What type of number cannot be expressed as a ratio of two integers or a repeating or terminating decimal?

The concept of irrational numbers is fundamental in understanding different classifications of numbers. An irrational number is defined as a number that cannot be expressed as a ratio of two integers, meaning it cannot be written in the form ( \frac{a}{b} ), where ( a ) and ( b ) are both integers and ( b ) is not zero. Additionally, irrational numbers are characterized by their non-repeating and non-terminating decimal expansion.

Famous examples of irrational numbers include ( \pi ) and the square root of 2. These numbers go on forever in their decimal form without repeating any sequence of digits. This is what differentiates them from rational numbers, which can either be represented as simple fractions or as decimals that terminate or repeat.

Whole numbers, such as 0, 1, 2, and any positive integers, as well as integers, which include all whole numbers and their negative counterparts, are always rational because they can be expressed as themselves over 1 (e.g., 3 is the same as ( \frac{3}{1} )). Therefore, it is clear that the type of number that cannot be expressed as a ratio of two integers or as a repeating or terminating decimal