.99999999999999999 = 1. The Article.
While they go on on ad nauseum about this, the basis of the argument is, as you guessed it, .99999999999999 really is 1. Don't bother to come up with an objection; the folks masterminding this article with brow beat you with an incredible 999999999 proofs as to why you are an idiot. Seriously, some of the deliciousness that is the article:
One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using long division, a simple division of integers like 1?3 becomes a recurring decimal, 0.3333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.3333… equals 0.9999…. But 3 × 1?3 equals 1, so 0.9999… = 1.[1]
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Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.9999… equals 9.9999…, which is 9 more than the original number. To see this, consider that subtracting 0.9999… from 9.9999… can proceed digit by digit; the result is 9 ? 9, which is 0, in each of the digits after the decimal separator. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.9999…, be called c. Then 10c ? c = 9. This is the same as 9c = 9. Dividing both sides by 9 completes the proof: c = 1.[1]
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As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing 1/3 as a number in its own right and to dealing with the set of natural numbers as a whole.[19]
I suppose if there is any doubt to the argument, Who Da Fuck Cares?; the answer would be... appearantly nerds on the internet. Yes, they prove that as well....
With the rise of the internet, debates about 0.999… have escaped the classroom and are commonplace on newsgroups and message boards including many that nominally have little to do with mathematics. In the newsgroup sci.math, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its FAQ[55] The FAQ briefly covers 1/3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.
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In the same vein, the question of 0.999… proved such a popular topic in the first seven years of Blizzard Entertainment''s Battle.net forums that the company's president, Mike Morhaime, announced at an April 1, 2004 press conference that it is 1:
- "We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers."[57]
Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.
Obviously we can state to a fact; that only on an the internet message board of mathnerds, does arguing over numbers equate a "sport." Yes I said it,Infinite arguments over numbers = 1 sport.
go Wikipedia!
