Is 0.9 Recurring Truly Considered Equal to 1 in Mathematics?

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SUMMARY

The discussion confirms that 0.9 recurring (0.999...) is mathematically equal to 1. This is established through various proofs, including the manipulation of infinite series and limits. Participants clarify that while 0.999... and 1 are different representations, they denote the same value in mathematics. The conversation also addresses misconceptions about the nature of infinity and the real numbers, emphasizing that 0.999... is a real number equal to 1.

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  • #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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