The Number System Worksheet - Free Printable

Let's solve this step by step. We need to classify each number into the appropriate sets:

Definitions:

- Natural Numbers: Positive integers starting from 1 (1, 2, 3, ...)
- Whole Numbers: Natural numbers plus zero (0, 1, 2, 3, ...)
- Integers: Whole numbers and their negatives (... -3, -2, -1, 0, 1, 2, 3, ...)
- Rational Numbers: Numbers that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$. Includes terminating or repeating decimals.
- Irrational Numbers: Numbers that cannot be written as a fraction. Non-repeating, non-terminating decimals (e.g., $\pi$, $\sqrt{2}$).
- Real Numbers: All rational and irrational numbers.

Now, let’s go through each number:

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1. $-\sqrt{17}$

- $\sqrt{17}$ is irrational (not a perfect square), so $-\sqrt{17}$ is also irrational.
- It's not a natural, whole, or integer.
- But it is a real number.
✔ Irrational Numbers, ✔ Real Numbers

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2. $-2$

- Negative integer.
- Not natural or whole.
- Integer, rational (since $-2 = \frac{-2}{1}$), and real.
✔ Integers, ✔ Rational Numbers, ✔ Real Numbers

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3. $-\frac{9}{37}$

- A fraction with integers numerator and denominator → rational.
- Not natural, whole, or integer (negative and not a whole number).
- But rational and real.
✔ Rational Numbers, ✔ Real Numbers

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4. $0$

- Zero is a whole number and an integer.
- Not natural (natural starts at 1).
- Rational ($0 = \frac{0}{1}$), real.
✔ Whole Numbers, ✔ Integers, ✔ Rational Numbers, ✔ Real Numbers

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5. $-6.06$

- Decimal, but terminates → can be written as a fraction: $-\frac{606}{100} = -\frac{303}{50}$
- So it's rational.
- Not natural, whole, or integer.
- Real.
✔ Rational Numbers, ✔ Real Numbers

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6. $4.\overline{56}$ (repeating decimal)

- Repeating decimal → rational.
- Can be written as a fraction.
- Not natural, whole, or integer.
- Real.
✔ Rational Numbers, ✔ Real Numbers

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7. $3.050050005...$ (non-repeating, non-terminating pattern)

- The decimal pattern increases in zeros: 3.050050005... → no repeating cycle.
- This is non-repeating and non-terminating, so it's irrational.
- Not rational, not integer, etc.
- But still a real number.
✔ Irrational Numbers, ✔ Real Numbers

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8. $18$

- Positive integer.
- Natural, whole, integer, rational ($\frac{18}{1}$), real.
✔ Natural Numbers, ✔ Whole Numbers, ✔ Integers, ✔ Rational Numbers, ✔ Real Numbers

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9. $\frac{-43}{0}$

- Division by zero → undefined.
- Not a number at all.
- So it doesn't belong to any of these sets.
✘ None (but if we must mark, leave blank)

> Note: This is undefined, so it does not belong to any number set.

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10. $\pi$

- Famous irrational number.
- Not rational, not integer, etc.
- But real.
✔ Irrational Numbers, ✔ Real Numbers

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11. $.634$ (repeating? Probably meant to be $\overline{634}$)

- Assuming it's $0.\overline{634}$ — repeating decimal → rational.
- Terminating or repeating → rational.
- Not natural, whole, integer.
- Real.
✔ Rational Numbers, ✔ Real Numbers

> If it's just 0.634 (terminating), still rational.

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12. $\sqrt{225}$

- $\sqrt{225} = 15$ → perfect square.
- So it's 15 → positive integer.
- So: natural, whole, integer, rational, real.
✔ Natural Numbers, ✔ Whole Numbers, ✔ Integers, ✔ Rational Numbers, ✔ Real Numbers

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13. $.634$ (same as #11?)

- Assuming it's a terminating decimal: 0.634
- Can write as $\frac{634}{1000} = \frac{317}{500}$
- Rational, real.
✔ Rational Numbers, ✔ Real Numbers

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14. $\sqrt{\frac{4}{49}}$

- $\sqrt{\frac{4}{49}} = \frac{\sqrt{4}}{\sqrt{49}} = \frac{2}{7}$
- Rational number.
- Not natural, whole, integer.
- Real.
✔ Rational Numbers, ✔ Real Numbers

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15. $-\sqrt{64}$

- $\sqrt{64} = 8$, so $-\sqrt{64} = -8$
- Integer, rational, real.
- Not natural or whole.
✔ Integers, ✔ Rational Numbers, ✔ Real Numbers

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Final Table:

| Number | Natural Numbers | Whole Numbers | Integers | Rational Numbers | Irrational Numbers | Real Numbers |
|--------|------------------|---------------|----------|--------------------|----------------------|--------------|
| 1. $-\sqrt{17}$ | | | | | ✔ | ✔ |
| 2. $-2$ | | | ✔ | ✔ | | ✔ |
| 3. $-\frac{9}{37}$ | | | | ✔ | | ✔ |
| 4. $0$ | | ✔ | ✔ | ✔ | | ✔ |
| 5. $-6.06$ | | | | ✔ | | ✔ |
| 6. $4.\overline{56}$ | | | | ✔ | | ✔ |
| 7. $3.050050005...$ | | | | | ✔ | ✔ |
| 8. $18$ | ✔ | ✔ | ✔ | ✔ | | ✔ |
| 9. $\frac{-43}{0}$ | | | | | | |
| 10. $\pi$ | | | | | ✔ | ✔ |
| 11. $.634$ | | | | ✔ | | ✔ |
| 12. $\sqrt{225}$ | ✔ | ✔ | ✔ | ✔ | | ✔ |
| 13. $.634$ | | | | ✔ | | ✔ |
| 14. $\sqrt{\frac{4}{49}}$ | | | | ✔ | | ✔ |
| 15. $-\sqrt{64}$ | | | ✔ | ✔ | | ✔ |

> Note: For #9, since division by zero is undefined, it belongs to none of the sets.

Let me know if you'd like this formatted as a table!