Is Gamma Irrational? Investigating the Irrationality of Pi and e

In summary, the conversation discusses the difficulty of proving that certain numbers, such as pi and e, are irrational. The speaker shares their approach of rewriting e as a series and proving each term to be irrational, but admits to being stuck when trying to show that the multiplied numbers are also irrational. The topic of whether pi+e is irrational is brought up, and the possibility of γ(gamma) being irrational is mentioned as well.
  • #1
Pjennings
17
0
I've been thinking about pi^e lately, and trying to prove that it is irrational. By rewriting e as 1+1+1/2+1/3!+...+1/n! I got it to pi^2*pi^(1/2)*pi^(1/3!)*...*pi^(1/n!), and proved that each of these terms is irrational. I'm stuck when it comes to showing that multiplied together these numbers are irrational. Any ideas?
 
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  • #2
Although we know a lot about certain forms, proving numbers irrational is generally a very, very hard thing to do. For example, I think we don't even know whether or not pi+e is irrational!
 
  • #3
I don't think that is a workable approach, since it's possible to form a rational number as the series where the partial sums are all irrational.
 
  • #4
No, we do not know if pi+e is irrational, but I don't think that one is very interesting. I think it would be interesting to know if γ(gamma) is irrational though.
 

FAQ: Is Gamma Irrational? Investigating the Irrationality of Pi and e

What is an irrational number?

An irrational number is a number that cannot be expressed as a ratio of two integers. These numbers have infinite non-repeating decimal representations and cannot be written as fractions.

How can you prove that a number is irrational?

One way to prove that a number is irrational is by contradiction. Assume that the number can be expressed as a ratio of two integers, then show that this leads to a contradiction. Another method is to show that the number has an infinite non-repeating decimal representation.

What are some examples of irrational numbers?

Some examples of irrational numbers are pi (3.141592...), the square root of 2 (1.414213...), and the golden ratio (1.618033...).

Can an irrational number be written as a fraction?

No, by definition, irrational numbers cannot be expressed as fractions. They are non-terminating and non-repeating decimals.

Why are irrational numbers important in mathematics?

Irrational numbers play a crucial role in mathematics, especially in geometry and physics. They help us understand and model real-world phenomena, and they are essential in many mathematical proofs and calculations.

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