Free Printable Rational and Irrational Numbers Worksheets - Free Printable

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

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🔹 Definitions:

- A rational number can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$. It includes terminating decimals and repeating decimals.
- An irrational number cannot be expressed as a simple fraction. Its decimal form is non-repeating and non-terminating.

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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. Any terminating decimal can be written as a fraction.
For example:
$$
0.21 = \frac{21}{100}
$$
Since it’s a ratio of two integers, it is rational.

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

- Type: Rational
- Reasoning: This is already a fraction of two integers. Simplify:
$$
\frac{3}{12} = \frac{1}{4}
$$
So it is clearly rational.

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#### 3. $\sqrt{12} - 2$

- Type: Irrational
- Reasoning: First, simplify $\sqrt{12}$:
$$
\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}
$$
So:
$$
\sqrt{12} - 2 = 2\sqrt{3} - 2
$$
$\sqrt{3}$ is irrational, and multiplying by 2 doesn't make it rational. Subtracting 2 (a rational number) from an irrational number results in an irrational number.

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

- Type: Irrational
- Reasoning: Again, $\sqrt{12} = 2\sqrt{3}$, so:
$$
\frac{\sqrt{12}}{4} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}
$$
$\sqrt{3}$ is irrational, and dividing by 2 (a rational number) keeps it irrational. So the result is irrational.

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#### 5. 4.125...

- Type: Irrational
- Reasoning: The notation “...” suggests a non-repeating, non-terminating decimal. However, we must be careful: if it were repeating, like 4.125125..., it would be rational. But here, the dots indicate no clear pattern, and since it's not specified to repeat, we assume it's non-repeating and non-terminating — which means it's irrational.

> ⚠️ Note: If the number was meant to be $4.\overline{125}$ (repeating), it would be rational. But since it's written as "4.125..." without a bar, we interpret it as non-repeating, hence irrational.

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

- Type: Rational
- Reasoning: This looks like a difference of squares:
$$
(\sqrt{12} - 4)(\sqrt{12} + 4) = (\sqrt{12})^2 - (4)^2 = 12 - 16 = -4
$$
The result is $-4$, which is an integer → rational.

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

- Type: Rational
- Reasoning: This is a terminating decimal (specifically, rounded to 2 decimal places). Even though it's a rounded value, the number 12.52 itself is:
$$
12.52 = \frac{1252}{100} = \frac{313}{25}
$$
So it's a ratio of integers → rational.

> Note: Rounding does not make a number irrational; it just approximates it. The resulting value (12.52) is still rational.

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✔ Final Table:

| Number | Reasoning |
|--------|---------|
| 0.21 | Terminating decimal → can be written as $\frac{21}{100}$ → rational |
| $\frac{3}{12}$ | Simplifies to $\frac{1}{4}$ → rational |
| $\sqrt{12} - 2$ | Equals $2\sqrt{3} - 2$; $\sqrt{3}$ is irrational → irrational |
| $\frac{\sqrt{12}}{4}$ | Equals $\frac{\sqrt{3}}{2}$; $\sqrt{3}$ is irrational → irrational |
| 4.125... | Non-repeating, non-terminating decimal → irrational |
| $(\sqrt{12} - 4)(4 + \sqrt{12})$ | Difference of squares: $12 - 16 = -4$ → rational |
| 12.52 (rounded to 2 d.p.) | Terminating decimal → can be written as fraction → rational |

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