If all real numbers were to be multiplied to each other, what would be the product?

(a) Undefined

(b) Imaginary Number

(c) None of these

If we exclude zero, product is undefined.

The pairing argument which seemed to show the product is 1, is incorrect. The result depends directly on how they are paired. Instead of pairing every real number with its reciprocal, pair it with half its reciprocal. Now we are multiplying 1/2 an infinite uncountable number of times. Or pair every number with twice its reciprocal. Now we are multiplying 2 an infinite uncountable number of times.