Irrational numbers worksheet - Free Printable

Here is the complete solution to classify each number as rational or irrational, with clear reasoning for each.

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🔢 1. 0.21

✔ Rational

Reasoning:
0.21 is a terminating decimal. Any terminating decimal can be written as a fraction of two integers.
→ 0.21 = 21/100, which is a ratio of integers → rational.

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🔢 2. 3/12

✔ Rational

Reasoning:
This is already written as a fraction of two integers (3 and 12). Even though it simplifies to 1/4, it’s still a ratio of integers → rational.

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🔢 3. √12 – 2

✘ Irrational

Reasoning:
√12 = √(4×3) = 2√3. So, √12 – 2 = 2√3 – 2.
√3 is irrational (it cannot be expressed as a fraction of integers), and multiplying or subtracting a rational number (like 2) from an irrational number still results in an irrational number.
→ irrational.

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🔢 4. √12 / 4

✘ Irrational

Reasoning:
√12 = 2√3, so √12 / 4 = (2√3)/4 = √3 / 2.
√3 is irrational, and dividing an irrational number by a rational number (2) still gives an irrational number.
→ irrational.

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🔢 5. 4.125...

⚠️ Needs clarification — but likely IRRATIONAL

Reasoning:
The ellipsis (...) suggests the decimal continues *indefinitely*. If it’s non-repeating and non-terminating, then it’s irrational.
But if the “...” means it repeats (e.g., 4.125125125...), then it would be rational. However, since no repeating pattern is indicated (like a bar over digits), we assume it’s non-repeating → irrational.

> 💡 *Note: In most math contexts, unless specified otherwise, “...” after a decimal without a pattern implies non-repeating → irrational.*

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🔢 6. (√12 – 4)(4 + √12)

✔ Rational

Reasoning:
This is a difference of squares:
(a – b)(a + b) = a² – b², where a = √12, b = 4.
So:
= (√12)² – 4²
= 12 – 16
= –4

–4 is an integer → rational.

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🔢 7. 12.52 (rounded to 2 d.p.)

✔ Rational

Reasoning:
Even though it says “rounded to 2 d.p.”, the number given is 12.52, which is a terminating decimal.
→ 12.52 = 1252/100 = 313/25 → ratio of integers → rational.

> 📝 *Important note:* The phrase “rounded to 2 d.p.” might imply that the original number was irrational (like π ≈ 3.14), but here the number *as presented* is 12.52 — a finite decimal — so we classify based on what’s written. If the question meant “this is a rounded version of an irrational number,” it should specify that. As written, 12.52 is rational.

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## ✔ Final Answers Table:

| Number | Classification | Reasoning |
|---------------------|----------------|-----------|
| 0.21 | Rational | Terminating decimal → can be written as 21/100 |
| 3/12 | Rational | Fraction of integers → simplifies to 1/4 |
| √12 – 2 | Irrational | Equals 2√3 – 2; √3 irrational → result irrational |
| √12 / 4 | Irrational | Equals √3 / 2; √3 irrational → result irrational |
| 4.125... | Irrational | Non-repeating, non-terminating decimal (assumed) |
| (√12 – 4)(4 + √12) | Rational | Simplifies to –4 (difference of squares) |
| 12.52 (rounded) | Rational | Terminating decimal → 1252/100 |

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