Question

among the following seven numbers: $3\frac{1}{33}$, $\sqrt{3}$, $3.\dot{3}$, $\frac{1}{3}$, $1$, $0.323323332\cdots$, and $3.3$, there is(are) ____ irrational number(s).\
a. 0\
b. 1\
c. 2\
d. 3

Explanation:

Step1: Define irrational numbers

Irrational numbers are non-repeating, non-terminating decimals that cannot be written as $\frac{p}{q}$ where $p,q$ are integers, $q
eq0$.

Step2: Classify each number

• $3$: $\frac{3}{1}$, rational.
• $\frac{1}{33}$: $\frac{1}{33}$, rational.
• $3.\overline{3}$: Repeating decimal, rational ($=3+\frac{1}{3}=\frac{10}{3}$).
• $\sqrt{3}$: Non-repeating/terminating, irrational.
• $\frac{1}{3}-1=-\frac{2}{3}$: $\frac{-2}{3}$, rational.
• $0.323323332\cdots$: Non-repeating/terminating, irrational.
• $3.3$: $\frac{33}{10}$, rational.

Step3: Count irrational numbers

Count of irrational numbers: 2.

Answer:

C. 2