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Showing posts from January, 2018

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 before, so

Radiation poisoning from reindeer

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I was doing a summer internship at the Finnish  Radiation and Nuclear Safety Authority  a few years back, where I got the opportunity to try my hand at writing. It was very interesting work and the internship went by reaaally fast. Oh, and I was also in charge of monitoring the radiation levels in the air of northern Finland, while my supervisors were on summer vacation (yay). Thankfully, the radiation stayed at safe levels throughout the summer, although it would have been interesting to see what happens when they are elevated. I did get to see some action as well, since we hosted a drill with other authorities, the emergency services and the military. The setting was that there was an unknown source of radioactivity spotted in Northern Finland and fighter jets were sent to collect samples from the clouds. Our job was to determine what the cause was and what needs to be done to avoid further damages. It was partly exciting and partly dull, since I was assigned to spectate and

Thermoelectric pumping - how to exceed 100 % efficiency

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Physicists have known for decades that it is possible, in principle, to get more light power from an LED than the input electrical power. In other words, an LED is capable of achieving an efficiency greater than 100 %, but how does that even make any sense? LED rainbow by steeph-k Let's start from how LED s work. The name LED is an abbreviation of the words Light Emitting Diode. More specifically, LEDs are usually p-n junction diodes . N-type diodes have an abundance of electrons, while p-types have holes (because if you take an electron out, it leaves a hole, right?). The electrons have a negative charge and holes positive, so they attract each other. When you make a junction out of these two types, you get a rather unique material, where the holes dominate one side and electrons the other. Now, you might think that when we slam these two materials together, the electrons will occupy the holes and the material as a whole becomes electrically neutral. Actually, this happens