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The Archimedean property: any point x before the finish line lies between two of the points P n (inclusive).. It is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series and limits.
This page is for mathematical arguments concerning 0.999... Previous discussions have been archived from the main talk page, which is now reserved for editorial discussions.
1 troy ounce of four nines fine gold (999.9) Nines are an informal logarithmic notation for proportions very near to one or, equivalently, percentages very near 100%. Put simply, "nines" are the number of consecutive nines in a percentage such as 99% (two nines) [1] or a decimal fraction such as 0.999 (three nines).
The correct notation is (0.9; 0.99; 0,999; ...). Q: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly?
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You have to be sure though when claiming that it is one, to point out that it is really (1 - .0000...1) While the difference between .999... and 1 is infinitely nothing, it cannot be dismissed because it is everywhere..000...1 is the first thing there is greater than zero, and it's between every change between every two numbers all the way up to one.
The real numbers can be defined synthetically as an ordered field satisfying some version of the completeness axiom.Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are non Archimedean fields ...
The notation 0.999..., as said, denotes a "repeating decimal", and is not prepared to be amended to 0.999...9, because this latter notation totally loses the meaning of a "repeating decimal". It denotes a decimal with an unspecified, but finite number of 9 s right to the decimal point.