How are irrational numbers characterized?

Irrational numbers are defined as numbers that cannot be expressed as a fraction of two integers. This means that there is no way to write them as ( \frac{a}{b} ), where ( a ) and ( b ) are both whole numbers, and ( b ) is not zero. Additionally, irrational numbers do not exhibit repeating patterns in their decimal representation; they continue indefinitely without settling into a repeating cycle.

For example, the square root of 2 and the number π (pi) are both considered irrational because their decimal forms do not terminate or repeat, making it impossible to represent them as fractions. This characteristic distinctly separates irrational numbers from rational numbers, which can be expressed in fractional form and often display terminating or repeating decimals.

Understanding this definition clarifies why irrational numbers belong to a separate category in the number system, setting them apart from whole numbers, fractions, and repeating decimals.