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Showing posts from July, 2018

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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Arrow of time - reversed or not?

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I wrote some weeks ago about the basic concepts behind entropy and the arrow of time . It also conveniently served as test of MathJax . If you have a blog or website where you want to show some equations, you can apply MathJax with a short string of html code, and voila, nice clean LaTeX typesetting becomes available! Okay, enough advertising and let's get on with it. The two main reasons I wrote about entropy last time, is because 1. it is one of the most fascinating concepts in all of physics and 2. there are some fairly recent studies I wish to write about, and one needs to understand some basics before I go deeper into those. There was this one study that was circulated widely in popular science channels, which got hyped into the form: "scientists reversed the arrow of time!" Spoiler alert, no they didn't. I'm not saying that what the group did wasn't seriously cool and a great advancement, it's just that they didn't do what it sai...
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Laser awesomeness - The European XFEL

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Lasers are awesome, and I guess most people would agree since they have become so common. Anyone can buy a laser pointer with pocket change, and laser shows are used as attractions at night clubs. Blu-ray discs got their name from the type of lasers they are recorded and played with, and almost all production lines utilize lasers at some point. You can even get a car with laser headlights ! But you know what kind of lasers are the most awesomest? The really big and powerful ones, and it doesn't get much bigger or more powerful than the European X-ray free electron laser, or European XFEL for short. European XFEL is an incredible research facility housing one of the most ridiculous lasers ever made by human kind. It's a humongous device, stretching over a length of 3.4 kilometers underground, and it's construction cost is an estimated 1.22 billion euros. The operation of the device sounds like it's taken straight from some sci-fi show. First, on the ...
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