What type of numbers includes both rational and irrational numbers?

The correct answer is real numbers, which encompass both rational and irrational numbers. Rational numbers include any numbers that can be expressed as the quotient of two integers, where the denominator is not zero. This means that fractions, integers, and finite or repeating decimals all fall under the category of rational numbers.

Irrational numbers, on the other hand, are numbers that cannot be expressed as a simple fraction; they have non-repeating, non-terminating decimal representations, such as the square root of 2 or pi.

By definition, the set of real numbers combines these two categories into one comprehensive set, which is essential for analysis in mathematics, particularly in calculus and other higher-level math topics. This unified grouping of rational and irrational numbers is critical for completing the number line and understanding the continuum of values between integers and fractions.

Other number sets, such as complex numbers, represent an extension that includes imaginary numbers; natural numbers and whole numbers refer only to specific types of integers and do not account for rational or irrational values. Thus, real numbers are the broad category that accurately encompasses both types of numbers.