- #1
Aditya89
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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
- Thread starter Aditya89
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In summary, e^pi is an irrational number that cannot be written as a fraction. This is because e, which is a transcendental number, when raised to the power of another transcendental number like pi, results in an irrational number. The irrationality of e^pi was proven by Johann Lambert in 1761 and has both mathematical and practical significance. It helps establish the existence of transcendental numbers and is useful for generating random numbers in computer programming.
- #2
devious_
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It's transcendental. This can be proved using Gelfond's theorem.
- #3
jim mcnamara
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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_
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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."
...
1. Is e^pi a rational number?
No, e^pi is an irrational number. This means that it cannot be expressed as a ratio of two integers.
2. Can e^pi be written as a fraction?
No, e^pi cannot be written as a fraction because it is an irrational number.
3. Why is e^pi considered an irrational number?
Euler's number, e, is a transcendental number, meaning it cannot be expressed as a root of a polynomial equation with integer coefficients. When e is raised to the power of a transcendental number, such as pi, the result is also transcendental and thus irrational.
4. How is the irrationality of e^pi proven?
The irrationality of e^pi was proven by Johann Lambert in 1761 using a proof by contradiction. He showed that if e^pi is rational, it would lead to a contradiction in the decimal representation of the number.
5. What is the significance of e^pi being irrational?
The irrationality of e^pi has both mathematical and practical significance. Mathematically, it helps to establish the existence of transcendental numbers, which have important applications in calculus and number theory. Practically, it means that the decimal representation of e^pi is non-repeating and non-terminating, making it useful for generating random numbers in computer programming.
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