Identifying Rational and Irrational Numbers WKS 05 | PDF | Complex ... - Free Printable

Problem Analysis:

The task involves identifying whether given numbers are rational or irrational and explaining the reasoning. Additionally, there is a section to identify rational numbers from a list and explain why \(3.14\) is rational while \(\pi\) is irrational.

---

Step-by-Step Solution:

#### Part 1: Determine Whether Each Number Is Rational or Irrational
We will analyze each number and classify it as rational or irrational, providing reasoning for each.

1. \(\frac{5}{6}\)
- Reasoning: A rational number is any number that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). Here, \(\frac{5}{6}\) is already in fractional form with integers \(5\) and \(6\).
- Classification: Rational

2. \(\frac{2}{5}\)
- Reasoning: Similar to the first number, \(\frac{2}{5}\) is a fraction with integers \(2\) and \(5\).
- Classification: Rational

3. \(0\)
- Reasoning: Zero can be expressed as \(\frac{0}{1}\), which is a fraction with integers \(0\) and \(1\).
- Classification: Rational

4. \(\frac{1}{2}\)
- Reasoning: This is a fraction with integers \(1\) and \(2\).
- Classification: Rational

5. \(\frac{2}{7}\)
- Reasoning: This is a fraction with integers \(2\) and \(7\).
- Classification: Rational

6. \(1\frac{1}{6}\)
- Reasoning: Mixed numbers can be converted to improper fractions. \(1\frac{1}{6} = \frac{7}{6}\), which is a fraction with integers \(7\) and \(6\).
- Classification: Rational

7. \(4.\overline{3}\)
- Reasoning: Repeating decimals can be expressed as fractions. For example, \(4.\overline{3} = 4 + \frac{1}{3} = \frac{13}{3}\), which is a fraction with integers \(13\) and \(3\).
- Classification: Rational

8. \(-3\)
- Reasoning: Integers are rational numbers because they can be written as fractions with a denominator of \(1\). Here, \(-3 = \frac{-3}{1}\).
- Classification: Rational

9. \(9.777...\)
- Reasoning: Repeating decimals are rational. \(9.777... = 9 + \frac{7}{9} = \frac{88}{9}\), which is a fraction with integers \(88\) and \(9\).
- Classification: Rational

10. \(-67\)
- Reasoning: Integers are rational numbers. \(-67 = \frac{-67}{1}\).
- Classification: Rational

---

#### Part 2: Circle All Rational Numbers
We will analyze each number in the list and determine if it is rational.

11.
- \(\frac{9}{12}\): Simplifies to \(\frac{3}{4}\), which is a fraction.
- \(\sqrt{32}\): Not a perfect square, so it is irrational.
- \(\pi\): Known to be irrational.
- \(\sqrt{5}\): Not a perfect square, so it is irrational.
- Rational: \(\frac{9}{12}\)

12.
- \(3\frac{1}{6}\): Mixed number, converts to \(\frac{19}{6}\), which is a fraction.
- \(\sqrt{8}\): Not a perfect square, so it is irrational.
- \(\infty\): Not a number, so it is neither rational nor irrational.
- \(\sqrt{16}\): Perfect square, equals \(4\), which is an integer (and thus rational).
- Rational: \(3\frac{1}{6}, \sqrt{16}\)

13.
- \(\sqrt{1}\): Equals \(1\), which is an integer.
- \(\sqrt{2}\): Not a perfect square, so it is irrational.
- \(\sqrt{3}\): Not a perfect square, so it is irrational.
- \(\sqrt{4}\): Perfect square, equals \(2\), which is an integer.
- Rational: \(\sqrt{1}, \sqrt{4}\)

14.
- \(64\): Integer, so rational.
- \(-64\): Integer, so rational.
- \(\sqrt{64}\): Perfect square, equals \(8\), which is an integer.
- \(\frac{1}{64}\): Fraction with integers \(1\) and \(64\).
- Rational: \(64, -64, \sqrt{64}, \frac{1}{64}\)

15.
- \(23.157\): Terminating decimal, so rational.
- \(\frac{17}{5}\): Fraction with integers \(17\) and \(5\).
- \(\sqrt{24}\): Not a perfect square, so it is irrational.
- \(\sqrt{25}\): Perfect square, equals \(5\), which is an integer.
- Rational: \(23.157, \frac{17}{5}, \sqrt{25}\)

16.
- \(0.5\): Terminating decimal, so rational.
- \(5.5\): Terminating decimal, so rational.
- \(\frac{14}{9}\): Fraction with integers \(14\) and \(9\).
- \(\sqrt{\pi}\): Square root of an irrational number, so it is irrational.
- Rational: \(0.5, 5.5, \frac{14}{9}\)

17.
- \(\sqrt{28}\): Not a perfect square, so it is irrational.
- \(\sqrt{29}\): Not a perfect square, so it is irrational.
- \(\sqrt{30}\): Not a perfect square, so it is irrational.
- \(\sqrt{31}\): Not a perfect square, so it is irrational.
- Rational: None

18.
- \(-16\): Integer, so rational.
- \(16\): Integer, so rational.
- \(\sqrt{16}\): Perfect square, equals \(4\), which is an integer.
- \(\sqrt[4]{16}\): Fourth root of \(16\), equals \(2\), which is an integer.
- Rational: \(-16, 16, \sqrt{16}, \sqrt[4]{16}\)

---

#### Part 3: Explain Why \(3.14\) Is Rational While \(\pi\) Is Irrational
- \(3.14\):
- \(3.14\) is a terminating decimal, meaning it has a finite number of digits after the decimal point. Any terminating decimal can be expressed as a fraction. For example, \(3.14 = \frac{314}{100}\), which is a fraction with integers \(314\) and \(100\). Therefore, \(3.14\) is rational.

- \(\pi\):
- \(\pi\) is a non-terminating, non-repeating decimal. Its decimal representation goes on infinitely without repeating (e.g., \(3.141592653589793...\)). Since it cannot be expressed as a fraction of two integers, \(\pi\) is irrational.

---

Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Part 1:} \\
1. \text{Rational} \\
2. \text{Rational} \\
3. \text{Rational} \\
4. \text{Rational} \\
5. \text{Rational} \\
6. \text{Rational} \\
7. \text{Rational} \\
8. \text{Rational} \\
9. \text{Rational} \\
10. \text{Rational} \\
\\
\text{Part 2:} \\
\text{Rational numbers: } \frac{9}{12}, 3\frac{1}{6}, \sqrt{16}, \sqrt{1}, \sqrt{4}, 64, -64, \sqrt{64}, \frac{1}{64}, 23.157, \frac{17}{5}, \sqrt{25}, 0.5, 5.5, \frac{14}{9}, -16, 16, \sqrt{16}, \sqrt[4]{16} \\
\\
\text{Part 3:} \\
3.14 \text{ is rational because it is a terminating decimal. } \pi \text{ is irrational because it is a non-terminating, non-repeating decimal.}
\end{array}
}
\]