Math worksheet challenging students to categorize various numbers into the correct subsets of the real number system.

Absolutely! Let’s solve this step by step.

This worksheet asks you to classify each number into the sets it belongs to from the following:

- Real — all numbers on the number line (includes rational and irrational)
- Rational — can be written as a fraction of two integers (including terminating or repeating decimals)
- Irrational — cannot be written as a fraction; non-repeating, non-terminating decimals
- Integer — whole numbers and their negatives (… -3, -2, -1, 0, 1, 2, 3…)
- Whole — non-negative integers (0, 1, 2, 3…)
- Natural — positive integers (1, 2, 3…)

> Note: Every natural number is also whole, integer, rational, and real.
> Every whole number is also integer, rational, and real.
> Every integer is also rational and real.
> Every rational is real.
> Irrational numbers are only real — they are NOT rational, integer, whole, or natural.

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Let’s go one by one:

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1. –√3 / 2

- √3 is irrational → dividing by 2 doesn’t make it rational.
- So this is irrational → therefore also real
- Not rational, not integer, not whole, not natural.

✔ Real, Irrational

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2. 55.88

- This is a terminating decimal → can be written as 5588/100 → so it’s rational
- Rational → also real
- Not an integer (has decimal part), not whole, not natural.

✔ Real, Rational

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3. 0

- Zero is an integer, whole, and rational (0 = 0/1)
- Also real
- Not natural (natural numbers start at 1)

✔ Real, Rational, Integer, Whole

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4. 4 1/8

- Mixed number → equals 33/8 → rational
- Also real
- Not integer (fractional part), not whole, not natural

✔ Real, Rational

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5. √5

- Square root of a non-perfect square → irrational
- Therefore also real

✔ Real, Irrational

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6. –83

- Negative whole number → integer
- Also rational (–83/1), real
- Not whole or natural (negative)

✔ Real, Rational, Integer

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7. 12/3

- Simplifies to 4 → which is a natural, whole, integer, rational, real

✔ Real, Rational, Integer, Whole, Natural

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8. –√100

- √100 = 10 → so this is –10 → integer
- Also rational, real
- Not whole or natural (negative)

✔ Real, Rational, Integer

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9. –√4

- √4 = 2 → so this is –2 → integer
- Also rational, real
- Not whole or natural

✔ Real, Rational, Integer

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10. 3.24

- Terminating decimal → 324/100 → rational
- Also real
- Not integer, whole, or natural

✔ Real, Rational

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11. 3π

- π is irrational → multiplying by 3 doesn’t make it rational → irrational
- Therefore also real

✔ Real, Irrational

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✔ Final Answers:

1. Real, Irrational
2. Real, Rational
3. Real, Rational, Integer, Whole
4. Real, Rational
5. Real, Irrational
6. Real, Rational, Integer
7. Real, Rational, Integer, Whole, Natural
8. Real, Rational, Integer
9. Real, Rational, Integer
10. Real, Rational
11. Real, Irrational

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💡 Tip: When classifying, always start from the most specific set (like natural) and work outward. If it’s natural → it’s automatically whole, integer, rational, real. If it’s irrational → only real. If it’s a fraction or decimal that terminates/repeats → rational.

Let me know if you’d like this in a printable format or with explanations for each number!