![]() If you take any of the infinitely many terms in the series 0.9 0.09 0.009 . cannot add up to anything else, so it must add up to 1. They used the trick of saying 0.9 0.09 . It seemed that the addition of infinitely many terms was impossible. Another problem is that if we add more terms then we have still only added a finite number of terms. all add-up to 1 would require an infinite amount of work. In order to determine that 9/10 9/100 . Mathematicians wanted all quantities to have a base 10 representation, so in the 16th century Stevin created the basis for modern decimal notation in which he allowed an actual infinity of digits.įor over 200 years mathematicians were troubled by infinite decimals. 1/3 could be represented in bases that had 3 as a factor, like base 12, but it could not be represented in base 10. Infinite decimals have a long and troubled history. (And some people do genuinely seem to mean something closer to what we call the hyperreals than to what we call the reals.) Reply Delete In fact, they usually mean the naive notion, ill-defined as it is. The accepted formal definition of the reals is a formalization imposed by mathematicians on an informal notion, and we should stop assuming that when non-experts talk about the reals, they automatically mean the same thing mathematicians do. With the benefit of a few centuries of work, we know to be skeptical of the naive theory of infinitesimals that people seem to find intuitive - there's no way to capture exactly the structure people seem to find intuitive. It's very clear that most of the people who struggle with this are actually feeling their way towards an alternate axiomitization of the real numbers. The primary issue isn't so much the complexity of the proof as the fact that it digs deeper into the formal definition of the real numbers than many people are used to. I appreciate that this makes the important point that the proof that 0.9˙=1 is actually non-trivial, and that the commonly given arguments substitute mathematical trickery for addressing what's actually hard about the proof.īut I think one should go further. On the bright side, you have a fascinating journey ahead of you if you decide to fill in the gaps. I'm afraid you were tricked, and it really is a bit more complicated than that. ![]() One side of the conversation insists that \(0.\dot=1\). In the way of perpetual internet conversations, although the participants change, the actual positions are pretty constant. If you follow any kind of mathematical discussions on the internet, you'll probably have noticed that there's one particular topic that refuses to die: ![]()
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