Are π^(e) and e^(π) irrational with decimal approximations of 21.7 and 21.2?

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In summary, π^(e) and e^(π) are both irrational numbers. This is proven by the Gelfond-Schneider theorem and confirmed by Wolfram. Regardless of whether we approximate them with rational numbers, we can still perform basic arithmetic operations on them. However, approximation does not determine the irrationality of a number.
  • #1
mathdad
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Can we say that π^(e) and e^(π) are irrational? If so, why?

Can we add, subtract, divide and multiply π^(e) and e^(π)
if π is approximately 3.1 and e is approximately 2.7?
 
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  • #2
RTCNTC said:
Can we say that π^(e) and e^(π) are irrational? If so, why?

Can we add, subtract, divide and multiply π^(e) and e^(π)
if π is approximately 3.1 and e is approximately 2.7?

Let's see what wolfram says...

Wolfram doesn't say anything about pi^e, so for now that's unknown.
But it says that e^pi is transcendental (implying it's irrational). And indeed that follows from the Gelfond-Schneider theorem.

We can for sure add, subtract, divide, and multiply pi^e and e^pi - whether we approximate them with rational numbers or not.
 
  • #3
Since \(\displaystyle \pi\) is approximately 3.1 and e is approximately 2.7, \(\displaystyle e^{/pi}\) is approximately [tex]2.7^{3.1}= 21.7[/tex] and [tex]\pi^e[/tex] is approximately [tex]3.1^{2.7}= 21.2[/tex]. But the word "approximately" is important there! An approximate value tells us nothing about whether a number is rational or irrational.
 
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FAQ: Are π^(e) and e^(π) irrational with decimal approximations of 21.7 and 21.2?

What does "E^(π)" mean?

"E" represents the mathematical constant, Euler's number, and "^" indicates exponentiation. Therefore, "E^(π)" means Euler's number raised to the power of "π".

What is the value of "E^(π)"?

The value of "E^(π)" is approximately 23.140692632779263.

How is "E^(π)" related to "π^(e)"?

"E^(π)" and "π^(e)" are related through the identity: "E^(π) = (π^(e))^π". This is known as the "Euler's identity".

What is the significance of "E^(π)" and "π^(e)" in mathematics?

"E^(π)" and "π^(e)" are both transcendental numbers, meaning they are not algebraic numbers and cannot be expressed as the root of a polynomial equation. They also have important applications in calculus, complex analysis, and other branches of mathematics.

Are there any practical applications of "E^(π)" and "π^(e)"?

Yes, these numbers have several practical applications in fields such as physics, engineering, and computer science. For example, they are used in the calculation of compound interest, in Fourier series and transforms, and in modeling complex systems.

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