Classifying Real Numbers Worksheet Answer Key Pdf - Fill Online ... - Free Printable

Problem Overview:

The task is to classify each given number into the appropriate subsets of the real number system. The subsets are:

1. Real Numbers (ℝ): All numbers that can be represented on the number line.
2. Rational Numbers (ℚ): Numbers that can be expressed as a ratio of two integers, \( \frac{p}{q} \), where \( q \neq 0 \).
3. Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers and have non-repeating, non-terminating decimal expansions.
4. Integers (ℤ): Whole numbers and their negatives, including zero.
5. Whole Numbers: Non-negative integers (0, 1, 2, 3, ...).
6. Natural Numbers (ℕ): Positive integers (1, 2, 3, ...).

We will analyze each number step by step.

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List of Numbers to Classify:

1. \( -\frac{5}{8} \)
2. \( 35.98 \)
3. \( 0 \)
4. \( \frac{4}{11} \)
5. \( \sqrt{5} \)
6. \( -93 \)
7. \( \frac{12}{3} \)
8. \( \sqrt{100} \)
9. \( -\sqrt{4} \)
10. \( 3.24 \)
11. \( 3\pi \)

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Classification:

#### 1. \( -\frac{5}{8} \)
- Real Number: Yes (all rational numbers are real).
- Rational Number: Yes (it is a fraction of two integers).
- Irrational Number: No (it is rational).
- Integer: No (it is not a whole number or its negative).
- Whole Number: No (it is negative).
- Natural Number: No (it is negative).

Classification: Real, Rational

#### 2. \( 35.98 \)
- Real Number: Yes.
- Rational Number: Yes (it can be written as \( \frac{3598}{100} \)).
- Irrational Number: No.
- Integer: No.
- Whole Number: No.
- Natural Number: No.

Classification: Real, Rational

#### 3. \( 0 \)
- Real Number: Yes.
- Rational Number: Yes (it can be written as \( \frac{0}{1} \)).
- Irrational Number: No.
- Integer: Yes.
- Whole Number: Yes.
- Natural Number: No (natural numbers start from 1).

Classification: Real, Rational, Integer, Whole

#### 4. \( \frac{4}{11} \)
- Real Number: Yes.
- Rational Number: Yes (it is a fraction of two integers).
- Irrational Number: No.
- Integer: No.
- Whole Number: No.
- Natural Number: No.

Classification: Real, Rational

#### 5. \( \sqrt{5} \)
- Real Number: Yes.
- Rational Number: No (the square root of 5 is an irrational number).
- Irrational Number: Yes.
- Integer: No.
- Whole Number: No.
- Natural Number: No.

Classification: Real, Irrational

#### 6. \( -93 \)
- Real Number: Yes.
- Rational Number: Yes (it can be written as \( \frac{-93}{1} \)).
- Irrational Number: No.
- Integer: Yes.
- Whole Number: No (it is negative).
- Natural Number: No (it is negative).

Classification: Real, Rational, Integer

#### 7. \( \frac{12}{3} \)
- Simplify: \( \frac{12}{3} = 4 \).
- Real Number: Yes.
- Rational Number: Yes (it simplifies to an integer).
- Irrational Number: No.
- Integer: Yes.
- Whole Number: Yes.
- Natural Number: Yes.

Classification: Real, Rational, Integer, Whole, Natural

#### 8. \( \sqrt{100} \)
- Simplify: \( \sqrt{100} = 10 \).
- Real Number: Yes.
- Rational Number: Yes (it simplifies to an integer).
- Irrational Number: No.
- Integer: Yes.
- Whole Number: Yes.
- Natural Number: Yes.

Classification: Real, Rational, Integer, Whole, Natural

#### 9. \( -\sqrt{4} \)
- Simplify: \( \sqrt{4} = 2 \), so \( -\sqrt{4} = -2 \).
- Real Number: Yes.
- Rational Number: Yes (it simplifies to an integer).
- Irrational Number: No.
- Integer: Yes.
- Whole Number: No (it is negative).
- Natural Number: No (it is negative).

Classification: Real, Rational, Integer

#### 10. \( 3.24 \)
- Real Number: Yes.
- Rational Number: Yes (it can be written as \( \frac{324}{100} \)).
- Irrational Number: No.
- Integer: No.
- Whole Number: No.
- Natural Number: No.

Classification: Real, Rational

#### 11. \( 3\pi \)
- Real Number: Yes.
- Rational Number: No (π is irrational, and multiplying it by 3 keeps it irrational).
- Irrational Number: Yes.
- Integer: No.
- Whole Number: No.
- Natural Number: No.

Classification: Real, Irrational

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Final Answer:

\[
\boxed{
\begin{array}{ll}
1. & \text{Real, Rational} \\
2. & \text{Real, Rational} \\
3. & \text{Real, Rational, Integer, Whole} \\
4. & \text{Real, Rational} \\
5. & \text{Real, Irrational} \\
6. & \text{Real, Rational, Integer} \\
7. & \text{Real, Rational, Integer, Whole, Natural} \\
8. & \text{Real, Rational, Integer, Whole, Natural} \\
9. & \text{Real, Rational, Integer} \\
10. & \text{Real, Rational} \\
11. & \text{Real, Irrational} \\
\end{array}
}
\]