Question
Marcelo claims that the statement "rational numbers are integers" is ALWAYS true.
Is Marcelo correct?
If yes, explain why.
If no, provide an example.
Is Marcelo correct?
If yes, explain why.
If no, provide an example.
Answers
GPT-4o mini
No, Marcelo is not correct. The statement "rational numbers are integers" is not always true.
To understand this, we need to clarify the definitions of rational numbers and integers:
- **Rational numbers** are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. For example, \( \frac{1}{2} \), \( \frac{-3}{4} \), and \( 5 \) (which can be expressed as \( \frac{5}{1} \)) are all rational numbers.
- **Integers** are whole numbers that can be positive, negative, or zero. They include numbers like -2, -1, 0, 1, 2, etc.
While all integers are rational numbers (since any integer \( n \) can be written as \( \frac{n}{1} \)), not all rational numbers are integers. For example, the number \( \frac{1}{2} \) is a rational number but not an integer.
Therefore, a counterexample is \( \frac{1}{2} \), which demonstrates that not all rational numbers are integers.
To understand this, we need to clarify the definitions of rational numbers and integers:
- **Rational numbers** are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. For example, \( \frac{1}{2} \), \( \frac{-3}{4} \), and \( 5 \) (which can be expressed as \( \frac{5}{1} \)) are all rational numbers.
- **Integers** are whole numbers that can be positive, negative, or zero. They include numbers like -2, -1, 0, 1, 2, etc.
While all integers are rational numbers (since any integer \( n \) can be written as \( \frac{n}{1} \)), not all rational numbers are integers. For example, the number \( \frac{1}{2} \) is a rational number but not an integer.
Therefore, a counterexample is \( \frac{1}{2} \), which demonstrates that not all rational numbers are integers.