0.999...= 1
Personally I&#x27;ve always thought &quot;proofs&quot; using &quot;arithmetic&quot; are right, but kind of stated backwards.<p>The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you <i>never</i> define &quot;...&quot;. Until you&#x27;ve defined &quot;...&quot;, it&#x27;s just a meaningless sequence of marks on paper. You can&#x27;t prove anything about it using arithmetic, or otherwise.<p>What the &quot;arithmetic proofs&quot; are really showing that if we want &quot;...&quot; to have certain extremely reasonable properties, then we must choose to define it in such a way that 0.999... = 1. Other definitions would be possible (for example, a stupid definition would be 0.999... = 42), just not useful.<p>What probably causes the flame wars over &quot;...&quot; is that most people never see how &quot;...&quot; <i>is</i> defined (which properly would require constructing the reals). They only see these indirect arguments about how &quot;...&quot; <i>should</i> be defined, which look unsatisfying. Or they grow so accustomed to writing down &quot;...&quot; in school that they think they already know how it&#x27;s defined, when it never has been!