Which statement best defines irrational numbers?

Which statement best defines irrational numbers?

• A. Numbers that can be expressed as a ratio of two integers.
• B. Numbers whose decimal expansions are nonterminating and nonrepeating.
• C. Numbers that are whole numbers.
• D. Numbers that are rational but not integers.

Answer: (C)

Irrational numbers are defined by their decimal behavior and by not being expressible as a fraction of integers. Specifically, they have decimal expansions that go on forever without terminating and without falling into a repeating pattern. That nonterminating, nonrepeating nature is what sets them apart from rational numbers, which either terminate or repeat in their decimal form and can be written as a ratio of integers. So the statement describing a decimal that continues forever without repeating captures the essence of irrational numbers. The other descriptions point to rational numbers or integers, which do have either terminating decimals or repeating patterns, and can be written as fractions of integers. For example, pi and the square root of 2 both have nonterminating, nonrepeating decimals, illustrating irrational behavior.

Irrational numbers are defined by their decimal behavior and by not being expressible as a fraction of integers. Specifically, they have decimal expansions that go on forever without terminating and without falling into a repeating pattern. That nonterminating, nonrepeating nature is what sets them apart from rational numbers, which either terminate or repeat in their decimal form and can be written as a ratio of integers. So the statement describing a decimal that continues forever without repeating captures the essence of irrational numbers. The other descriptions point to rational numbers or integers, which do have either terminating decimals or repeating patterns, and can be written as fractions of integers. For example, pi and the square root of 2 both have nonterminating, nonrepeating decimals, illustrating irrational behavior.