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)
List three numbers that are integers but not natural numbers.

Key Concepts

Natural Numbers

Natural numbers typically refer to the counting numbers starting from 1, although some definitions include 0. The key distinction in this context is that natural numbers do not include negative numbers, which is why certain integers are not considered natural numbers.

Integers

Integers include all whole numbers as well as their negatives and zero. They form a complete set that does not include fractions or decimals, making them fundamental in many areas of mathematics.

Irrational Numbers

Irrational numbers cannot be written as a simple fraction of two integers. Their decimal expansions are nonterminating and nonrepeating, meaning no finite set of digits repeats periodically.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Their decimal representations either terminate or eventually repeat, providing a clear pattern or ending point.

Nonterminating, Nonrepeating Decimal

A nonterminating, nonrepeating decimal is a decimal that goes on forever without any repeating pattern. This characteristic is typical of irrational numbers, which cannot be expressed as a simple fraction of two integers.

Repeating Decimal

A repeating decimal is one in which a digit or a group of digits repeats infinitely after a certain point. It is also the decimal representation of a rational number in cases where the prime factors of the denominator (in lowest terms) include primes other than 2 or 5, leading to a repeating pattern.

Terminating Decimal

A terminating decimal is a number whose decimal representation ends after a finite number of digits. It comes from fractions where the denominator (once reduced) has only the prime factors 2 and/or 5, making the conversion to a finite decimal possible.

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