What type of numbers cannot be expressed as the ratio of two integers and include examples such as pi?

Irrational numbers are defined as numbers that cannot be expressed as the ratio of two integers. This means that there is no way to write them in the form of a fraction where both the numerator and denominator are whole numbers. A well-known example of an irrational number is pi (π), which represents the ratio of the circumference of a circle to its diameter and continues infinitely without repeating.

Other examples of irrational numbers include the square root of 2 and the mathematical constants e and the golden ratio. These numbers have non-terminating, non-repeating decimal expansions, setting them apart from rational numbers, which can be fully characterized as fractions.

In contrast, rational numbers can be clearly defined as those that can be expressed in the form of a fraction. Whole numbers are a subset of rational numbers that include all non-negative integers (0, 1, 2, ...), and natural numbers are positive integers starting from 1. Both whole and natural numbers can be expressed as fractions (for instance, the whole number 4 can be expressed as 4/1). Therefore, these categories do not represent numbers that are unable to be expressed as ratios of integers, which reinforces why irrational numbers are the correct answer.