Classification of Numbers worksheet - Free Printable

Let's solve the worksheet step by step and explain each answer.

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Section A: Classify the Numbers

We are to classify each group of numbers as Natural, Whole, Rational, or Irrational.

#### 1) 8, 0, 2, 5...

- These are counting numbers (positive integers starting from 1), but include 0.
- Natural numbers: positive integers starting from 1 → {1, 2, 3, ...}
- Whole numbers: Natural numbers + 0 → {0, 1, 2, 3, ...}
- Since 0 is included, this set is not natural, but it is whole.
- All these numbers are also rational (they can be written as fractions).
- But the question asks for the best classification based on the list.

✔ Answer: Whole numbers

> ✔ *Explanation:* The sequence includes 0, so it’s not natural. It matches the definition of whole numbers.

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#### 2) 0.355353, √2, √5

- 0.355353: This is a terminating decimal → can be written as a fraction → rational
- √2: Cannot be expressed as a fraction; non-repeating, non-terminating → irrational
- √5: Same as above → irrational

So, we have both rational and irrational numbers.

But the question asks to classify the group — so we need to see what category they all belong to.

- Not all are rational (because √2 and √5 are irrational)
- Not all are irrational (because 0.355353 is rational)
- They are not natural, whole, or integers

But all real numbers are either rational or irrational.

So, the best classification here is that the group contains both rational and irrational numbers, but since the question likely wants a single label, we say:

✔ Answer: Rational and Irrational numbers

> ✔ *Explanation:* The group includes both types, so the correct classification is that it contains rational and irrational numbers.

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#### 3) 2/3, 1/2, 1/8

- All are fractions (ratios of integers)
- Can be written as decimals: 0.666..., 0.5, 0.125
- Terminating or repeating decimals → rational

They are not natural, not whole, not integers

✔ Answer: Rational numbers

> ✔ *Explanation:* All three are fractions of integers, so they are rational.

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#### 4) 7, 4, 1, 9

- All are positive integers
- Start from 1 → these are natural numbers
- Also whole numbers (since whole numbers include natural numbers and 0)
- But since 0 is not in the list, they are natural, which is a subset of whole

✔ Answer: Natural numbers

> ✔ *Explanation:* These are positive integers starting from 1 → natural numbers.

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Section B: True/False

#### 1) Integers are both Natural and Whole numbers.

- False
- Integers include negative numbers, zero, and positive numbers: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Natural numbers: 1, 2, 3, ...
- Whole numbers: 0, 1, 2, 3, ...
- So integers are not all natural or whole (e.g., -1 is an integer but not natural or whole)

✔ Answer: F

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#### 2) Rational & Irrational numbers are both Real Numbers.

- True
- Real numbers include all rational and irrational numbers
- There are no numbers that are rational or irrational that aren't real

✔ Answer: T

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#### 3) 1 is a Whole number.

- Whole numbers: 0, 1, 2, 3, ...
- Yes, 1 is included

✔ Answer: T

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#### 4) All Whole numbers are Natural Numbers except 0.

- Whole numbers: 0, 1, 2, 3, ...
- Natural numbers: 1, 2, 3, ...
- So yes, all whole numbers except 0 are natural numbers

✔ Answer: T

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#### 5) −1, −6, −3, −7 are examples of Natural numbers.

- Natural numbers are positive integers only (1, 2, 3, ...)
- Negative numbers are not natural

✔ Answer: F

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Bonus: Symbols for Number Classification

These are standard mathematical symbols:

| Type | Symbol |
|------|--------|
| Whole Numbers | W |
| Integers | ℤ (Z) |
| Counting Numbers (same as Natural) | ℕ (N) |
| Real Numbers | ℝ (R) |
| Rational Numbers | ℚ (Q) |
| Irrational Numbers | No standard symbol, but often denoted as ℝ \ ℚ (real numbers minus rational numbers) |

However, commonly accepted:

- Whole Numbers: W
- Integers: ℤ
- Counting Numbers: ℕ
- Real Numbers: ℝ
- Rational Numbers: ℚ
- Irrational Numbers: No universal symbol, but sometimes I or ℝ\ℚ

But in many textbooks:

- Irrational numbers: no specific symbol, but you may write Irrational or use ℝ \ ℚ

✔ Answers:

- Whole Numbers: W
- Integers: ℤ
- Counting Numbers: ℕ
- Real Numbers: ℝ
- Rational Numbers: ℚ
- Irrational Numbers: (no standard symbol) → typically left blank or written as "Irrational"
But if required: ℝ \ ℚ or I

> However, in most school contexts, the expected answers are:
>
> - Whole Numbers: W
> - Integers: ℤ
> - Counting Numbers: ℕ
> - Real Numbers: ℝ
> - Rational Numbers: ℚ
> - Irrational Numbers: I (sometimes used informally)

But standard is:

✔ Final Bonus Answers:

- Whole Numbers: W
- Integers: ℤ
- Counting Numbers: ℕ
- Real Numbers: ℝ
- Rational Numbers: ℚ
- Irrational Numbers: (no standard symbol) → Not applicable or I

But if the teacher expects a symbol, use I or leave blank.

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✔ Final Answer Summary:

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Section A:

1) Whole numbers
2) Rational and Irrational numbers
3) Rational numbers
4) Natural numbers

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Section B: True/False

1) F
2) T
3) T
4) T
5) F

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Bonus: Symbols

- Whole Numbers: W
- Integers: ℤ
- Counting Numbers: ℕ
- Real Numbers: ℝ
- Rational Numbers: ℚ
- Irrational Numbers: (no standard symbol) → I or ℝ \ ℚ

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