What's an irrational number?
Rational number
A real number that is not rational is called irrational. Irrational numbers include , , , and . The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.
Intermediate value theorem
The theorem depends on, and is equivalent to, the completeness of the real numbers. The intermediate value theorem does not apply to the rational numbers ℚ because gaps exist between rational numbers; irrational numbers fill those gaps. For example, the function formula_16 for formula_17 satisfies formula_18 and formula_19. However, there is no rational number formula_20 such that formula_21, because formula_22 is an irrational number.