Numbers can be classified as rational or irrational depending on how they can be written and represented.

A rational number is any number that can be written exactly as a fraction in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). This includes whole numbers, integers, fractions and decimals that terminate or recur.

For example, \( \frac{3}{4} \), \( -5 \) and \( 0.25 \) are all rational numbers. A terminating decimal such as \( 0.8 \) is rational because it can be written as \( \frac{4}{5} \). A recurring decimal such as \( 0.\dot{6} \) is also rational because it is equal to \( \frac{2}{3} \).

An irrational number cannot be written exactly as a fraction. Its decimal representation does not terminate and does not recur. The digits continue forever without forming a repeating pattern.

A common example is \( \sqrt{2} \). Its decimal form continues indefinitely without repetition. Another example is \( \pi \), which represents the ratio of the circumference of a circle to its diameter. These numbers cannot be expressed exactly as fractions.

The key difference is that rational numbers have decimals that either end or repeat, while irrational numbers have decimals that go on forever without repeating. Recognising this distinction is important when deciding whether a value is exact, approximate or suitable for use in algebraic and numerical problems.