's transcendental. This can be proved using Gelfond's theorem.
- #3
jim mcnamara
Mentor
- 4,789
- 3,852
Since you seem to be going in this direction, you might like Euler's formula
which relates 0, e, pi, i
http://mathworld.wolfram.com/EulerFormula.html
which relates 0, e, pi, i
http://mathworld.wolfram.com/EulerFormula.html
- #4
devious_
- 312
- 3
From the link jim posted:
"Gauss is reported to have commented that if [[itex]e^{i \pi} + 1 = 0[/itex]] was not immediately obvious, the reader would never be a first-class mathematician."
...
"Gauss is reported to have commented that if [[itex]e^{i \pi} + 1 = 0[/itex]] was not immediately obvious, the reader would never be a first-class mathematician."
...
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