How to prove the irrationality of an expression

[bjgawp]

I've been thinking about this for a bit and I was wondering how one would go proving the irrationality of an expression such as pi + 2 or e - root(2). At a first (seemingly intuitive) thought, I figured that the sum of two irrational numbers (call them x and y) should be irrational as well but then the values x = pi and y = 1 - pi came to mind. Just a curious thought from a first-year university student.

 

[mathman]

I suspect (as your examples show) that there is no general approach. For your two examples, it is trivially obvious that they are irrational (if pi+2 was rational, then pi would be, if e-root(2) was rational, the e would be algebraic - it isn't).

 

[AlephZero]

Proofs that some numbers are irrationals are easy. Euclid gave a proof that root(2) is irrational - legend has it this was first discovered by Pythagoras, who thought it was such a subversive fact that his students were threatened with death if they told anybody about it!

Suppose root(2) is rational and equals p/q, where p and q have no common factor.
Then 2 = p^2/q^2, so p^2 = 2q^2, so p^2 is an even number
If p^2 is even, then p must be even. Write p = 2k.
Then 4k^2 = 2q^2, so 2k^2 = q^2.
Therefore q^2 is even and q is even.

So p and q must have a common factor 2, which contradicts the assumption. Therefore root(2) is irrational.

Proofs for pi and e are more complicated - but I expect Google will find something.

 

[Feldoh]

http://pi314.at/math/irrational.html
http://www.math.clemson.edu/~simms/neat/math/pi/piproof.html

http://mathforum.org/isaac/problems/eproof.html
http://en.wikipedia.org/wiki/Proof_that_e_is_irrational

 

[mathwonk]

these questions can be unbelievably hard. e.g. i think it is still unknown whether say e and pi are independent transcendentals//////////.?? sounds crazy. anyone remember?

 

[bjgawp]

Thanks everyone! I'm aware of the irrationality of the transcendental numbers e and pi, just proving them when they are involved in operations with other irrational numbers was my question. I find it interesting that the two aforementioned numbers are irrational yet we still aren't sure (or rather, haven't proven I think) whether if e + pi or e - pi are irrational or not.

 

[Curious3141]

A lot of the "standard" proofs for "nice" irrational constants involve assuming the opposite, then showing it leads to an integer between zero and one (an obvious contradiction).

Proof of e's irrationality is very easy using the series expansion. Proof of pi's irrationality is rather more involved - the simplest version I've seen is the proof of the irrationality of pi^2 (a stronger result than proving pi is irrational), and even that involved showing that assuming rational pi^2 led to some definite integral yielding an integer between 0 and 1.

Proof of transcendence can be very, very difficult without assuming non-elementary theorems like Gelfond-Schneider and Lindemann-Weierstrass. But once those are assumed it becomes surprisingly easy to prove the transcendence of e, pi and e^pi (but not pi^e - which is still a mystery as to its irrationality/transcendence).

The few times when a proof of transcendence becomes trivially easy is when the number is constructed to be transcendental, like Liouville's constant.