why is the product of a rational number and an irrational number irrational?

because the product is always a non-terminating, non-repeating decimal
because the product is always a fraction
because the product is always a negative number
because the product is always a repeating or a terminating decimal

Question

why is the product of a rational number and an irrational number irrational?

because the product is always a non-terminating, non-repeating decimal
because the product is always a fraction
because the product is always a negative number
because the product is always a repeating or a terminating decimal

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "direct_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Rational Numbers", "Irrational Numbers", "Real Number Properties" ], "current_concepts": [ "Rational Numbers", "Irrational Numbers", "Real Number Properties" ] } ## Define rational and irrational numbers $$ \begin{aligned} &\text{Rational: } \frac{a}{b} \quad (a, b \in \mathbb{Z}, b eq 0) \implies \text{terminating or repeating decimal} \ &\text{Irrational: cannot be written as } \frac{a}{b} \implies \text{non-terminating, non-repeating decimal} \end{aligned} $$ ## Analyze the product of a non-zero rational and an irrational number $$ \begin{aligned} &r \in \mathbb{Q} \setminus \{0\}, \quad i otin \mathbb{Q} \ &p = r \cdot i \implies p otin \mathbb{Q} \implies \text{non-terminating, non-repeating decimal} \end{aligned} $$ ## Match with the given options $$ \text{Option 1 matches the definition of an irrational product.} $$

(A) because the product is always a non-terminating, non-repeating decimal

(B) because the product is always a fraction

(C) because the product is always a negative number

(D) because the product is always a repeating or a terminating decimal

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Real Number Properties" ] }

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Question

Explanation:

Define rational and irrational numbers

Analyze the product of a non-zero rational and an irrational number

Match with the given options

Answer: