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

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

$\frac{1}{2}$, $3 = \frac{3}{1}$, $0 = \frac{0}{5}$, $-2.5 = \frac{-5}{2}$.

• $\sqrt{5}$: 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.

Irrational.

• $\sqrt{8}$:

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

Irrational.

• $\sqrt{9}$:

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}

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

Rational (repeating decimal).

Rational.

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

Irrational.

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)