The product of a non-zero rational number and an irrational number can always be written as

Multiplying an irrational number by a non-zero rational number gives an irrational number. If the product were rational, dividing it by the non-zero rational number would make the original number rational, which is a contradiction. Irrational numbers have decimal expansions that do not terminate or repeat, so the product is a non-repeating, non-terminating decimal.

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