What is defined as any real number that cannot be expressed as a fraction of two integers?

The concept at the heart of this question revolves around the classification of numbers. An irrational number is specifically defined as any real number that cannot be expressed as a ratio of two integers, meaning that it cannot be written in the form ( \frac{a}{b} ), where ( a ) and ( b ) are integers and ( b ) is not zero.

Irrational numbers include values such as ( \sqrt{2} ), ( \pi ), and ( e ). These numbers have non-terminating, non-repeating decimal expansions. This property clearly distinguishes them from rational numbers, which can be expressed as a simple fraction.

In contrast, rational numbers can indeed be expressed as such a fraction, while whole numbers and natural numbers are specific subsets of integers. Whole numbers include zero and all positive integers, and natural numbers consist strictly of positive integers. Neither of these categories includes the concept of being unable to express a number as a fraction of two integers.

Thus, identifying irrational numbers correctly emphasizes an important aspect of real numbers, characterizing them by their inability to fit into the rational framework.