Use a calculator ^* to express each number in decimal form....

Use a calculator $^{*}$ to express each number in decimal form. Classify each decimal mumber as terminating, repeating, or nonrepeating and nonterminating. Identify the pattern of repeated digits in any repeating decimal numbers. (A) $\frac{13}{6}$ (B) $\sqrt{21}$ (C) $\frac{7}{16}$ (D) $\frac{29}{111}$

Use a calculator $^{*}$ to express each number in decimal form. Classify each decimal mumber as terminating, repeating, or nonrepeating and nonterminating. Identify the pattern of repeated digits in any repeating decimal numbers.
(A) $\frac{13}{6}$
(B) $\sqrt{21}$
(C) $\frac{7}{16}$
(D) $\frac{29}{111}$

Key Concepts

Nonrepeating and Nonterminating Decimals (Irrational Numbers)

Nonrepeating and nonterminating decimals are characteristic of irrational numbers, which cannot be expressed exactly as a ratio of two integers. Their decimal expansions never settle into a permanently repeating pattern and extend infinitely, making them inexact representations that are best approximated using calculators.

Repeating Decimals

Repeating decimals are decimals in which one or more digits recur infinitely after the decimal point. They result from rational numbers whose denominators have prime factors other than 2 and 5. Identifying the repeating pattern within these decimals is crucial in representing them succinctly, often using a bar to denote the sequence that repeats.

Pattern Recognition in Repeating Decimals

This concept involves identifying the group of digits that repeat indefinitely in the decimal representation of a number. Recognizing the repeated sequence facilitates a clear and concise representation of the number, such as through the use of a vinculum (bar) over the repeating digits.

Decimal Representation of Rational Numbers

This concept covers the transformation of fractions into their decimal equivalents. Rational numbers, when divided, yield decimal forms that are either terminating or repeating, allowing for an analysis of the pattern or structure of the resulting sequence of digits.

Terminating Decimals

Terminating decimals are numbers that have a finite number of digits after the decimal point. They occur when the denominator of a fraction, after being simplified, contains only the prime factors 2 and/or 5. Recognizing terminating decimals aids in determining the precise numeric value without an infinitely extending sequence.

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