Separate the terminating decimal, repeating decimal, non-terminating decimal.

Question 1: Definition of rational number with examples

Step 1: Definition.
A rational number is a real number that can be expressed as the ratio $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

Step 2: Examples of rational numbers.
$\frac{1}{2}$, $3 = \frac{3}{1}$, $0 = \frac{0}{5}$, $-2.5 = \frac{-5}{2}$.

Step 3: Classify the given numbers as rational or irrational.

• $\sqrt{5}$: 5 is not a perfect square, so irrational.

$\sqrt{5} \approx 2.236\dots$ (non-terminating, non-repeating decimal)

Irrational.

• $\sqrt{2}$: 2 is not a perfect square, so irrational.

$\sqrt{2} \approx 1.414\dots$ (non-terminating, non-repeating)

Irrational.

• $\sqrt{3}$: 3 is not a perfect square, so irrational.

$\sqrt{3} \approx 1.732\dots$

Irrational.

• $\sqrt{8}$:

$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

Since $\sqrt{2}$ is irrational, $2\sqrt{2}$ is irrational.

Irrational.

• $\sqrt{9}$:

$\sqrt{9} = 3 = \frac{3}{1}$

Rational (example of rational number).

• $\sqrt{3.14}$: 3.14 is not a perfect square, so irrational.

Irrational.

Final classification:
Rational: $\sqrt{9}$
Irrational: $\sqrt{5}, \sqrt{2}, \sqrt{3}, \sqrt{8}, \sqrt{3.14}$

Rational: \sqrt{9
Irrational: \sqrt{5}, \sqrt{2}, \sqrt{3}, \sqrt{8}, \sqrt{3.14}}

Question 2: Separate rational and irrational decimals (non-terminating given)

Assume the decimals are $0.333\dots$, $2.010010001\dots$ (and typical third like $\sqrt{2} \approx 1.414213562\dots$).

Step 1: $0.333\dots = 0.\overline{3}$ (repeating non-terminating).

Let $x = 0.\overline{3}$.

$10x = 3.\overline{3}$

$10x - x = 3.\overline{3} - 0.\overline{3}$

$9x = 3$

$x = \frac{3}{9} = \frac{1}{3}$

Rational (repeating decimal).

Rational.

Step 2: $2.010010001\dots$

Pattern: 2.01 001 0001 ... (1's separated by increasing zeros: 1 zero, 2 zeros, 3 zeros, etc.).
Non-terminating, non-repeating.

Irrational.

Step 3: Decimal expansion of $\sqrt{2} \approx 1.414213562\dots$ (if included).
Non-terminating, non-repeating.

Irrational.

Final classification:
Rational: $0.333\dots = \frac{1}{3}$
Irrational: $2.010010001\dots$, $1.414213562\dots$ (for $\sqrt{2}$)

Rational: 0.333\dots
Irrational: 2.010010001\dots
(and \sqrt{2 decimal if listed)}