Question

which of the following is true?

• $sqrt{2}$ is an integer.
• $4.8overline{33}$ is a rational number but not an integer.
• 0 is neither a rational number nor an irrational number.
• -6.175 is an irrational number.

Explanation:

Step1: Recall number - type definitions

Rational numbers can be written as $\frac{p}{q}$ where $p,q\in\mathbb{Z}$ and $q
eq0$. Integers are whole numbers (positive, negative, or 0). Irrational numbers cannot be written as a fraction.

Step2: Analyze - 6.175

• 6.175 can be written as $\frac{- 6175}{1000}=\frac{-247}{40}$, so it is a rational number, not an irrational number.

Step3: Analyze 0

0 is an integer and can be written as $\frac{0}{1}$, so it is a rational number.

Step4: Analyze $4.\overline{833}$

Let $x = 4.\overline{833}$. Then $10x=48.\overline{333}$ and $1000x = 4833.\overline{333}$. Subtracting $10x$ from $1000x$ gives $990x=4785$, so $x=\frac{4785}{990}=\frac{1063}{220}$, it is a rational number but not an integer.

Step5: Analyze $\sqrt{2}$

$\sqrt{2}$ cannot be written as a fraction $\frac{p}{q}$ where $p,q\in\mathbb{Z}$ and $q
eq0$, so it is an irrational number, not an integer.

Answer:

4.833 (with the bar) is a rational number but not an integer.