Classifying & Ordering Real Numbers-Part 1 interactive worksheet ... - Free Printable

Problem Analysis:

The task involves classifying real numbers into different categories based on their properties. The categories are:
- Natural Numbers: Positive integers starting from 1 (e.g., 1, 2, 3, ...).
- Whole Numbers: Non-negative integers starting from 0 (e.g., 0, 1, 2, 3, ...).
- Integers: All whole numbers and their negatives (e.g., ..., -2, -1, 0, 1, 2, ...).
- Rational Numbers: Numbers that can be expressed as a ratio of two integers \( \frac{p}{q} \) where \( q \neq 0 \) (e.g., fractions, terminating decimals, repeating decimals).
- Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers and have non-terminating, non-repeating decimal expansions (e.g., \( \sqrt{2}, \pi \)).
- Real Numbers: All rational and irrational numbers combined.

Step-by-Step Solution:

#### Part 1: Drag each number into the Venn Diagram
We need to classify each number into the appropriate category in the Venn Diagram.

1. -19: This is an integer but not a whole number or natural number. It is also rational.
- Category: Integers (and Rational).

2. \( 1.\overline{2} \): This is a repeating decimal, which means it is rational.
- Category: Rational.

3. 0: This is a whole number, integer, and rational.
- Category: Whole, Integers, Rational.

4. 3: This is a natural number, whole number, integer, and rational.
- Category: Natural, Whole, Integers, Rational.

5. \( \sqrt{10} \): This is an irrational number because \( \sqrt{10} \) cannot be expressed as a fraction and has a non-terminating, non-repeating decimal expansion.
- Category: Irrational.

6. \( -\sqrt{81} \): Simplify \( \sqrt{81} = 9 \), so \( -\sqrt{81} = -9 \). This is an integer and rational.
- Category: Integers, Rational.

7. 3.456: This is a terminating decimal, which means it is rational.
- Category: Rational.

8. \( -\frac{6}{11} \): This is a fraction, which means it is rational.
- Category: Rational.

9. -1.4828...: If this is a non-repeating, non-terminating decimal, it is irrational. If it repeats, it is rational. Assuming it is non-repeating (as indicated by the ellipsis):
- Category: Irrational.

10. -44: This is an integer but not a whole number or natural number. It is also rational.
- Category: Integers, Rational.

11. \( \pi \): This is a well-known irrational number.
- Category: Irrational.

#### Part 2: Indicate which grouping each real number belongs to
We need to fill in the table for the given numbers.

a) \( \sqrt{81} \):
- Simplify: \( \sqrt{81} = 9 \).
- Natural: Yes (9 is a natural number).
- Whole: Yes (9 is a whole number).
- Integer: Yes (9 is an integer).
- Rational: Yes (9 is rational).
- Irrational: No (9 is not irrational).
- Real: Yes (9 is real).

b) \( 1.\overline{2} \):
- Natural: No (not a natural number).
- Whole: No (not a whole number).
- Integer: No (not an integer).
- Rational: Yes (repeating decimal).
- Irrational: No (not irrational).
- Real: Yes (real number).

c) 0:
- Natural: No (0 is not a natural number).
- Whole: Yes (0 is a whole number).
- Integer: Yes (0 is an integer).
- Rational: Yes (0 is rational).
- Irrational: No (0 is not irrational).
- Real: Yes (0 is real).

d) 13:
- Natural: Yes (13 is a natural number).
- Whole: Yes (13 is a whole number).
- Integer: Yes (13 is an integer).
- Rational: Yes (13 is rational).
- Irrational: No (13 is not irrational).
- Real: Yes (13 is real).

#### Part 3: Select a number that is an integer but not a whole number
An integer that is not a whole number must be negative. From the given numbers:
- -19, -44, and -\(\sqrt{81}\) (which simplifies to -9) are integers but not whole numbers.
- Choose one: -19.

#### Part 4: Select a rational number that is not an integer
A rational number that is not an integer must be a fraction or a decimal that is not a whole number. From the given numbers:
- \( 1.\overline{2} \), 3.456, and -\(\frac{6}{11}\) are rational but not integers.
- Choose one: \( 1.\overline{2} \).

Final Answers:

1. Venn Diagram Classification:
- -19: Integers, Rational.
- \( 1.\overline{2} \): Rational.
- 0: Whole, Integers, Rational.
- 3: Natural, Whole, Integers, Rational.
- \( \sqrt{10} \): Irrational.
- \( -\sqrt{81} \): Integers, Rational.
- 3.456: Rational.
- \( -\frac{6}{11} \): Rational.
- -1.4828...: Irrational.
- -44: Integers, Rational.
- \( \pi \): Irrational.

2. Table Filling:
- \( \sqrt{81} \): Natural (Yes), Whole (Yes), Integer (Yes), Rational (Yes), Irrational (No), Real (Yes).
- \( 1.\overline{2} \): Natural (No), Whole (No), Integer (No), Rational (Yes), Irrational (No), Real (Yes).
- 0: Natural (No), Whole (Yes), Integer (Yes), Rational (Yes), Irrational (No), Real (Yes).
- 13: Natural (Yes), Whole (Yes), Integer (Yes), Rational (Yes), Irrational (No), Real (Yes).

3. Integer but not a whole number: -19.
4. Rational number but not an integer: \( 1.\overline{2} \).

Final Boxed Answers:

\[
\boxed{-19, 1.\overline{2}}
\]