Repeated and Nonrepeated Decimals

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Discussion Overview

The discussion revolves around the classification of repeated and non-repeated decimals as rational or irrational numbers. Participants explore definitions and provide examples, focusing on the implications of these classifications within the broader context of number theory.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants assert that repeating decimals are rational numbers, providing examples such as $$0.\overline{154}=\frac{154}{999}$$.
  • Others argue that non-repeating decimals are irrational, citing $\sqrt{2}$ as an example.
  • One participant points out that if all decimals are either repeating or non-repeating, then claiming both types are rational would imply there are no irrational numbers, challenging the initial premise.

Areas of Agreement / Disagreement

Participants express disagreement regarding the classification of non-repeating decimals, with some asserting they are irrational while others challenge the implications of the definitions provided.

Contextual Notes

The discussion highlights potential limitations in the definitions of rational and irrational numbers, particularly in relation to the completeness of the classifications presented.

mathdad
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My question concerns repeated and nonrepeated decimals. Are both rational numbers? Can you give an example for each?
 
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RTCNTC said:
My question concerns repeated and nonrepeated decimals. Are both rational numbers? Can you give an example for each?

A repeating decimal number is rational because you can always express such a number as the string of repeating digits over an equal number of 9's (one of the tricks my father taught me as a child). For example, we may write:

$$0.\overline{154}=\frac{154}{999}$$

A non-repeating decimal is irrational since it cannot be expressed as the ratio of one integer to another. $\sqrt{2}$ is an example of a non-repeating decimal.
 
You realize, I hope, that all decimals are either "repeating" or "non-repeating". So if it were true that "all repeated and non-repeated decimals are rational numbers" then there would be no irrational numbers!
 
MarkFL said:
A repeating decimal number is rational because you can always express such a number as the string of repeating digits over an equal number of 9's (one of the tricks my father taught me as a child). For example, we may write:

$$0.\overline{154}=\frac{154}{999}$$

A non-repeating decimal is irrational since it cannot be expressed as the ratio of one integer to another. $\sqrt{2}$ is an example of a non-repeating decimal.

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