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Showing posts with the label light speed

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 ...

Le Grand K announces retirement, at last

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Starting from 1889 and scheduled to become emeritus (emerita?) sometime in 2019, the International Prototype Kilogram (also known as Le Grand K) had a service of 130 years. Which is a remarkably long time for a standard of measurement based on a metallic cylinder. The change is rather massive (pun intended), because it affects every SI-system unit. However, the biggest change is not the kilogram, but the fact that some of the old SI units had a dependence on measured values. The new definition relates the base units to constants of nature, which will be redefined as being exact. The speed of light is a good example of setting some constant to be exact. We used to measure it's value, which leads to experimental error. But then in the 15th General Conference on Weights and Measures in 1975 it was decided to set it at exactly 299 792 458 meters per second . This also allowed us to define the meter in a very precise way, by establishing some temporal yardstick. Simil...

Speed of light - the ultimate speed limit?

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One thing that we are taught very early on in physics lectures is that the speed of light in vacuum, c, is the fastest anything can ever move, period. But is that really true? Well, the short answer is yes, and the long answer has some ifs and buts. Probably the most popular reasoning behind this statement is that Einsteins theory of relativity does not allow for anything to exceed the speed of light. But what it actually says is that information or energy cannot travel faster than light. Of course, even this is a very strict condition, since every particle, physical system and any radiation can be used to carry information and/or energy, which means that none of those can travel faster than light. To achieve a speed faster than c, you need something that doesn't contain information or energy, but what could that possibly be? One entity that potentially carries neither, is the group velocity of light. In optical telecommunication, group velocity is the speed of the...