Real Number System worksheet with definitions and examples for students to fill in.

Let’s go step by step to fill in the blanks and label each number correctly.

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Step 1: Fill in the definitions

#### REAL NUMBERS
Definition:
“A number that can represent the position along a number line. (It is not imaginary.)”

> Why? Real numbers are all the numbers you can plot on a number line — positives, negatives, zero, fractions, decimals, etc. They’re “real” as opposed to imaginary numbers like √-1.

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#### IRRATIONAL NUMBERS
Definition:
“A real number that cannot be made by dividing two integers. It is a non-ending, non-repeating decimal.”

> Why? Irrational numbers can’t be written as a fraction of two whole numbers. Their decimals go on forever without repeating — like π or √2.

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#### RATIONAL NUMBERS
Definition:
“A real number that can be made by dividing two integers.”

> Why? Rational = ratio of integers. Like 3/4, -5/2, even 7 (which is 7/1).

Special Types under Rational Numbers:
- Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Whole Numbers: 0, 1, 2, 3, ... (non-negative integers)
- Counting Numbers: 1, 2, 3, 4, ... (positive integers, also called natural numbers)

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Step 2: Label each number with all possible categories

We use these abbreviations:
- R = Real
- I = Irrational
- RA = Rational
- IN = Integer
- W = Whole
- C = Counting

Remember: All numbers here are real unless they’re imaginary (none are). So every answer will include “R”.

Also: If a number is irrational, it CANNOT be rational, integer, whole, or counting.

If a number is rational, check if it’s also an integer, then whole, then counting.

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#### 1. -12
- Real? Yes → R
- Irrational? No → not I
- Rational? Yes → RA (because -12 = -12/1)
- Integer? Yes → IN
- Whole? No → whole numbers start at 0
- Counting? No → counting starts at 1

✔ Labels: R, RA, IN

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#### 2. √7
√7 ≈ 2.64575... — doesn’t terminate or repeat → irrational

- Real? Yes → R
- Irrational? Yes → I
- Rational? No → not RA
- Integer? No
- Whole? No
- Counting? No

✔ Labels: R, I

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#### 3. 3/5
This is a fraction → rational

- Real? Yes → R
- Irrational? No → not I
- Rational? Yes → RA
- Integer? No → not a whole number
- Whole? No
- Counting? No

✔ Labels: R, RA

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#### 4. 0.42681392...
The dots mean it goes on forever. But does it repeat? The problem doesn’t say it repeats — and since no pattern is shown, we assume it’s non-repeating → irrational

Wait — actually, let’s think carefully.

In math problems, if a decimal is written with “...” and no indication of repetition (like a bar over digits), and it looks random, it’s usually meant to be irrational.

But sometimes, if it’s just long but terminating, it would stop. Here it says “...” so it continues.

Since no repeating pattern is indicated, and it’s not obviously terminating, we treat it as irrational.

BUT — wait! Actually, in many school contexts, if they write a long decimal with “...” and don’t specify it’s repeating, they might still expect you to recognize it could be rational if it eventually repeats — but here there’s no clue.

Actually, looking again: 0.42681392... — this looks like it’s meant to be non-repeating. In fact, this exact decimal appears in some textbooks as an example of an irrational number.

So:

- Real? Yes → R
- Irrational? Yes → I
- Rational? No
- Others? No

✔ Labels: R, I

*(Note: If this were a repeating decimal, it would be rational — but since no repetition is marked, we go with irrational.)*

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#### 5. √16
√16 = 4 → which is a whole number

- Real? Yes → R
- Irrational? No → not I
- Rational? Yes → RA (4 = 4/1)
- Integer? Yes → IN
- Whole? Yes → W
- Counting? Yes → C (since 4 > 0)

✔ Labels: R, RA, IN, W, C

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#### 6. 0
Zero is special.

- Real? Yes → R
- Irrational? No → not I
- Rational? Yes → RA (0 = 0/1)
- Integer? Yes → IN
- Whole? Yes → W (whole numbers include 0)
- Counting? No → counting starts at 1

✔ Labels: R, RA, IN, W

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## Final Answer:

Here are the completed definitions and labels:

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REAL NUMBERS:
A number that can represent the position along a number line. (It is not imaginary.)

IRRATIONAL NUMBERS:
A real number that cannot be made by dividing two integers. It is a non-ending, non-repeating decimal.

RATIONALE NUMBERS:
A real number that can be made by dividing two integers.

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Labels for each number:

1. -12 → R, RA, IN
2. √7 → R, I
3. 3/5 → R, RA
4. 0.42681392... → R, I
5. √16 → R, RA, IN, W, C
6. 0 → R, RA, IN, W

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✔ All answers checked and verified.