Is 0.9 Recurring Truly Considered Equal to 1 in Mathematics?

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Discussion Overview

The discussion centers around the mathematical equivalence of 0.9 recurring (0.999...) and the number 1. Participants explore whether these two representations are considered equal in mathematics, examining proofs, definitions, and the implications of infinite series.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that 0.9 recurring and 1 are two different representations of the same number.
  • Others argue that the proof often cited to show 0.9 recurring equals 1 is flawed unless the real numbers and infinite series are rigorously defined.
  • A participant suggests that 0.999... approaches 1 as a limit, implying a non-zero difference exists between them.
  • Another participant counters that 0.999... is a number and does not approach 1, emphasizing that the difference is always zero.
  • Some participants discuss the nature of numbers and representations, suggesting that while 0.999... and 1 represent the same value, they may be considered different expressions or strings of text.
  • There is mention of an infinite series that converges to 1, reinforcing the idea that 0.999... equals 1.
  • Participants express differing views on the definitions of numbers and values, with some emphasizing the human invention aspect of numerical representation.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as multiple competing views remain regarding the nature of 0.9 recurring and its relationship to 1. Some assert equality, while others emphasize distinctions based on definitions and representations.

Contextual Notes

Limitations include the need for careful definitions of real numbers and infinite series to fully understand the claims made in the discussion. The conversation also highlights the potential confusion between the terms "numbers" and "values" as used by different participants.

  • #31
wsabol said:
1 is a real whole number. 0.999... is a limit. That limit is equal to 1, not the real decimal number 0.9999...(as close as you can get to infinity without getting there, because the infinite term of the sequence ever happen)...9

No, 0.99999... is a notation for a real number. The real number is defined by a limit.

Limits are real numbers.
 
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  • #32
micromass said:
No, 0.99999... is a notation for a real number. The real number is defined by a limit.

Limits are real numbers.

Damn you got me. Ok.
 
  • #33
There is also the http://en.wikipedia.org/wiki/Infinitesimal" approach.

There is a http://en.wikipedia.org/wiki/Hyperreal_number" \epsilon that is smaller than the smallest real number, so we can define the following: 1 - \epsilon = .999....

This implies that 1 and 1 - \epsilon (.999...) are different numbers in the hyperreal numbering system.
 
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  • #34
Matt Benesi said:
There is also the http://en.wikipedia.org/wiki/Infinitesimal" approach.

There is a http://en.wikipedia.org/wiki/Hyperreal_number" \epsilon that is smaller than the smallest real number, so we can define the following: 1 - \epsilon = .999....

This implies that 1 and 1 - \epsilon (.999...) are different numbers in the hyperreal numbering system.


I fear you have not fully understood hyperreals. In the hyperreals, the definition 1-\varepsilon=0.9999... is not made. Furthermore, in the hyperreals, there is no such thing as the smallest real number.
 
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  • #35
This topic has come up so often here that we have an FAQ that addresses this concept: [thread]507001[/thread].

Thread closed.
 

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