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

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 ...

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...

The series of tubes that we call the Internet

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Maybe some people still remember the infamous statement made by the late Republican Senator Ted Stevens during the net neutrality debates in 2006, that the Internet is " a series of tubes ." He used this statement to criticize the bill on net neutrality , because according to him, the Internet is not a "big truck" and tubes can get clogged. After the statement he was met with ridicule, because of a very limited understanding of the Internet, even though he was in charge of its regulation. To me, the extreme criticism that Stevens got for the tube statement is a little weird, because let's face it, the Internet is, in all actuality, a series of tubes. And no, I don't mean in a metaphorical sense. Although, with the word tube, I don't mean plumbing pipes or whatever. I have no idea what Stevens meant with tubes , and I don't think that he made a good argument against net neutrality. The argument was just ridiculed for all the wrong reasons. I mea...

Photons, what are they?

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Probably everybody has once or twice in their lifetime heard the word "photon" together with the phrase "oh, a photon is just a particle of light," or something along those lines. That's all well and good, but what does "particle of light" actually mean? Can you touch it? How big is it? What does it look like? What can you do with it? And who cares? If someone talks about particles, you would expect them to mean that this thing is something solid, like a ping pong ball. You would also expect it to have some other properties that particles usually have, like it says on the wikipedia page about particles : " In the physical sciences, a particle (or corpuscule in older texts) is a small localized object to which can be ascribed several physical or chemical properties such as volume or mass. " Let's also take a look at what wikipedia says about photons : " A photon is a type of elementary particle, the quantum of the electro...

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...

Blog under construction.

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Welcome to my blog! I am Matias Koivurova, a researcher from the University of Eastern Finland and my main area of interest is photonics and coherence of light, although I dabble in other areas as well.  I am sorry to disappoint, but if you are looking for some really smart person who will write about string theory, quantum gravity, or some other mediasexy topic, you will not find that here (those places can be found  here  and  here ). What I will be covering is research that I find particularly interesting, and hopefully I can make some sense of it to some less science inclined people. I will also update the blog with info on my own research and I will give a short description of each of my published articles. A pretty test picture. The name of the blog was inspired by my main area of research, which I think also describes my writing style. Partial coherence in photonics is often undesirable but can produce interesting phenomena; it remains to b...