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 ...

Forced correlations & In search of lost rationality, part 2.

As promised, I'm back for some more lost rationality! And ranting. Yeah, mostly ranting.


But before I go there, I'll lay out the research that this rant is related to. This one was about temporal coherence, where we look at correlations between different points in time instead of space.

To measure those correlations, you need a device such as a Michelson interferometer where you split the incoming light to two, delay one of the copies, and then recombine them. This simple setup is sufficient only for some special cases, and measuring the correct correlations is way more complicated. There actually isn't a general method to do that, modern detectors are far too slow!

Let's just say that if you need to measure the temporal coherence of a pulse train, then one way to do it is to isolate individual pulses and measure them with a FROG (yes, that is a real scientific instrument) and then study their correlations from the measured data.

Anyhow, if you have a device that measures the temporal coherence of light, how do you know it gives correct results? You need to test it in practice, and to do that, you need some light that is not completely temporally coherent.

However, finding suitable sources that feature decreased temporal coherence is very difficult, so I needed to reduce it forcibly. When I started looking into this, there were exactly zero methods to decrease the temporal coherence of light in a controlled manner, so I had to come up with my own way.

Oh, how I tried. I spent countless hours in the lab with our femtosecond laser trying to degrade the coherence in such a way that I could reproduce it at will, and so that there would be no danger to the user or the machine. It was much harder than anticipated!

The biggest problem was that if I found a way to reduce the temporal coherence, it decreased way too much. From almost completely coherent to nearly incoherent, which was no good. I tried propagation through scattering and turbid media, and at one point we were pondering with my supervisors if we should try to boil the photons (is anyone else hungry?).

Then one day it hit me. Every single pulse shaper can be used to do what I wanted! You just need to change the shape of the pulse periodically and in a suitable manner. And, as you probably guessed it, changing pulse shape directly implies reduced temporal coherence. From that revelation, this paper was born, the end. Well not really the end, I still need to do lots of experiments with this setup.

The above picture shows one type of pulse shaper, which can be used to modulate the temporal coherence, if the spatial light modulator (SLM) is driven with a time dependent signal. At the input the pulses are exactly similar and at the output they are different when compared to each other, thus reducing coherence. Simple, right?

Now, we get to the ranting part. We faced unbelievable difficulties when we tried to submit it to our first choice journal. Our manuscript was assigned to an editor who flat out rejected it without peer review. He even refused the theory we used, which is the cutting edge of coherence research and widely used. It's even central to many published articles in the said journal!

Then he sent us an e-mail where he tried to argue that the theory we use does not work at all. He had even attached a Mathetica file where he tried to do some calculations. That did not work. At all. He made some very elementary mistakes, such as taking absolute values before averaging (cause \( |z_1+z_2+...+z_N|\), is not the same as \( |z_1|+|z_2|+...+|z_N|\), as you probably know) and his logic was full of holes.

We tried to explain our point as clearly as possible but he kept rambling on about how we somehow believe that there exist completely coherent lasers (something we never said) and that even our experimental methodology is completely off and we know nothing about measuring ultrashort pulses and so on.

In short, he had decided that we were wrong and he was right. And there was nothing that could change that.

While dealing with this editor, he referred to himself as "the editor from hell" and said that we have probably never met anyone like him before. His words, not mine. Yeah. But he was right, since my professors who have authored several hundreds of papers each in peer reviewed journals, told me that this was completely unprecedented.

At one point "the editor from hell" called my professor directly and they argued on the phone for almost two hours. It got really loud and ugly, and there were no winners. But you know what was the most interesting plot twist in all of this? Before this debacle, he had gotten the recognition of being an "outstanding referee."

Finally, after a loooong time going back and forth with this, we sent the manuscript to a different journal, where it was accepted almost as is, and it was even featured as the article of the week.

Go figure.


By C Ding, M Koivurova, J Turunen, T Setälä, A T Friberg
Published in Journal of Optics 19 (9), 095501

Links:
Journal of optics (published version)
ResearchGate (accepted version)

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