The Real Number System Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve this step by step.

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Part 1: Fill in the Definitions and Examples

We'll complete the table for each type of number.

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#### 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: -5, 0, 3.14, √2, π, etc.

> ✔ Real numbers include all rational and irrational numbers.

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#### 2. IRRATIONAL NUMBERS
- Definition: A real number that cannot be made by dividing two integers. It is a non-ending, non-repeating decimal.
- Examples: √2, √3, π, e, √7

> ✔ Irrational numbers cannot be expressed as fractions.

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#### 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, ...

> ✔ Rational numbers include fractions, terminating decimals, and repeating decimals.

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Now let’s move to the examples section.

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Part 2: Classify Each Number

We are to give all possible labels for each number using:
- R = Real
- I = Irrational
- RA = Rational
- IN = Integer
- W = Whole
- C = Counting

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#### 1. -12
- Is it a real number? ✔ Yes → R
- Is it rational? ✔ Yes, because it can be written as -12/1 → RA
- Is it an integer? ✔ Yes → IN
- Is it a whole number? ✘ No (whole numbers start at 0) → No
- Is it a counting number? ✘ No (counting numbers are positive integers starting from 1) → No

✔ Labels: R, RA, IN

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#### 2. √7
- Is it real? ✔ Yes → R
- Is it rational? ✘ No, because √7 is not a perfect square and cannot be expressed as a fraction → Not RA
- Is it irrational? ✔ Yes → I
- 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 integer, whole, or counting? ✘ No (it's a fraction less than 1)

✔ Labels: R, RA

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#### 4. 0.42681392... *(non-repeating, non-terminating)*
- The ellipsis (...) suggests it goes on forever without repeating.
- This is a non-repeating, non-terminating decimal → Irrational
- So it's real, but not rational

✔ Labels: R, I

> ⚠️ Note: If the decimal were repeating or terminating, it would be rational. But since it's described with "..." and no pattern, we assume it's irrational.

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#### 5. √16
- √16 = 4 (perfect square)
- So it's real → R
- It's rational (since 4 = 4/1) → RA
- It's an integer → IN
- It's a whole number → W
- It's a counting number → C

✔ Labels: R, RA, IN, W, C

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#### 6. 0
- Real? ✔ Yes → R
- Rational? ✔ Yes (0 = 0/1) → RA
- Integer? ✔ Yes → IN
- Whole number? ✔ Yes → W
- Counting number? ✘ No (counting starts at 1) → No

✔ 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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✔ Completed Table Summary

#### REAL NUMBERS
- Definition: A number that can represent the position along a number line. (It is not imaginary.)
- Examples: -12, √7, 3/5, 0.4268..., 4, 0

#### IRRATIONAL NUMBERS
- Definition: A real number that cannot be made by dividing two integers. It is a non-ending, non-repeating decimal.
- Examples: √7, √2, π, 0.42681392...

#### RATIONAL NUMBERS
- Definition: A real number that can be made by dividing two integers.
- Special Types:
- Integers: -12, 4, 0
- Whole Numbers: 0, 4
- Counting Numbers: 4

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