Question

which of the following statements is false? 1.666... is a rational number. $-sqrt{3}$ is an irrational number. $\frac{7}{5}$ is a rational number. 0 is neither a rational number nor an irrational number.

Explanation:

Step1: Recall rational/irrational definitions

Rational numbers: can be expressed as $\frac{p}{q}$ ($p,q\in\mathbb{Z}, q
eq0$), including integers, fractions, terminating/repeating decimals. Irrational: non - repeating, non - terminating decimals, like $\sqrt{prime}$.

Step2: Analyze each option

• Option 1: $1.666... = 1.\overline{6}=\frac{5}{3}$ (repeating decimal), so rational. Correct.
• Option 2: $-\sqrt{3}$: 3 is prime, $\sqrt{3}$ is irrational, so $-\sqrt{3}$ is irrational. Correct.
• Option 3: $\frac{7}{5}$ is a fraction ($p = 7,q = 5$), so rational. Correct.
• Option 4: 0 can be written as $\frac{0}{1}$, so it is a rational number (all integers are rational). So "0 is neither a rational number nor an irrational number" is false.

Answer:

The option "0 is neither a rational number nor an irrational number" (the last option with the circle next to "0 is neither a rational number nor an irrational number")