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Is 0.999... = 1? (spoiler alert: no it is not)

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You may have encountered the popular claim that \( 0.999... = 1 \), where the three dots signify that the decimal continues forever. This is a somewhat weird claim, since it would mean that mathematics is broken. There should be no way for two different numbers to have the same value. What makes it weirder is that this is quite popular claim. I've even seen mathematicians say that it's true! But is it though? One popular proof is to first denote \( S = 0.999...\) and then multiply by \(10\) to get \( 10S = 9.999...\) and subtract \( S \) from it, to get  \( 10S - S = 9.000...\) and finally dividing by \(9\) yields  \( S = 1.000... = 1 \) and we see that  \(0.999... = 1\)! However, there's a problem. This short derivation is not strictly speaking correct. It is veeeery close to being correct, and to see why let's look at finite decimals first. Let's say that \(S = 0.999\) (note that this is not the same as \(S = 0.999...\) ). Let's do the same trick as ...
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The year 2017 in optics and photonics

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The year 2017 is very nearly over, and it is time to look back at what news the year brought in photonics. I will be covering here some of the things I found the most interesting. First of all, being a Finnish citizen, I must mention that Finland turned 100 years old on 6.12.2017! To celebrate 100 years of independence, there were several events all over the country and abroad as well. Company called Valoparta made the biggest art installation of all time, lighting up the whole Saana fell (a type of mountain in Lapland). Picture from  http://www.valoparta.com/luminousfinland100/ This is the first art installation of its kind, with 2.5 million square meters of illuminated area and it was visible from three countries, Finland, Sweden and Norway. And to top it all off, we had the good fortune of seeing northern lights above Saana at the time of the installation. Truly spectacular! Hard science Nobel prize The Nobel prize in physics  went  to the...
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Why photonics? A brief history of me

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Hello to you and welcome to my blog! This is my first (actual) post here and I would like to start by telling you why I want to write about this particular topic. Like it says in the title, this blog is about physics and math, with an emphasis on photonics, so I will be writing about a large variety of things but keep coming back to that one special topic. I will not be talking (at least not too much) about the topics that are the most popular (and maybe the most controversial) in science, such as string theory, quantizing gravity or about the search for a theory of everything. Why, then, do I want to tell people about photonics? As you may have guessed, it is because I actually am qualified to talk about it, since it is my main area of expertise and I happen to have a formal education in it. The arXiv universe! The white dots are papers on optics, notice where they are centered? ( paperscape.org ) The way I came to do this line of work is rather unconventional. When I wa...
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