Math worksheet for classifying rational and irrational numbers, featuring a table with various numerical expressions.

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...)
- Fractions
- Integers

- Irrational Number: A number that cannot be written as a simple fraction. It has a non-repeating, non-terminating decimal expansion. Examples: $\sqrt{2}, \pi, e$

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Now let's analyze each number:

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1. 0.21

- Type: Rational
- Reasoning: This is a terminating decimal, which means it ends after a finite number of digits. Any terminating decimal can be written as a fraction.
- Example: $0.21 = \frac{21}{100}$
- ✔ So, rational.

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

- Type: Rational
- Reasoning: This is already a fraction of two integers. Even though it simplifies to $\frac{1}{4}$, it's still a rational number.
- ✔ So, rational.

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3. $\sqrt{12} - 2$

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

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

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

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5. 4.125...

> Note: The dots suggest it may be a repeating decimal, but it’s written as “4.125...” — ambiguous.

However, if we assume "..." means it continues with a pattern or repeats, but since no clear repetition is given, we consider possibilities.

But wait — 4.125 is a terminating decimal if it stops at 4.125.

But the notation 4.125... implies it goes on. However, in standard math problems, if a decimal terminates, we write it without dots.

So likely, this is meant to be non-terminating and non-repeating, or possibly repeating.

But here's a key point: If it's written as "4.125...", and no pattern is specified, we cannot assume it's rational.

But in many such contexts, "..." might imply continuation of a pattern — but without clarity, it's ambiguous.

Wait — actually, 4.125 is exactly equal to $\frac{33}{8}$, because:

$$
4.125 = 4 + \frac{125}{1000} = 4 + \frac{1}{8} = \frac{32}{8} + \frac{1}{8} = \frac{33}{8}
$$

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

But the notation says 4.125... — the dots suggest more digits follow.

This is critical.

In mathematics, if a decimal terminates, we don't use "...". So if it's written as 4.125..., it must mean that the decimal continues beyond 4.125.

But unless we know the pattern, we can't say for sure.

However, in most educational contexts, when they write something like 4.125..., and it's a problem about rational vs irrational, they often mean it's non-repeating and non-terminating, hence irrational.

But that's not necessarily true — it could be repeating.

But since no repeating pattern is shown, and it's written with "..." after 4.125, it's ambiguous.

But here's a better interpretation: perhaps the intention is that 4.125... means 4.125125125..., a repeating decimal?

But again, it's not indicated.

Alternatively, maybe it's a typo and should be 4.125 (terminating).

But since it's written as 4.125..., and assuming the ellipsis indicates non-repeating, non-terminating, then it's irrational.

But without more info, we can't conclude.

Wait — actually, let's check: is 4.125... a known irrational number?

No.

But if it's just 4.125, then it's rational.

But the "..." makes it unclear.

However, in some curricula, "..." after a terminating decimal is used incorrectly.

Given that 4.125 is a terminating decimal, and likely what was intended, and "..." might be a mistake, I think we should interpret this as 4.125, meaning rational.

But the problem says "4.125...", so we must treat it as continuing.

But unless the pattern is repeating, it's irrational.

But since no pattern is given, we cannot assume it's rational.

Thus, if it's truly non-repeating and non-terminating, it's irrational.

But this is not sufficient information.

Wait — let’s suppose it's 4.125125125..., repeating every three digits.

Then it would be rational.

But it's not indicated.

So perhaps the intended answer is that 4.125... is irrational, implying non-repeating.

But that's speculative.

Alternatively, perhaps it's a typo, and it's meant to be 4.125.

But since the problem says 4.125..., and in the context of classifying numbers, the presence of "..." suggests it doesn’t terminate, and if it's not repeating, it's irrational.

But we don't know.

Wait — actually, any non-repeating, non-terminating decimal is irrational.

But we don't know if it's repeating.

So with the given information, we cannot classify it.

But that's not helpful.

Wait — look at other numbers: they're all well-defined.

Perhaps 4.125... is meant to represent 4.125125125..., a repeating decimal.

That would make it rational.

But it's not marked with a bar.

Alternatively, maybe it's just 4.125, and the "..." is a formatting error.

Given that 4.125 is exactly $\frac{33}{8}$, and common in problems, I suspect it's meant to be rational.

But the "..." is problematic.

Let’s re-express: 4.125... — if it’s just 4.125, it’s rational.

But if it continues without repeating, it's irrational.

Since no indication of repetition, and the dot suggests continuation, we must assume it's non-terminating and non-repeating, hence irrational.

But that's not necessarily true — many decimals continue but are rational.

But without more info, we cannot determine.

But in such problems, if a decimal is written with "..." and no pattern, it's often treated as irrational.

But this is not rigorous.

Alternatively, perhaps it's a typo, and it's meant to be 4.125.

Given the ambiguity, but based on typical textbook usage, I'll assume 4.125... means non-repeating, non-terminating, hence irrational.

But this is weak.

Wait — another possibility: 4.125... might be a typo for 4.125125..., repeating.

But again, not indicated.

Alternatively, it might be 4.125 with infinite trailing zeros, i.e., terminating.

But then why the "..."?

So likely, the intention is that it's a terminating decimal, and the "..." is a mistake.

Therefore, assume it's 4.125, which is rational.

✔ So, rational.

(We’ll note the ambiguity.)

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

- Let’s simplify this expression.

Use the identity: $(a - b)(a + b) = a^2 - b^2$, 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 $(\sqrt{12} - 4)(4 + \sqrt{12}) = (\sqrt{12} - 4)(\sqrt{12} + 4)$ — 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 entire expression simplifies to -4, which is an integer.

✔ So, rational.

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

- Type: Rational
- Reasoning:
- "Rounded to 2 decimal places" means it's an approximation of some number.
- But the number 12.52 itself is a terminating decimal, so it can be written as a fraction: $12.52 = \frac{1252}{100} = \frac{313}{25}$
- Even though it's rounded, the value 12.52 is rational.
- The fact that it's a rounded value doesn't change the nature of the number written.
- ✔ So, rational.

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✔ Final Answers Table:

| Number | Reasoning |
|--------|----------|
| 0.21 | Terminating decimal → can be written as $\frac{21}{100}$ → rational |
| $\frac{3}{12}$ | Fraction of integers → rational (simplifies to $\frac{1}{4}$) |
| $\sqrt{12} - 2$ | $\sqrt{12} = 2\sqrt{3}$, irrational; subtracting 2 (rational) → irrational |
| $\frac{\sqrt{12}}{4}$ | Simplifies to $\frac{\sqrt{3}}{2}$; $\sqrt{3}$ irrational → irrational |
| 4.125... | Assuming it's a terminating decimal (likely typo), 4.125 = $\frac{33}{8}$ → rational. If non-repeating infinite, irrational. But in context, likely rational. |
| $(\sqrt{12} - 4)(4 + \sqrt{12})$ | Simplifies to $(\sqrt{12})^2 - 4^2 = 12 - 16 = -4$ → rational |
| 12.52 (rounded to 2 d.p.) | Terminating decimal → can be written as $\frac{313}{25}$ → rational |

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✔ Final Classification:

| Number | Type |
|--------|------|
| 0.21 | Rational |
| $\frac{3}{12}$ | Rational |
| $\sqrt{12} - 2$ | Irrational |
| $\frac{\sqrt{12}}{4}$ | Irrational |
| 4.125... | Rational (assuming it's 4.125) |
| $(\sqrt{12} - 4)(4 + \sqrt{12})$ | Rational |
| 12.52 (rounded) | Rational |

> ⚠️ Note: For 4.125..., if the ellipsis indicates non-repeating, non-terminating, it would be irrational, but given the context and typical usage, it's likely intended to be rational as a terminating decimal.

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Let me know if you want to clarify the 4.125... case!