Real Numbers

Real numbers are all the numbers you have learned so far. They include whole numbers, integers, fractions, decimals, rational numbers, and even irrational numbers (though we are skipping their detailed tutorial).

In short: Real numbers are all the numbers that can be placed on a number line.

Real numbers include:

• Natural numbers
• Whole numbers
• Integers
• Even and odd numbers
• Rational numbers (like fractions and decimals that repeat or end)
• Irrational numbers (like \(\sqrt{2}\), π — cannot be written as fractions)

In mathematical notation:

\( \mathbb{R} = \{ x : x \text{ is rational or irrational} \} \)

Think of a number line stretching infinitely in both directions.

Every point on this line represents a real number.

Examples along the line:

• -5
• -1.2
• 0
• 5
• \(\sqrt{3}\)
• 3.14

Real numbers are divided mainly into two groups:

All real numbers can be plotted on a number line. The line is continuous, meaning it has no gaps. Every tiny point represents a number.

Even numbers that look strange (like √7 or 0.333...) still belong somewhere on the number line.

Real numbers follow important properties used in arithmetic and algebra.

• 5
• -12
• 0
• 3.75
• −2.5
• 0.333...
• √11
• π

All these are real numbers because they appear somewhere on the number line.

Real numbers are used in every corner of daily life because almost every measurement or quantity we use is a real number.

All rational numbers are real numbers, but not all real numbers are rational.