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

Let's solve this step by step.

---

Part 1: Classify each number

We need to classify each number under the correct categories:

- Real: All numbers on the number line (includes rational and irrational).
- Rational: Can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers, $q \ne 0$. Includes terminating and repeating decimals.
- Irrational: Cannot be expressed as a fraction; non-repeating, non-terminating decimals.
- Integer: Whole numbers and their negatives: ..., -2, -1, 0, 1, 2, ...
- Whole: Non-negative integers: 0, 1, 2, 3, ...
- Counting: Positive integers: 1, 2, 3, ...

---

#### 1. -19
- Real: ✔
- Rational: ✔ (can be written as $-\frac{19}{1}$)
- Irrational: ✘
- Integer: ✔
- Whole: ✘ (negative)
- Counting: ✘

---

#### 2. π (pi)
- Real: ✔
- Rational: ✘
- Irrational: ✔ (non-repeating, non-terminating decimal)
- Integer: ✘
- Whole: ✘
- Counting: ✘

---

#### 3. $-\frac{2}{3}$
- Real: ✔
- Rational: ✔ (it’s a fraction)
- Irrational: ✘
- Integer: ✘
- Whole: ✘
- Counting: ✘

---

#### 4. 42
- Real: ✔
- Rational: ✔
- Irrational: ✘
- Integer: ✔
- Whole: ✔
- Counting: ✔

---

#### 5. $\sqrt{8}$
- $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$ → irrational
- Real: ✔
- Rational: ✘
- Irrational: ✔
- Integer: ✘
- Whole: ✘
- Counting: ✘

---

#### 6. 0.582
- Terminating decimal → can be written as $\frac{582}{1000}$
- Real: ✔
- Rational: ✔
- Irrational: ✘
- Integer: ✘
- Whole: ✘
- Counting: ✘

---

#### 7. 0
- Real: ✔
- Rational: ✔ ($\frac{0}{1}$)
- Irrational: ✘
- Integer: ✔
- Whole: ✔
- Counting: ✘ (counting starts at 1)

---

#### 8. 2.51821... (with "..." suggesting non-repeating, non-terminating)
- This is a non-repeating, non-terminating decimal → irrational
- Real: ✔
- Rational: ✘
- Irrational: ✔
- Integer: ✘
- Whole: ✘
- Counting: ✘

> Note: If it were repeating (e.g., 2.51821821...), it would be rational. But since it's just shown as "...", we assume it's non-repeating → irrational.

---

#### 9. $\sqrt{25}$
- $\sqrt{25} = 5$
- So it's just 5
- Real: ✔
- Rational: ✔
- Irrational: ✘
- Integer: ✔
- Whole: ✔
- Counting: ✔

---

✔ Final Table (Checkmarks):

| Number | Real | Rational | Irrational | Integer | Whole | Counting |
|---------------|------|----------|------------|---------|-------|----------|
| -19 | ✔ | ✔ | ✘ | ✔ | ✘ | ✘ |
| π | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |
| $-\frac{2}{3}$ | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
| 42 | ✔ | ✔ | ✘ | ✔ | ✔ | ✔ |
| $\sqrt{8}$ | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |
| 0.582 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
| 0 | ✔ | ✔ | ✘ | ✔ | ✔ | ✘ |
| 2.51821... | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |
| $\sqrt{25}$ | ✔ | ✔ | ✘ | ✔ | ✔ | ✔ |

---

Part 2: True or False Statements

#### 5. The square root of a non-perfect square integer is always irrational.
✔ True

- Example: $\sqrt{2}, \sqrt{3}, \sqrt{5}$ are all irrational.
- A perfect square has an integer square root; others do not.
- Proven by contradiction in math: if $\sqrt{n}$ is rational for non-perfect square $n$, then $n$ must be a perfect square.

---

#### 6. A never-ending decimal is always irrational.
✘ False

- Counterexample: $0.\overline{3} = \frac{1}{3}$ → rational but never-ending (repeating)
- Only non-repeating, non-terminating decimals are irrational.
- So, repeating decimals are rational even if they go on forever.

---

#### 7. Integers are always rational.
✔ True

- Any integer $n$ can be written as $\frac{n}{1}$ → rational.

---

#### 8. The square root of a perfect square integer is always rational.
✔ True

- Example: $\sqrt{9} = 3$, $\sqrt{16} = 4$, $\sqrt{1} = 1$, etc.
- All are integers → rational.

---

#### 9. All fractions are rational.
⚠️ Careful: This depends on definition.

But in standard math:
- A fraction means $\frac{a}{b}$ where $a$ and $b$ are integers, $b \ne 0$ → that’s the definition of a rational number.
- So yes, all such fractions are rational.

✔ True

> Note: If someone says "fraction" like $\frac{\pi}{2}$, that’s not a rational number — but technically, that's not a *rational* fraction. In this context, “fraction” refers to a ratio of two integers.

So assuming “fraction” means rational expression with integers → True

---

#### 10. Whole numbers are always rational.
✔ True

- Whole numbers: 0, 1, 2, 3, ...
- Each can be written as $\frac{n}{1}$ → rational.

---

✔ Final Answers for #5–10:

| Statement | Answer | Explanation |
|---------|--------|-----------|
| 5. The square root of a non-perfect square integer is always irrational. | True | Yes, proven mathematically. |
| 6. A never-ending decimal is always irrational. | False | Repeating decimals like $0.\overline{3}$ are rational. |
| 7. Integers are always rational. | True | Can be written as $\frac{n}{1}$. |
| 8. The square root of a perfect square integer is always rational. | True | Result is an integer → rational. |
| 9. All fractions are rational. | True | Fractions of integers are rational. |
| 10. Whole numbers are always rational. | True | Whole numbers are integers → rational. |

---

Let me know if you'd like this formatted as a filled-in worksheet!