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)
$$\sqrt{6}$$

Key Concepts

Repeating Decimal

A repeating decimal is a decimal number in which a digit or a group of digits repeats infinitely. This type of decimal can be represented as a fraction, meaning it is a rational number, and the repeating sequence is often denoted with a bar over the repeating digits.

Terminating Decimal

A terminating decimal is a decimal number that has a finite number of digits after the decimal point. These decimals do not continue infinitely and can be expressed exactly as a fraction with a denominator that is a power of 10, or equivalently, a product of 2's and 5's only.

Nonterminating, Nonrepeating Decimal

A nonterminating, nonrepeating decimal is a decimal expansion that goes on forever without exhibiting any repeated pattern. Such decimals cannot be written as a ratio of two integers, classifying them as irrational numbers.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is non-zero. Their decimal representations are either terminating or repeating, which distinguishes them from irrational numbers.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of integers. Their decimal expansions are nonterminating and nonrepeating, and common examples include many square roots of non-perfect squares.

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