Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
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Mathematics Assessment Project (MAP)
Why is the product of rational numbers rational? (Explanation)
Any rational number can be written as one integer divided by another, right? That's the definition of a rational number. Now, suppose you have two of them, and you multiply them together. What happens? a c a * c - * - = ----- b d b * d Now, if a and c are integers, then a*c is also an integer, right? Similarly for b and d. So the product of two rational numbers must be a rational number.