REAL NUMBERS | Irrational Numbers | Part -1

Understanding Irrational Numbers

To understand irrational numbers, it’s essential to first explore the broader category they belong to — real numbers. Real numbers consist of all values that can be located on the number line, including both rational and irrational types.

The World of Real Numbers

The set of real numbers is made up of two main types:

• Rational numbers: These are values that can be represented as a ratio of two whole numbers, where the bottom number (denominator) is not zero.
• Irrational numbers: These are numbers that cannot be written as a simple fraction and have non-terminating, non-repeating decimal forms.

Together, these numbers fill the entire number line, representing all possible magnitudes in continuous quantities.

Rational Numbers and Their Decimal Representation

Rational numbers cover integers as well as decimals that either end or repeat in a predictable pattern. For example:

• 3 (which can be written as 3/1)
• 0.75 (which equals 3/4)
• 0.333... (a repeating decimal representing 1/3)

The decimal expansion of rational numbers either ends after a finite number of digits or repeats a pattern indefinitely.

Irrational Numbers and Their Decimal Representation

Irrational numbers are numbers that cannot be written as a ratio of two integers. Their decimal form neither terminates nor repeats. Examples include:

• π (pi), approximately 3.14159...
• √2 (the square root of 2), which is roughly equal to 1.41421 and continues without repeating.

These decimal expansions continue infinitely without any repeating sequence, making irrational numbers impossible to write exactly as fractions or finite decimals, though they can be approximated.

In conclusion, real numbers are composed of rational and irrational numbers. Rational numbers have decimals that terminate or repeat, while irrational numbers have infinite, non-repeating decimals. Both together form the real number system that models all continuous numerical values on the number line.