Real Number

Define real number with operation on real numbers.

A real number is a value that represents a quantity along a continuous line, often called the number line. Real numbers include both rational numbers (like integers, fractions) and irrational numbers (numbers that cannot be expressed as a simple fraction, like √2 or π). The set of real numbers is typically denoted by the symbol ℝ.

Examples :- 

1. Real numbers:

• 5
• −3
• 0
• ½
• 2.75
• √2
• π

1. Not a real number:

• √−1

Why? Because there is no real number whose square is −1.

1. Categories of Real Numbers :-

Real numbers are divided into two main groups:

Real Numbers → Rational Numbers + Irrational Numbers 

And rational numbers can be further divided into integers, whole numbers, and natural numbers.

Category	Meaning	Examples
Natural Numbers (N)	Counting numbers	1, 2, 3, 4, 5…
Whole Numbers (W)	Natural numbers + 0	0, 1, 2, 3, 4…
Integers (Z)	Positive and negative whole numbers, including 0	…−3, −2, −1, 0, 1, 2, 3…
Rational Numbers (Q)	Numbers that can be written as p/q, where q ≠ 0	½, −3, 0.75, 5
Irrational Numbers	Cannot be written as p/q	√2, √3, π
Real Numbers (R)	Rational + Irrational numbers	2, −5, ½, √2, π

2. Natural Numbers :-

Natural numbers are the numbers we use for counting.

Examples:

1, 2, 3, 4, 5, 6, 7, …

Usually, natural numbers start from 1.

Example

If there are 5 apples:

🍎 🍎 🍎 🍎 🍎

We count them as:

1, 2, 3, 4, 5

So, 1, 2, 3, 4, 5 are natural numbers.

Symbol: N

N = {1, 2, 3, 4, 5, …}

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3. Whole Numbers:-

Whole numbers include 0 and all natural numbers.

Examples:

0, 1, 2, 3, 4, 5, …

The important difference is:

Natural numbers start from 1, but whole numbers include 0.

Number	Natural?	Whole?
0	❌	✅
1	✅	✅
5	✅	✅
10	✅	✅
−2	❌	❌

Symbol: W

W = {0, 1, 2, 3, 4, …}

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4. Integers:-

Integers include:

• Positive whole numbers
• Negative whole numbers
• Zero

Examples:

…, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …

Example

Temperature can be:

• +5°C
• 0°C
• −5°C

All three are integers.

Symbol: Z

Z = {…, −3, −2, −1, 0, 1, 2, 3, …}

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5. Rational Numbers:-

A rational number is a number that can be written in the form:

p/q

where:

• p = an integer
• q = a non-zero integer

Examples

½, ¾, −5/2, 7, 0.25, 1.5

Even whole numbers and integers are rational numbers because they can be written as fractions.

For example:

5 = 5/1

−3 = −3/1

0 = 0/1

So, 5, −3 and 0 are all rational numbers.

Decimal Examples

Rational numbers have decimals that are either:

1. Terminating
2. Repeating

Examples:

0.5 = ½

0.25 = ¼

0.3333… = ⅓

0.6666… = ⅔

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6. Irrational Numbers:-

Irrational numbers cannot be written as p/q, where p and q are integers and q ≠ 0.

Their decimal expansion:

Never ends and never repeats in a fixed pattern.

Examples

√2 = 1.41421356…

√3 = 1.73205080…

π = 3.14159265…

These decimals continue forever without repeating in a fixed pattern.

Important Example

Is √4 irrational?

No.

Because:

√4 = 2

And 2 is a rational number.

But:

√2

cannot be simplified into a rational number, so it is irrational.

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7. Rational vs Irrational Numbers:-

Rational Numbers	Irrational Numbers
Can be written as p/q	Cannot be written as p/q
Decimal may terminate	Decimal never terminates
Decimal may repeat	Decimal does not repeat in a fixed pattern
Example: ½	Example: √2
Example: 0.75	Example: π
Example: ⅓	Example: √3

Easy Trick

Ask yourself:

"Can I write this number as a fraction of two integers?"

• Yes → Rational
• No → Irrational

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8. Relationship Between Number Categories:-

Think of the number system like boxes inside boxes:

Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real

This means:

• Every natural number is a whole number.
• Every whole number is an integer.
• Every integer is a rational number.
• Every rational number is a real number.

But the reverse is not always true.

For example:

½ is rational but not an integer.

−3 is an integer but not a whole number.

0 is a whole number but usually not a natural number.

Examples of Classification

Number	Category
7	Natural, Whole, Integer, Rational, Real
0	Whole, Integer, Rational, Real
−4	Integer, Rational, Real
½	Rational, Real
2.5	Rational, Real
0.333…	Rational, Real
√2	Irrational, Real
√5	Irrational, Real
π	Irrational, Real

Example 1

Classify −8.

−8 → Integer → Rational → Real

Therefore:

−8 is an integer, rational number and real number.

Example 2

Classify ¾.

¾ → Rational → Real

Therefore:

¾ is a rational and real number.

Example 3

Classify √7.

√7 cannot be simplified into a rational number.

Therefore:

√7 → Irrational → Real

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Number Line:-

Every real number has a position on the number line.

A simple number line looks like:

← negative numbers | 0 | positive numbers →

For example:

−3 ← −2 ← −1 ← 0 → 1 → 2 → 3

Numbers on the left of 0 are negative.

Numbers on the right of 0 are positive.

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⭐ Quick Revision

Remember	Meaning
N	Natural numbers
W	Whole numbers
Z	Integers
Q	Rational numbers
R	Real numbers
Rational	Can be written as p/q
Irrational	Cannot be written as p/q
Real numbers	Rational + Irrational

🧠 Memory Trick

N → W → Z → Q → R

Think:

Natural → Whole → Integer → Rational → Real

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✏️ Practice Questions

A. Identify the category

1. 5 → __________
2. 0 → __________
3. −7 → __________
4. ½ → __________
5. √2 → __________
6. π → __________
7. 0.75 → __________
8. √25 → __________

B. Rational or Irrational?

Write Rational or Irrational.

1. ¾
2. √2
3. 0.5
4. π
5. √9
6. √7
7. −12
8. 0.121212…

C. True or False

1. Every integer is a rational number.
2. Every rational number is an integer.
3. Every irrational number is a real number.
4. 0 is a whole number.
5. √4 is irrational.
6. π is irrational.

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🎯 Answers

A

1. 5 → Natural, Whole, Integer, Rational, Real
2. 0 → Whole, Integer, Rational, Real
3. −7 → Integer, Rational, Real
4. ½ → Rational, Real
5. √2 → Irrational, Real
6. π → Irrational, Real
7. 0.75 → Rational, Real
8. √25 = 5 → Natural, Whole, Integer, Rational, Real

B

1. ¾ → Rational
2. √2 → Irrational
3. 0.5 → Rational
4. π → Irrational
5. √9 = 3 → Rational
6. √7 → Irrational
7. −12 → Rational
8. 0.121212… → Rational

C

1. True
2. False
3. True
4. True
5. False
6. True

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📌 One-Minute Revision

Real numbers are all numbers that can be represented on a number line.

Real Numbers = Rational Numbers + Irrational Numbers

Rational: Can be written as p/q

Irrational: Cannot be written as p/q

Natural: 1, 2, 3, …

Whole: 0, 1, 2, 3, …

Integers: … −2, −1, 0, 1, 2 …

Rational: ½, −3, 0.75, 2.5 …

Irrational: √2, √3, π …

⭐ Most important relationship:

Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real

Rational ∪ Irrational = Real Numbers

 

Operations on Real Numbers

1. Addition (+): 

   - The sum of two real numbers is also a real number. 

   - Example: 3 + 4.5 = 7.5 

2. Subtraction (−):

   - The difference between two real numbers is also a real number.

   - Example: 7.5 - 2.3 = 5.2 

3. Multiplication × or \:

   - The product of two real numbers is also a real number.

   - Example: 3 × 4 = 12 

4. Division ÷ or /:

   - The quotient of two real numbers is a real number, provided the divisor is not zero.

   - Example: 10 ÷ 2 = 5 

   - Division by zero is undefined in real numbers.

5. Exponentiation (^):

   - Raising a real number to the power of another real number results in a real number, depending on the values involved.

   - Example: 23 = 8

Properties of Operations on Real Numbers

- Commutative Property:

  - Addition: a + b = b + a 

  - Multiplication:  a × b = b × a 

- Associative Property:

  - Addition: (a + b) + c = a + (b + c) 

  - Multiplication: (a × b) × c = a × (b × c)

- Distributive Property:

  - Multiplication over Addition: ( a × (b + c) = a × b + a × c 

- Identity Elements:

  - Addition: ( a + 0 = a )

  - Multiplication: a × 1 = a 

- Inverse Elements:

  - Additive Inverse: ( a + (-a) = 0)

  - Multiplicative Inverse: ( a × 1/a = 1 (for  a ≠ 0 )

Real numbers and their operations are fundamental in mathematics, forming the basis for algebra, calculus, and many other areas of study.