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Showing posts from May, 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

The many problems of time travel

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Time travel is a popular concept in works of fantasy, with many different flavors. In the Back to the Future trilogy, Doc Brown built a time machine out of a DeLorean just because he could. In Harry Potter and the Prisoner of Azkaban, Hermione Granger uses a time-turner  so she could study every subject in Hogwarts, and even Superman turns back time to undo the death of Lois Lane. Seems like Superman really is quite super. But scientifically speaking, how much sense does time travel actually make? Let's start from the most obvious type of time travel: forward in time. Clearly we are all automatically traveling towards future time by default, since we age and seasons shift and whatever. But the interesting thing is that according to the theory of relativity, it is possible to slow down the passage of time for yourself, while everything else around you continues to age normally. This effect is called time dilation , and it is an important effect in the universes dynamics.

Entropy and the arrow of time

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When you whisk an egg and leave it alone, why doesn't the egg white and yolk separate on their own? If you light a fire, why does it keep burning? And why do we perceive that time never changes it's direction? All of these questions are answered by a single concept in physics: entropy. Simply put, entropy is a measure of the microscopic disorder of a system, and the second law of thermodynamics says that it can only increase or stay constant in a closed system. And all of this is just simple statistics. Let's put this in more concrete terms with a simple example: we flip three identical coins and record the outcome. First of all, what does this have to do with anything? In thermodynamics, we are interested in the energy of the studied system. That energy is made up of all of the individual energy states of the particles that make up the said system. The two sides of a coin represent the possible states of a two level system, which makes it quite a popular example.

Update on the double-slit experiment

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After two papers on something that I did not receive an education for, I finally got a topic in photonics, my bread and butter to be. I started working as a part of the coherence coherence group of UEF in the summer of 2015 and my first task was to design and build an interferometer, which would be used to quantify spatial coherence. Thin film interference. Physics is pretty, ain't it? Now, one might ask, what is spatial coherence? The idea behind this concept is simple: it describes how well different parts from a beams cross-section correlate with each other. If the correlation is strong, then we talk about high coherence and low when the correlations are weak. A very good overview of spatial coherence and its measurement can be found  here . The usual way to measure it is to use the  Young's interferometer  (a double slit experiment, variations of this are used in quantum mechanics), but there are some problems. First of all, a single measurement is not enough.