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Euler famously showed that ∏ ( 1 − x n ) = 1 − x − x 2 + x 5 + x 7 − x 1 2 − x 1 5 + . . . for ∣ x ∣ < 1 , where the exponents are the generalised pentagonal numbers. In our case, x = 0 . 1 , we get the overestimate 0 . 8 9 0 0 1 0 1 if we go up to exponent 7 , with an error of less than 1 0 − 1 1 . So, no, the product fails to be rational.