Determine whether each statement is true or false. If the sta your reasoning. If the statement is false, provide a countere

Question

12. A number cannot be irrational and an integer.

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13. All integers are rational.

GPT-4o mini · Answer

**12. A number cannot be irrational and an integer.** **True.** An integer is a whole number that can be positive, negative, or zero (e.g., -2, -1, 0, 1, 2). An irrational number, on the other hand, is a number that cannot be expressed as a fraction of two integers, meaning it cannot be written in the form $ \frac{a}{b} $ where $ a $ and $ b $ are integers and $ b \neq 0 $. Since all integers can be expressed as a fraction (for example, the integer 2 can be expressed as $ \frac{2}{1} $), they are rational. Therefore, a number cannot be both irrational and an integer. --- **13. All integers are rational.** **True.** By definition, a rational number is any number that can be expressed as the quotient of two integers, where the denominator is not zero (i.e., in the form $ \frac{a}{b} $ with $ b \neq 0 $). All integers can be written as a fraction where the denominator is 1 (e.g., the integer 3 can be expressed as $ \frac{3}{1} $). Hence, all integers are indeed rational numbers.

Answer

Simplify each

GPT-4o mini · Answer

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Answer

No the response

GPT-4o mini · Answer

Sure! Here’s a simplified response to your previous statements regarding irrationality and integers:

12. A number cannot be irrational and an integer.

True. Integers are whole numbers (like -1, 0, 1, 2) that can be expressed as fractions with 1 as the denominator (e.g., $ \frac{2}{1} $). Irrational numbers cannot be expressed as such fractions. Therefore, no integer can be irrational.

13. All integers are rational.

True. Rational numbers can be expressed as fractions of integers (e.g., $ \frac{a}{b} $ where $ b \neq 0 $). Since every integer (like 5) can be represented as a fraction (e.g., $ \frac{5}{1} $), all integers are rational numbers.