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The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. ... every number larger ...
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 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.
Q: You guys talk a lot about real analysis, limits, and calculus; shouldn't this just be about arithmetic? A: Unfortunately, in order to formally prove many qualities of numbers, one often has to resort to higher mathematics: real analysis in the case of real numbers, number theory in the case of integers, and so forth.
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The string "0.000...1" is not a meaningful real decimal because, although a decimal representation of a real number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit d being k places past the decimal point is that the digit contributes d · 10-k ...
However, you are willing to wave your hands, and say "there's a number just less than 0.999..., which I'll call 0.999...98", without defining what those symbols mean. If you're trying to present an alternative viewpoint to the real numbers, then for gosh sakes, present it already!
Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations.. Assume .Then for every integer there is a finite decimal =. such that: