<p>I will bite also… To help you decide whether this is worth paying attention to, I’ll state my qualifications: I’m a junior and a pure math major at Caltech; I’ve taken three courses in advanced real and complex analysis, in which questions of this sort are treated.</p>
<p>Anyway, usually questions like this that come up on non-mathematical forums are nonsense, but this one is actually not. The answer is yes.</p>
<p>There is a formal definition of decimal expansions. If w is the integer part of the number, and d_k is the kth digit after the decimal point in base 10, then an unterminating decimal is defined in standard analysis to be</p>
<p>w + \sum<em>{i=1}^\infty d</em>k/10^k.</p>
<p>For a decimal expansion to be well-defined, there must be a function from the natural numbers {1,2,3,…} to the digits {0,1,2,3,4,5,6,7,8,9} assigning to each place k a digit d_k. </p>
<p>A decimal expansion is said to terminate if there is an integer K such that for all k > K, d_k=0.</p>
<p>When w=0 and d_k = 9 for all k, easy calculus shows that this is a geometric series equal to 1. </p>
<p>As an aside, it is pretty easy to prove that for any set {d_k}, the sum is well-defined. (All non-terminating decimals converge.) However, it is also true there are real numbers that have multiple decimal expansions in base 10. The multiplicative identity is equivalently represented by .999… and by 1. No real number has more than two equivalent decimal expansions. There is a proof, which is easy to construct (a good exercie) but longer than one or two lines… feel free to ask if you want to see it. </p>
<p>In fact, it turns out that there are two expansions if and only if one of them terminates; thus, the set of real numbers with two decimal expansions has measure zero on the real line, which means in some sense that it is very small.</p>
<p>And to answer Hriundeli’s worry, the existence of two decimal representations for some number does not create any paradoxical results, so no need to worry.</p>
<p>To answer this nonsense about .0000 with a 1 infinitely far away, that decimal expansion is not well-defined (see above for definition) since there is no way to make that into a function from the natural numbers {1,2,3,…} to the digits {0,1,2,3,4,5,6,7,8,9}. I will ask you, what is the natural number in the domain which gives the image 1, and you will not be able to answer, showing that your decimal expansion does not have 1 in any place at all. You cannot say “the infinitely large natural number” is the element of the domain that maps to 1, because there is no maximal natural number – this follows direclty from Peano’s axioms.</p>
<p>So, it is true that there is no need to form “beliefs” about this, since in the standard mathematical framework of real analysis there is a clear provable answer, namely: yes, 1.00 does equal .9999999…</p>
<p>If someone has alternative definitions for infinite decimal expansions different from the standard ones, please cite at least one textbook or scholarly article using that definition.</p>
(sry, River, i told ya im dumb ;))</p>