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glebovg
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Is there a proof that there exist (or does not exist) integers m and n such that [itex]{e^{m/n}} = \pi[/itex]? How would one prove such a statement?
Transcendental numbers are real numbers that are not algebraic, meaning they cannot be expressed as a root of a polynomial equation with integer coefficients. They are infinite, non-repeating, and cannot be represented as a fraction. Examples of transcendental numbers include pi and e.
All transcendental numbers are irrational, but not all irrational numbers are transcendental. Irrational numbers include both algebraic and transcendental numbers, while transcendental numbers are specifically those that cannot be expressed as roots of polynomial equations.
The concept of transcendental numbers was first introduced by mathematician Joseph-Louis Lagrange in 1751, but the term "transcendental" was coined by German mathematician Carl Louis Ferdinand von Lindemann in 1882.
Transcendental numbers play an important role in mathematics, particularly in the field of analysis. They are used in the study of continuous functions and infinite series, and are essential for understanding and proving many mathematical theorems.
Yes, there are infinitely many transcendental numbers. In fact, it has been proven that the majority of real numbers are transcendental. However, specific transcendental numbers are difficult to identify and there are still many unsolved questions about their distribution and properties.