0.999... Totally Explained

Everything about 999 totally explained

In mathematics, the recurring decimal 0.999…, which is also written as

0.ar = -1.

(Compare with the series above.) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof above in the opposite direction: if x = …999 then 10x = …990, so 10x = x − 9, hence x = −1 again. one may add the two equations and arrive at …999.999… = 0. This equation doesn't make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.

Related questions

Zeno's paradoxes, particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it isn't clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.

Division by zero occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as complex analysis, where the extended complex plane, for example the Riemann sphere, has a "point at infinity". Here, it makes sense to define ¹/₀ to be infinity; and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.

Negative zero is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0. Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the sign and magnitude or one's complement formats, or floating point numbers as specified by the IEEE floating-point standard).Further Information

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