Question

among the following eight numbers: $2.71828$, $sqrt{2}$, $4.overline{56}$, $pi - 2$, $\frac{22}{7}$, $sqrt{9}$, $pi$, and $\frac{1}{pi}$, there is(are) ____ rational number(s).
a. 1
b. 2
c. 3
d. 4

Explanation:

Step1: Define rational numbers

Rational numbers can be written as $\frac{p}{q}$ where $p,q$ are integers, $q
eq0$.

Step2: Check each number

1. $2.71828$: Terminating decimal, rational.
2. $\sqrt{2}$: Irrational (non-repeating, non-terminating).
3. $4.\overline{56}$: Repeating decimal, rational.
4. $\pi - 2$: $\pi$ is irrational, so result is irrational.
5. $\frac{22}{7}$: Fraction of integers, rational.
6. $\sqrt{9}=3$: Integer, rational.
7. $\pi$: Irrational (non-repeating, non-terminating).
8. $\frac{1}{\pi}$: Reciprocal of irrational, irrational.

Step3: Count rational numbers

Count: $2.71828, 4.\overline{56}, \frac{22}{7}, \sqrt{9}$ → 4 numbers.

Answer:

D. 4