Classifying Rational and Irrational Numbers worksheet with a table for students to determine if numbers are rational or irrational and explain their reasoning.

Let's go through each number and determine whether it is rational or irrational, with clear reasoning.

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🔹 Definitions:

- Rational Number: A number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$. Rational numbers include:
- Terminating decimals (e.g., 0.5)
- Repeating decimals (e.g., 0.333...)
- Integers and fractions

- Irrational Number: A number that cannot be written as a fraction of two integers. These have non-repeating, non-terminating decimal expansions (e.g., $\pi$, $\sqrt{2}$).

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Now, let’s analyze each number:

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1. 0.21

- Classification: Rational
- Reasoning: This is a terminating decimal. Any terminating decimal can be written as a fraction.
- $0.21 = \frac{21}{100}$
- Since it's a ratio of two integers, it's rational.

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2. $\frac{3}{12}$

- Classification: Rational
- Reasoning: This is already a fraction of two integers. Even though it simplifies to $\frac{1}{4}$, it's still a rational number because it can be expressed as a ratio of integers.

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3. $\sqrt{12} - 2$

- Classification: Irrational
- Reasoning:
- First, simplify $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$
- So $\sqrt{12} - 2 = 2\sqrt{3} - 2$
- $\sqrt{3}$ is irrational, and multiplying by 2 keeps it irrational.
- Subtracting 2 (a rational number) from an irrational number results in an irrational number.
- Therefore, $2\sqrt{3} - 2$ is irrational.

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4. $\frac{\sqrt{12}}{4}$

- Classification: Irrational
- Reasoning:
- $\sqrt{12} = 2\sqrt{3}$, so:
$$
\frac{\sqrt{12}}{4} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}
$$
- $\sqrt{3}$ is irrational, and dividing it by 2 (a rational number) does not make it rational.
- So $\frac{\sqrt{3}}{2}$ is irrational.

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5. 4.125...

> Note: The "..." suggests the decimal continues, but we need to interpret what this means.

- Classification: Possibly Irrational — but depends on context.
- But wait: If the number is written as 4.125..., and no pattern is given, we must assume it's non-repeating and non-terminating unless otherwise specified.

However, 4.125 is actually a terminating decimal, and if it ends there, it's rational.

But since it says "4.125...", the ellipsis implies continuation.

If it's meant to be 4.125125125... (repeating), then it would be rational.

But without a bar or indication of repetition, "..." usually means non-repeating.

But here’s the key: In standard math notation, if it's written as 4.125..., it's ambiguous.

However, if it's just 4.125, it terminates → rational.

But since it has "...", it likely implies non-terminating, non-repeating → irrational.

Wait — let’s check: Is 4.125 exactly equal to a fraction?

Yes:
$$
4.125 = \frac{4125}{1000} = \frac{33}{8}
$$

So if it's just 4.125, it's rational.

But the notation "4.125..." suggests more digits follow.

Assuming the "..." means it continues indefinitely without repeating, then it's irrational.

But this is ambiguous.

Let’s assume the intention is that it's a non-repeating, non-terminating decimal, so:

- Classification: Irrational
- Reasoning: The decimal expansion is non-terminating and non-repeating (implied by "..."), so it cannot be expressed as a fraction of integers → irrational.

> ⚠️ However, if the intention was just 4.125 (terminating), then it would be rational.

But since it's written as 4.125..., and no repeating pattern is indicated, we treat it as non-repeating and non-terminating → irrational.

But this is a common ambiguity.

Let’s suppose instead that it's 4.125125125... — repeating every three digits. Then it's rational.

But again, no bar or indication.

So best interpretation: "4.125..." with no pattern = possibly irrational.

But in most educational contexts, when such notation appears, it may be a typo or meant to be 4.125 (terminating).

Let’s re-evaluate: Is 4.125... a known number?

No. Without a defined pattern, we cannot assume it's rational.

Therefore, to be safe:

> ✔ Classification: Irrational
> ✔ Reasoning: The decimal is non-terminating and non-repeating (as implied by "..."), so it cannot be expressed as a fraction of two integers → irrational.

But note: This is only valid if the decimal doesn't repeat.

Alternatively, if it's meant to be 4.125 (i.e., terminating), then it's rational.

Given the notation, we’ll assume it's non-repeating → irrational.

But let’s double-check: In many textbooks, "4.125..." might be used to mean 4.125 only.

To avoid confusion, perhaps the intended number is 4.125, which is rational.

But since it says "4.125...", I'll go with irrational, assuming non-repeating.

But let's consider another possibility: Maybe it's 4.125125125..., repeating.

Then it would be rational.

But again, no bar.

So safest answer: Unclear, but likely irrational due to the ellipsis.

But for clarity, let's assume it's not repeating.

✔ Final decision: Irrational, because the decimal continues infinitely without repeating.

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6. $(\sqrt{12} - 4)(4 + \sqrt{12})$

Let’s simplify this expression.

This is of the form $(a - b)(a + b) = a^2 - b^2$, but here it's $(\sqrt{12} - 4)(4 + \sqrt{12})$

Note: $(\sqrt{12} - 4)(\sqrt{12} + 4)$ would be difference of squares.

But here it's $(\sqrt{12} - 4)(4 + \sqrt{12})$ — same thing!

Because addition is commutative: $4 + \sqrt{12} = \sqrt{12} + 4$

So:
$$
(\sqrt{12} - 4)(\sqrt{12} + 4) = (\sqrt{12})^2 - (4)^2 = 12 - 16 = -4
$$

So the result is -4, which is an integer.

- Classification: Rational
- Reasoning: The expression simplifies to $-4$, which is an integer and can be written as $\frac{-4}{1}$, so it's rational.

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7. 12.52 (rounded to 2 d.p.)

- Classification: Rational
- Reasoning: Rounding to 2 decimal places gives us 12.52, a terminating decimal.
- $12.52 = \frac{1252}{100} = \frac{313}{25}$
- So it's a ratio of integers → rational.

Even though it's a rounded value (so it may represent an approximation of an irrational number), the number 12.52 itself is rational because it's a terminating decimal.

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✔ Final Table:

| Number | Reasoning |
|--------|----------|
| 0.21 | Terminating decimal; can be written as $\frac{21}{100}$ → rational |
| $\frac{3}{12}$ | Fraction of integers; simplifies to $\frac{1}{4}$ → rational |
| $\sqrt{12} - 2$ | $\sqrt{12} = 2\sqrt{3}$, irrational; subtracting 2 (rational) from irrational → irrational |
| $\frac{\sqrt{12}}{4}$ | Simplifies to $\frac{\sqrt{3}}{2}$; $\sqrt{3}$ is irrational → irrational |
| 4.125... | Non-terminating, non-repeating decimal (implied by "...") → irrational |
| $(\sqrt{12} - 4)(4 + \sqrt{12})$ | Simplifies to $-4$, an integer → rational |
| 12.52 (rounded to 2 d.p.) | Terminating decimal; equals $\frac{1252}{100}$ → rational |

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🟩 Summary:

| Number | Type |
|-------|------|
| 0.21 | Rational |
| 3/12 | Rational |
| √12 − 2 | Irrational |
| √12 / 4 | Irrational |
| 4.125... | Irrational |
| (√12 − 4)(4 + √12) | Rational |
| 12.52 (rounded) | Rational |

✔ All explained!