Are the numbers ##eπ## and ##e+ π## transcendental?

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In summary, the conversation discusses the concepts of transcendental and algebraic numbers, specifically in relation to ##e## and ##π##. It is mentioned that these numbers are irrational and cannot be solved for in polynomial equations. The conversation also raises the question of whether ##\pi+e## or ##e\pi## is transcendental.
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chwala
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TL;DR Summary
A number is called an algebraic number if it is a solution of a polynomial equation ##a_0 z^n+a_1z^{n-1} + ... a_{n-1}z + a_n =0## where ##a_0,a_1 ...a_n## are integers...otherwise transcendental.
My question is [following the example on the attachment which is apparently clear to me].
1. Are the numbers ##eπ## and ##e+ π## Transcendental?
2. Algebraic numbers can also be rational and not necessarily integers? is that correct?
 

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All rational numbers are algebraic: [itex]n/m[/itex] is the solution of [itex]mz - n = 0[/itex].
 
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chwala said:
TL;DR Summary: A number is called an algebraic number if it is a solution of a polynomial equation ##a_0 z^n+a_1z^{n-1} + ... a_{n-1}z + a_n =0## where ##a_0,a_1 ...a_n## are integers...otherwise transcendental.

My question is [following the example on the attachment which is apparently clear to me].
1. Are the numbers ##eπ## and ##e+ π## Transcendental?
Presumably. The standard theorems about transcendency don't apply to them, but I haven't checked in detail. More interesting is the question: Who cares? These numbers do not occur naturally and I haven't seen any theorem that needed to know whether ##\mathbb{Q}[e,\pi ]## is of transcendental degree one or two. I guess literally nobody will expect it to be one.

chwala said:
2. Algebraic numbers can also be rational and not necessarily integers? is that correct?
##\mathbb{Z}\subsetneq \mathbb{Q} \subsetneq \mathbb{A}\subsetneq \mathbb{C}## if ##\mathbb{A}## is the field of all algebraic numbers over the rationals.
 
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What i would add on this is that ##e## and ## π## are irrational thus we cannot solve for them in a given equation say,

##z-e- π=0##

as is the case with algebraic terms in any given polynomial.
 
  • #5
chwala said:
What i would add on this is that ##e## and ## π## are irrational thus we cannot solve for them in a given equation say,

##z-e- π=0## as is the case with polynomials.
You must be careful. ##\pi, -\pi## and ##\pi^{-1}## are transcendent, but neither is ##(\pi)+(-\pi)## nor ##(\pi)\cdot (\pi^{-1}).##
 
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chwala said:
What i would add on this is that ##e## and ## π## are irrational
So is ## \sqrt 2 ## (but it is not of course transcendental, by definition).
fresh_42 said:
You must be careful. ##\pi, -\pi## and ##\pi^{-1}## are transcendent, but neither is ##(\pi)+(-\pi)## nor ##(\pi)\cdot (\pi^{-1}).##
And nor is ## e^{i \pi} ##.
 
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I believe it is an open problem: are ##\frac{\pi}{e}, e\pi, e+\pi## irrational? Transcendental?
See no.22

It is known that either ##\pi+e## or ##e\pi## is transcendental.
If both are algebraic, then ##(\pi+e)^2 - 4e\pi## is algebraic. So, ##\pi-e## is algebraic, which implies
[tex]
\frac{1}{2}((\pi+e) - (\pi-e)) = e
[/tex]
is algebraic, a contradiction.
 
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FAQ: Are the numbers ##eπ## and ##e+ π## transcendental?

Are the numbers ##eπ## and ##e+ π## both transcendental?

Yes, both ##eπ## and ##e+ π## are transcendental numbers. This means that they cannot be expressed as a ratio of two integers and have an infinite number of non-repeating digits after the decimal point.

How do we know that ##eπ## and ##e+ π## are transcendental?

The proof of the transcendence of ##eπ## and ##e+ π## was first given by Charles Hermite in 1873. He showed that these numbers cannot be algebraic, meaning they cannot be the root of any non-zero polynomial with rational coefficients.

Are there other numbers that are both irrational and transcendental?

Yes, there are many other numbers that are both irrational and transcendental. Some examples include ##\pi##, ##e##, and the square root of 2. In fact, the majority of real numbers are both irrational and transcendental.

Can we calculate the exact value of ##eπ## and ##e+ π##?

No, we cannot calculate the exact value of ##eπ## and ##e+ π##. These numbers have an infinite number of digits after the decimal point, so we can only approximate their values to a certain degree of accuracy.

What is the significance of transcendental numbers like ##eπ## and ##e+ π##?

Transcendental numbers have many important applications in mathematics, particularly in calculus and number theory. They also have connections to other areas of science, such as physics and engineering. Additionally, the study of transcendental numbers has led to important developments in the field of mathematical logic.

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