![]() ![]() It’s the same as saying 1 gets closer and closer to 2, but never quite reaches 2. So phrases like “it gets close to 1, but never reaches 1” are meaningless. Mathematicians say that 0.999… is a number, just like 2 and 3. But this is not quite what mathematicians mean with 0.999… This reasoning appears a lot and apparently, many people see 0.999… as some kind of process that gets close to 1. The way I see it, is that 0.999… gets closer and closer to 1, but never quite reaches 1. Not having that 1=0.999… would make our number system much uglier! Does this make our number system ugly? I understand that you might think that, but that’s just something we need to accept. The same thing happens to 1=0.999… really, it’s just another way to write the same number. \frac=0.333… ,īut somehow, many people don’t have any problems with this thing. Well, the thing is that this is just a misconception that is simply not true. A more rigorous proof is given in the post following this one.īut 1 cannot equal 0.999…, as every number can only have one representation! For example, how do we know that 3 \times (0.333…) = 0.999…? This is not that obvious if we think about it. \ 1=3\times (1/3) = 3\times (0.333…) = 0.999…Īll of these proofs are correct, but they are not rigorous. If you don’t accept that, and you think that two different numbers can have a difference of zero, then you’re in an ‘extended number system’ which has more numbers than we normally use. If you accept that two numbers whose difference is 0 must be the same, then that proves that 0.9999… = 1. So the difference between 0.9999… and 1 is less than 0.00001, 0.000000001, or 0.any number of 0s followed by 1 … so the difference must be zero. But it’s also larger that 0.9999999, so it’s less than 0.0000001 less than 1. If you accept that 0.9999… is a number, then how much less than 1 is it? It’s larger that 0.9999, so it’s less than 0.0001 less than 1. But there is no way to write a number that is greater that 0.999… and less than 1 in decimal notation. and 1 are different, there must be another number in between them. (That is intuitively obvious, and can be pictured on a number line, which will be familiar to many people.) Therefore, if 0.999…. Here are some non-rigorous proofs that 1=0.999…:įor any two unequal numbers, there is always another number in between them. No, they really are the same number, though this is often very counterintuitive to many beginning students. Why do people say 1 and 0.999… are equal? Aren’t they two different numbers? ![]()
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