Structured worksheet for students to define and classify real, rational, and irrational numbers.

Let's solve this step by step.

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Part 1: Fill in the Definitions and Examples

#### 1. REAL NUMBERS
- Definition: A number that can represent the position along a number line. (It is not imaginary.)
- Examples: Any number you can think of on the number line — like -3, 0, 2.5, √2, π, etc.
> Example: $ \pi, \sqrt{2}, 0, -5, 3.14 $

#### 2. IRRATIONAL NUMBERS
- Definition: A real number that cannot be made by dividing two integers. It is a non-ending, non-repeating decimal.
- Examples: $ \pi, \sqrt{2}, \sqrt{3}, e, 0.1010010001\ldots $
> These decimals go on forever without repeating.

#### 3. RATIONAL NUMBERS
- Definition: A real number that can be made by dividing two integers.
- Special Types:
- Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Whole Numbers: 0, 1, 2, 3, ...
- Counting Numbers: 1, 2, 3, ...

> Note: All integers are rational, all whole numbers are rational, and all counting numbers are rational.

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Part 2: Classify Each Number with All Possible Labels

We are to label each number with all applicable categories:
- Real (R)
- Irrational (I)
- Rational (RA)
- Integer (IN)
- Whole (W)
- Counting (C)

Let’s go through each:

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#### 1. -12
- Is it real? ✔ Yes → R
- Is it rational? ✔ Yes, because it can be written as $ \frac{-12}{1} $ → RA
- Is it an integer? ✔ Yes → IN
- Is it a whole number? ✘ No (whole numbers are 0, 1, 2, 3, ...) → Not W
- Is it a counting number? ✘ No (counting numbers start at 1) → Not C

✔ Labels: R, RA, IN

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#### 2. √7
- Is it real? ✔ Yes → R
- Is it rational? ✘ No. √7 is approximately 2.645..., but it's a non-repeating, non-terminating decimal → I
- Since it's irrational, it cannot be rational → Not RA
- Not an integer, whole, or counting number.

✔ Labels: R, I

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#### 3. 3/5
- Is it real? ✔ Yes → R
- Is it rational? ✔ Yes, it's a ratio of two integers → RA
- Is it an integer? ✘ No → Not IN
- Not whole or counting → Not W, not C

✔ Labels: R, RA

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#### 4. 0.42681392...
- The dots suggest it continues indefinitely and doesn’t repeat.
- If it's non-repeating and non-terminating, then it's irrational.
- But we must be cautious: if it were a repeating decimal (like 0.333...), it would be rational.
- Since it says "..." without indicating repetition, we assume it's non-repeating, so:
- Real? ✔ Yes → R
- Rational? ✘ No → I
- Not integer, whole, or counting.

✔ Labels: R, I

> ⚠️ Note: This is a bit ambiguous. If the decimal eventually repeats, it's rational. But since no pattern is given, we assume it's non-repeating → Irrational

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#### 5. √16
- Simplify: $ \sqrt{16} = 4 $
- So, it's just 4.
- Real? ✔ Yes → R
- Rational? ✔ Yes → RA
- Integer? ✔ Yes → IN
- Whole number? ✔ Yes → W
- Counting number? ✔ Yes → C

✔ Labels: R, RA, IN, W, C

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#### 6. 0
- Real? ✔ Yes → R
- Rational? ✔ Yes → RA
- Integer? ✔ Yes → IN
- Whole number? ✔ Yes → W
- Counting number? ✘ No (counting starts at 1) → Not C

✔ Labels: R, RA, IN, W

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✔ Final Answers:

| Number | Labels |
|--------------|--------|
| 1. -12 | R, RA, IN |
| 2. √7 | R, I |
| 3. 3/5 | R, RA |
| 4. 0.42681392... | R, I |
| 5. √16 | R, RA, IN, W, C |
| 6. 0 | R, RA, IN, W |

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✔ Summary Table (Completed):

#### REAL NUMBERS
- Definition: A number that can represent the position along a number line. (It is not imaginary.)
- Examples: $ \pi, \sqrt{2}, -3, 0, 5.7 $

#### IRRATIONAL NUMBERS
- Definition: A real number that cannot be made by dividing two integers. It is a non-ending, non-repeating decimal.
- Examples: $ \pi, \sqrt{3}, \sqrt{7}, e $

#### RATIONAL NUMBERS
- Definition: A real number that can be made by dividing two integers.
- Special Types:
- Integers: $ ..., -2, -1, 0, 1, 2, ... $
- Whole Numbers: $ 0, 1, 2, 3, ... $
- Counting Numbers: $ 1, 2, 3, ... $

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