Is the Product of All Irrationals Rational or Irrational?

[Loren Booda]

Some speculation:

Given that irrational numbers can be grouped in products of 2, 3...or N-->oo members, the products themselves being irrational,

and

given that irrational numbers can be grouped in products of 2, 3...or N-->oo members, the products themselves being rational,

it would seem that the product of all irrationals would be both irrational and rational, something like the limiting value of the sine function.

What do you think?

 

[matt grime]

The product of all irrationals? mm, you may want to read the thread in general maths about adding all the numbers between 0 and 1.

anyway, this alleged product, how on Earth are you defining it? I mean, I know how to multiply 2, 3 or finitely many numbers, and I know how to define the product of a sequence (1+x_1),(1+x_2),... , which exists exactly when the sum of the x_i's exists (and none of them is -1) but multiplying together an uncountable unordered set of numbers?

 

[Loren Booda]

Thanks for opening my eyes, matt. Apparently it was late at night when I baked my 1/2 idea.