FOM: (ir)rational numbers

Peter Schuster pschust at
Mon Jun 19 09:06:44 EDT 2000

Comment on Harvey Friedman Fri, 16 Jun 2000 13:20:16 -0400

>Comment on Richman 6/16/00 10:13AM.
>>This situation is similar to the classic example of the existence of
>>two irrational numbers, x and y, such that x^y (exponentiation) is
>>rational. Let a = sqrt 2 and consider the two candidates a^a and
>>(a^a)^a = 2. If a^a is rational, the first candidate does it; if not,
>>the second does it. This is not a constructive proof even though the
>>construction is staring us in the face.
>This is not such a good example since it is well known that the existence
>of two irrationals, x,y, such that x^y is rational can be proved
>constructively. Take x = e and y = ln(2).

That's a good point: one has to alter the example in the following way: 
Are there two *algebraic* irrational numbers x,y such that x^y is rational? 

>A better example, not due to me, is the following.
>THEOREM. e + pi is irrational or e - pi is irrational.
>Proof: Suppose e + pi and e - pi are rational. Then by addition, 2e is
>rational. Contradiction.
>One can also prove that e + pi is transcendental or e - pi is
>transcendental in this way.

Well, one has proven that not both e+pi and e-pi can be rational/algebraic. 
That's all, a simple result with an appropriately simple proof. What else? 
Presumably not a proof of "either e+pi or e-pi is irrational/transcendental. 

>This raises the following question. Does the above proof "contain" a proof
>that e + pi is irrational? Obviously not. But how do you prove that?
>FIrst of all, what does it mean?

"A real number r is irrational" should mean that r is distinct 
from each rational number, where "to be distinct from" ought to 
be understood as "has postive distance from". My feeling is that 
this is just what we have in mind when we talk about irrationals; 
it's at least this what we expect from them.  

Similarly, any real r should be called transcendental if it is 
distinct from all algebraic numbers or, presumably equivalent  
but at least much nicer, if P(r) is distinct from 0 for all non-zero 
polynomials P with rational coefficients. 

Peter Schuster. 

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