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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Blog is still alive!

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Dang it, I just realized that I haven't updated the blog in over half a year. Oops. 

The blog is speaking to you.

The last few months have been hectic as hell, and in the beginning of the year I was busy writing my PhD thesis. I got a severe case of "blargh, I never want to write anything again" because, well, I kind of overdid it. Not to brag (as if), but my thesis ended up containing enough material to cover three or four average theses in physics. Oops.

Although it was really challenging to write something like that, it was totally worth it. I got my PhD right before the summer vacation; with the highest possible grade might I add (okay, now I am bragging). Also, at the end of the summer I moved across Finland and started a job as a postdoc at Tampere University, yay! Now I am working on things I'm familiar with (coherence of light) and stuff I'm not so familiar with (metamaterials). This research direction seems really interesting, and I will definitely post about my research in the months to come.

But my next post will be about something completely different, which I will (hopefully) publish in the next few days. Stay tuned!

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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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