Problem preview

N-RN.3 Warmup M2-062-A11-V03

Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.

Classify always, sometimes, or never statements about rational and irrational sums and products

Problem

Is this statement always, sometimes, or never true? "The product of a nonzero rational number and an irrational number is rational."

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Never, because if a nonzero rational times an irrational were rational, then dividing by the nonzero rational would make the irrational number rational, which is impossible.

Sometimes, because multiplying can cancel the irrational part, like how irrational numbers can cancel when you add or subtract them.

Sometimes, because if the rational number is 0, then the product is 0, which is rational, but other rational numbers can give irrational products.

Always, because rational numbers are closed under multiplication, so multiplying by a rational number makes the product rational.

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Curriculum context

Course: Math II
Standard: N-RN.3
Category: Number and Quantity
Domain: The Real Number System
Objective: Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Problem type: Classify always, sometimes, or never statements about rational and irrational sums and products