Worksheet for classifying numbers as rational or irrational.

To solve the problem, we need to classify each number as either rational or irrational. Here's a step-by-step explanation for each number:

Definitions:

- Rational Numbers: Numbers that can be expressed as the ratio of two integers \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Rational numbers include integers, fractions, terminating decimals, and repeating decimals.
- Irrational Numbers: Numbers that cannot be expressed as the ratio of two integers. Irrational numbers include non-terminating, non-repeating decimals and certain square roots (e.g., \( \sqrt{2} \)).

Classification of Each Number:

1. 0.8
- This is a terminating decimal, which can be expressed as a fraction: \( 0.8 = \frac{8}{10} = \frac{4}{5} \).
- Classification: Rational (R)

2. \( -\frac{3}{10} \)
- This is already in the form of a fraction.
- Classification: Rational (R)

3. \( \sqrt{40} \)
- The square root of 40 is not a perfect square. It cannot be simplified to a rational number.
- Classification: Irrational (I)

4. \( \sqrt{81} \)
- The square root of 81 is 9, which is an integer.
- Classification: Rational (R)

5. \( 2 \frac{1}{3} \)
- This is a mixed number, which can be converted to an improper fraction: \( 2 \frac{1}{3} = \frac{7}{3} \).
- Classification: Rational (R)

6. 0.35
- This is a terminating decimal, which can be expressed as a fraction: \( 0.35 = \frac{35}{100} = \frac{7}{20} \).
- Classification: Rational (R)

7. 0.33333...
- This is a repeating decimal, which can be expressed as a fraction: \( 0.33333... = \frac{1}{3} \).
- Classification: Rational (R)

8. -9
- This is an integer, which can be expressed as a fraction: \( -9 = \frac{-9}{1} \).
- Classification: Rational (R)

9. 3.4
- This is a terminating decimal, which can be expressed as a fraction: \( 3.4 = \frac{34}{10} = \frac{17}{5} \).
- Classification: Rational (R)

10. \( \sqrt{2} \)
- The square root of 2 is a well-known irrational number. It cannot be expressed as a fraction and is non-terminating, non-repeating.
- Classification: Irrational (I)

Final Answer:

\[
\begin{array}{|c|c|}
\hline
\text{Number} & \text{Rational or Irrational?} \\
\hline
0.8 & R \\
\hline
-\frac{3}{10} & R \\
\hline
\sqrt{40} & I \\
\hline
\sqrt{81} & R \\
\hline
2 \frac{1}{3} & R \\
\hline
0.35 & R \\
\hline
0.33333 \ldots & R \\
\hline
-9 & R \\
\hline
3.4 & R \\
\hline
\sqrt{2} & I \\
\hline
\end{array}
\]

\boxed{
\begin{array}{|c|c|}
\hline
\text{Number} & \text{Rational or Irrational?} \\
\hline
0.8 & R \\
\hline
-\frac{3}{10} & R \\
\hline
\sqrt{40} & I \\
\hline
\sqrt{81} & R \\
\hline
2 \frac{1}{3} & R \\
\hline
0.35 & R \\
\hline
0.33333 \ldots & R \\
\hline
-9 & R \\
\hline
3.4 & R \\
\hline
\sqrt{2} & I \\
\hline
\end{array}
}