Describe each number as (a) a terminating decimal, (b) a...

Describe each number as (a) a terminating decimal, (b) a repeating decimal, or (c) a nonterminating, nonrepeating decimal. Then classify the number as a rational number or as an irrational number. (see Example 2)
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Key Concepts

Irrational Numbers

Irrational numbers are those that cannot be written as the ratio of two integers. Their decimal expansions are nonterminating and nonrepeating, reflecting their inherent complexity and the fact that they cannot be expressed exactly through a simple fraction.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, with the denominator not equal to zero. Their decimal expansions are either terminating or repeating, which is a direct consequence of their fractional representation.

Nonterminating, Nonrepeating Decimal

A nonterminating, nonrepeating decimal is an infinite decimal expansion that does not form a repeating pattern. This lack of periodicity is a hallmark of irrational numbers, which cannot be represented as a simple fraction.

Repeating Decimal

A repeating decimal is one where a sequence of one or more digits repeats infinitely after the decimal point. This repeating pattern indicates that the number can be precisely expressed as a fraction of two integers.

Terminating Decimal

A terminating decimal is one that has a finite number of digits after the decimal point. In other words, its decimal representation ends after a certain number of digits, making it straightforward to express exactly.

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