Revision as of 15:00, 8 May 2005 edit4.250.177.162 (talk) →See also: just trying to help.← Previous edit | Revision as of 15:05, 8 May 2005 edit undo4.250.177.162 (talk) →Three 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> | ||
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==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. | |||
x = 0.999... | |||
10x-x = 9.999... - 0.999... = 9x = 9 | |||
x = 1 | |||
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. | ||
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). | 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). | ||
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) ? | 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 == | == See also == |
Revision as of 15:05, 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
Explanation
The key step to understand here is that the following infinite geometric series is convergent:
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.
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?