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Revision as of 15:05, 8 May 2005 edit4.250.177.162 (talk) →Three alternative ways of explaining this truth← Previous edit Revision as of 15:27, 8 May 2005 edit undoBradBeattie (talk | contribs)6,888 edits →Some alternative ways of explaining this truthNext edit →
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:<math>\sum_{k=0}^\infty \left( \frac{1}{10} \right)^k = \frac{1}{1 - \frac{1}{10}}.</math> :<math>\sum_{k=0}^\infty \left( \frac{1}{10} \right)^k = \frac{1}{1 - \frac{1}{10}}.</math>


== Alternate proof ==
==Some alternative ways of explaining this truth==


{|-
Let x equal 0.999... Therefore 10x-x equals 9.999... - 0.999... which equals 9x = 9 and so x equals 1.
|<math>x</math>
|<math> = 0.999\ldots</math>
|-
|<math>10 \times x</math>
|<math> = 9.999\ldots</math>
|-
|<math>10 \times x - x</math>
|<math> = 9.999\ldots - 0.999\ldots</math>
|-
|<math>9 \times x</math>
|<math> = 9\,</math>
|-
|<math>x\,</math>
|<math>= 1\,</math>
|}


Another: What is 1-0.99999... ? You get 0.000000... which is the same as zero. Another: What is 1-0.99999... ? You get 0.000000... which is the same as zero.

Revision as of 15:27, 8 May 2005

In mathematics, one could easily fall in the trap of thinking that while 0.999... is certainly close to 1, nevertheless the two are not equal. Here's a proof that they actually are.

Proof

0.999 … {\displaystyle 0.999\ldots } = 9 10 + 9 100 + 9 1000 + ⋯ {\displaystyle ={\frac {9}{10}}+{\frac {9}{100}}+{\frac {9}{1000}}+\cdots }
= − 9 + 9 1 + 9 10 + 9 100 + 9 1000 + ⋯ {\displaystyle =-9+{\frac {9}{1}}+{\frac {9}{10}}+{\frac {9}{100}}+{\frac {9}{1000}}+\cdots }
= − 9 + 9 × ∑ k = 0 ∞ ( 1 10 ) k {\displaystyle =-9+9\times \sum _{k=0}^{\infty }\left({\frac {1}{10}}\right)^{k}}
= − 9 + 9 × 1 1 − 1 10 {\displaystyle =-9+9\times {\frac {1}{1-{\frac {1}{10}}}}}
= 1. {\displaystyle =1.\,}

Explanation

The key step to understand here is that the following infinite geometric series is convergent:

∑ k = 0 ∞ ( 1 10 ) k = 1 1 − 1 10 . {\displaystyle \sum _{k=0}^{\infty }\left({\frac {1}{10}}\right)^{k}={\frac {1}{1-{\frac {1}{10}}}}.}

Alternate proof

x {\displaystyle x} = 0.999 … {\displaystyle =0.999\ldots }
10 × x {\displaystyle 10\times x} = 9.999 … {\displaystyle =9.999\ldots }
10 × x − x {\displaystyle 10\times x-x} = 9.999 … − 0.999 … {\displaystyle =9.999\ldots -0.999\ldots }
9 × x {\displaystyle 9\times x} = 9 {\displaystyle =9\,}
x {\displaystyle x\,} = 1 {\displaystyle =1\,}

Another: What is 1-0.99999... ? You get 0.000000... which is the same as zero.

Try this: Divide one by three (one third) and you get .333333(an unending series of threes). Three one-thirds is one so three times .3333(an unending series of threes) is .99999999999(an unending series of nines).

Finally: If you don't have a problem with 1.00000(an unending series of zeros), why should there be a problem with 0.9999(an unending series of nines) ?

If you think there is a difference, in what way is that difference different from nil, nada, nothing, zilch, zero?

See also

External proofs


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