A different approach at convincing someone why 0.9 repeating is equal to 1

For some reason some people have an extremely difficult time accepting that 0.9 repeating is equal to 1. Not that it merely "approaches" 1, but that it's exactly equal to 1. They are just two different ways to write down the same value.

Some people are so incredibly obsessed with trying to prove that they are not equal that they will go to incredible lengths to try to do so. They will start arguing semantics, they will try to muddle the definition of an infinitely repeating decimal, and some may even attempt to invent completely new mathematics in order to somehow make the two things not equal. They are so obsessed with this that there's absolutely nothing you can tell them that would convince them otherwise. Nothing. You can try, but you will fail.

Regardless, even if it's rather moot (and will never, ever convince these people), here are two slightly different approaches at showing the equality of the two things. Instead of trying to prove it yourself, try to make them do the work.

Approach 1: Repeating decimal patterns as a fraction

It's a well-known result that every real number which decimal representation has an infinitely repeating pattern starting at some point after the decimal point is a rational number, and this is actually relatively easy to prove. And, in fact, this is a (well-known) one-to-one relationship: In other words, if the decimal representation of a number has an infinitely repeating pattern after the decimal point (not necessarily starting immediately after the decimal point, but from some point forward after that), it is a rational number, and if it doesn't have such a pattern, it's an irrational number.

This can be more succinctly (and mathematically) expressed as: A real number is rational if and only if its decimal expansion is eventually periodic.

And since such a value is a rational number, it can be written as a fraction, ie. the ratio between two integers. And, indeed, all values whose decimal representation has an infinitely repeating pattern starting at some point after the decimal point can be written as a ratio of two integers, ie. a fraction.

As an example 0.4 repeating is a rational number and can be written as 4/9.

Since this is a proven mathematical fact (and it's actually relatively easy to prove yourself), that means that 0.9 repeating is also a rational number which can be written as a fraction, ie. the ratio between two integers.

So the question is: Given that proven mathematical fact, find out what those two integers are. In other words, what is the fraction that gives 0.9 repeating.

If you want to present the argument to someone succinctly, it could be something like this:

"It's a known result that a real number is rational if and only if its decimal expansion is eventually periodic. This is easy to prove. That means that 0.9 repeating is a rational number. This also means that, as a rational number, it can be written as the ratio of two integers. Calculate what those two integers are."

Approach 2: Calculate the difference

If two values are equal, then their difference, ie. one subtracted from the other, is 0, pretty much by definition.

If two values are not equal, then their subtraction will differ from 0, again pretty much by definition.

Thus, if the real number 1 is different from the real number 0.9 repeating, calculate their difference, ie. the result of their subtraction. If they are indeed not equal, then the result has to be a real number that's different from 0. What is that real number?

(Note: There is no such a thing as "the smallest real number larger than zero". Such a thing does not exist, and it's logically and mathematically impossible to exist, especially in the set of real numbers. This is a quite famous and extremely trivially provable fact of arithmetic.)

This could be succinctly presented as:

"By axiomatic definition, if two real numbers are equal, their subtraction results in 0. Conversely, if two real numbers are not equal, their subtraction results in a non-zero real number. Calculate the subtraction of 1 and 0.9 repeating."

Why is the triple-slit experiment so uninteresting to scientists?

Some time in the early 2000's I read an article about how a laboratory had conducted a three-slit version of the famous double-slit experiment for the first time in literally 200 years that the double-slit experiment was a thing that physicists were interested in. The experiment appeared to confirm the predictions.

Even then, the experiment was flawed: It turned out that one of the slits didn't close completely, and left a tiny gap even when it was supposed to be closed. But instead of fixing the mechanical issue and redoing the experiment, they just mathematically compensated for the flaw.

For reasons that were never explained in the article.

All of the above is completely incomprehensible to me.

The double-slit experiment is one of the most famous experiments in quantum mechanics, and in fact in the entirety physics. It's one of the most studied and researched experiments in human history. Thousands and thousands of research papers have been written about it, and it's one of the most fundamental experiments that underline the entirety of quantum mechanics and particle physics, and has immensely profound implications to our understanding of the universe. It's an experiment that has been repeated probably hundreds of thousands if not millions times over the last 200 years. Expensive high-tech labs conduct the experiment, physicists conduct the experiment, physics students conduct the experiment, probably by the thousands every single day.

The double-slit experiment is probably one of the single experiments that has received most work and research in the entire history of humanity.

Thus, one would think that the triple-slit version of the experiment would be of similar interest to physicists.

But astonishing that doesn't appear to be so. It took a whopping 200 years before someone did the experiment in a laboratory setting. And this even though the technology to do so has existed for something like a hundred years. It took 200 years for anybody to do the experiment and publish a paper about it. The article quite specifically mentioned that this was the first time that anybody had done so.

And even then, the experiment was flawed, but apparently the authors were so uninterested in the entire thing that they couldn't even be bothered to fix the flaw and run the experiment again. That's how utterly unimportant they seemed to think it was.

On top of that, this paper in question, as well as the article talking about it, was not considered any sort of landmark experiment worthy of notoriety. The paper (and article) in question appears to be so non-notorious, so forgotten, that I can't even find it anymore, no matter how much googling I do. From all I have found, it appears to have completely disappeared from the internet.

I cannot even begin to comprehend this. This complete and utter lack of any interest in the triple-slit experiment, considering how fundamental and ground-breaking the double-slit version is. It just doesn't make any sense.

The vast majority of black holes do NOT have an accretion disc

The video game Elite: Dangerous depicts the Milky Way galaxy, including the numerous black holes that are known to exist there. If you go to such a black hole, it will be essentially invisible, the only way that it exhibits itself is by how it distorts the background (ie. gravitational lensing.)

Many people believe this to be unrealistic because they have this misconception, no doubt spread most prominently by movies like Interstellar, that all black holes have a very prominent bright accretion disc around them. In fact, when mentioning black holes, most people probably envision the image of the black hole in Interstellar in their minds.

In fact, there are some YouTube videos that take footage from Elite: Dangerous depicting a black hole and adding an accretion disc in post-processing, and the comment section of such videos will invariably be full of people commenting on how much more realistic it looks and wishing that the game implemented that kind of visuals.

The problem? The way that the game depicts stellar-mass black holes is actually realistic!

The fact is that stellar-mass black holes do not have any sort of visible accretion disc. They are too small for that. They are, indeed, pretty much like the game depicts them: Completely invisible, them being "visible" only in how they distort the background. There's nothing visible orbiting them.

If a stellar-mass black hole has a very closeby normal star, gas from the star may be falling into the black hole in a spiral pattern forming a sort of accretion disc, but even then it would be extremely faint, probably too faint to be seen by eye.

Only supermassive black holes in the centers of galaxies (and a few other places) may have a prominent visible accretion disc. And even with those it's theorized that not all of them might have one (for example, it's currently unknown if Sagittarius A*, the supermassive black hole in the center of our Milky Way galaxy, has a visible accretion disc. It's not a given that it has one.)

In other words, the depiction of the black hole in the movie Interstellar is not unrealistic either, but that's because it's a supermassive black hole, not a stellar-mass one.