Question

question
mark says that the number \\( \sqrt{2} \\) is a rational number because he can write it as the fraction \\( \frac{\sqrt{2}}{1} \\).
is mark correct? why or why not?
\\( \circ \\) mark is correct because the fraction \\( \frac{\sqrt{2}}{1} \\) is equal to \\( \sqrt{2} \\).
\\( \circ \\) mark is correct because the denominator, 1, is an integer.
\\( \circ \\) mark is not correct because \\( \sqrt{2} \\) is not equal to the fraction \\( \frac{\sqrt{2}}{1} \\).
\\( \circ \\) mark is not correct because a fraction is rational only if the numerator and the denominator are both integers.

Explanation:

To determine if Mark is correct, we recall the definition of a rational number: a rational number is a number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q
eq0\).

• For the fraction \(\frac{\sqrt{2}}{1}\), the numerator \(\sqrt{2}\) is not an integer (since \(\sqrt{2}\approx1.414\) and is irrational), while the denominator \(1\) is an integer.
• The first option is wrong because just being equal as a number doesn't make it rational; the definition requires integer numerator and denominator.
• The second option is wrong because the denominator being an integer is not sufficient— the numerator must also be an integer.
• The third option is wrong because \(\frac{\sqrt{2}}{1}\) is equal to \(\sqrt{2}\) numerically, but that's not the issue with rationality.
• The fourth option is correct because a rational number's fraction form requires both numerator and denominator to be integers, and \(\sqrt{2}\) is not an integer, so \(\frac{\sqrt{2}}{1}\) does not satisfy the definition of a rational - number fraction.

Answer:

D. Mark is not correct because a fraction is rational only if the numerator and the denominator are both integers.