There is a typo in the headline: Conterintutive -> Counterintuitive
Conterintuitive facts in mathematics, CS, and physics
151–160 of 335 posts
Re: Conterintuitive facts in mathematics, CS, and physics
#152> 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another. > 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood This one, on the other…
16 is like the money hall problem. I understand the answer, the answer makes sense to me. And yet but when I think of it how I think of it initially ... it still makes no sense.
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Re: Conterintuitive facts in mathematics, CS, and physics
#153Earlier quoted context omitted.
In this situation, the mass and weight are proportional and irrelevant to the problem. Other than proper respect of units, why would it really matter? I would agree that using mass+kg would remain correct and be less unusual, but it doesn't matter a lot.
It does not matter, it is just a pet peeve of mine. Might be due psychological trauma from when I was a kid and arguing with an older cousin about how pounds and kilograms are not units of the same thing, and the older cousin "winning" the argument in the eyes of the elders because the cousin was quite a few years older than me and considered to be smart in school.
To me (Netherlands) a pound is simply 0.5kg. Force is expressed in Newton.
Re: Conterintuitive facts in mathematics, CS, and physics
#154Number 19 is a clinker. Banach-Tarski applies only to objects in real-number space, but there are no such objects. For that to work, objects have to be infinitely divisible, but all of our objects are made out of atoms. Real-numbered space is a good enough approximation to our experience that we hardly ever encounter a model failure like this one.
Also this only works when the highly contested Axiom of Choice is used.
Re: Conterintuitive facts in mathematics, CS, and physics
#155Earlier quoted context omitted.
Hm, I'm not so sure that just because the definition of a bijection is technical that it is not intuitive. I'll start an enumeration of the rationals 1 1/3 2 1/4 3 1/5 ... If you can prove that you can do this, is that really so non-colloquial? It is certainly what we mean by "as many" for all finite sets, so what is wrong with doing this for infinitely many sets?
If every whole positive number is a fraction, but not every fraction is a whole positive number, then colloquially, I wouldn't define them as having "as many" elements as each other. Now, if you want to say they have the same cardinality (and you define cardinality as existing a bijection), then I would agree fully.
Wouldn't there be exactly twice (or twice + 1, if you allow negative fractions) as much fractions, since fractions are represented as two positive numbers (plus a bit if you consider the sign).
(The encoding could be "represent both numbers in binary, put the denominator in the odd bits (LSB = first bit), and the numerator in the even bits" so 2/3 => 10/11 => 1110 => 14)
Re: Conterintuitive facts in mathematics, CS, and physics
#156> 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another. > 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood This one, on the other…
Yes, it doesn't explicitly state the rate or distribution of events. But it is a good reminder of what happens when your whatevers/second are pretty high - see the famous "One in a million is next Tuesday" [1]. "Rare" is soon if you roll the dice fast enough. Any time your service has a high TPS, your API gets a lot of calls, a button in your app gets pressed a lot, ... this applies. Critically, "a lot" is defined re…
"Given the scale that Twitter is at, a one-in-a-million chance happens 500 times a day."
https://www.ted.com/talks/del_harvey_protecting_twitter_user...
Re: Conterintuitive facts in mathematics, CS, and physics
#157Here is one of my favorites. Take a light source and shine it at a filter that is polarized up/down that is in front of a filter that is polarized left/right. None of the light will get through: the first filter removes all of the L/R components of the light, and the second filter removes all the remaining light. Now add a third filter, between the two, which is polarized at 45 degrees. Now some of the light goes thr…
It makes more sense if you think of the polariser sheets as fences with vertical bars, and light like a string going through between two bars and being wiggled.
If the direction of the waves in the string are at right-angles to the slit, the wave can't propagate through the bars.
If the wave is at an angle, then some of the wave gets through, with a maximum of 100% when the wave wiggles up and down in the same direction as the slit.
In this model you can visualise how three 45-degree slits will allow some light to go through even though two 90 degree slits don't -- the light is transformed by its passage through the polarisers.
Where QM people get themselves confused is that they assume that the light is not transformed, even though it clearly is...
I.e.: If the polarisers didn't rotate the angle of the light waves, then successive aligned polarisers would result in very close to 0 light making it through, but this is not what happens. (Excluding losses due to non-polariser-related effects)
All of this goes back to a fundamental confusion between two possibilities:
1) Is it the EM field in free space that is quantised?
OR
2) Is it just the interactions of the EM field with matter that are quantised?
If you actually run the experiments to check which one, it turns out that (2) is true, but most QM people think (1) is true because most of the time they're indistinguishable.
See this video for such an experiment: https://www.youtube.com/watch?v=SDtAh9IwG-I
Re: Conterintuitive facts in mathematics, CS, and physics
#158Earlier quoted context omitted.
That one got me good, so my future at College Board is as bright as yours. However, I don't think the argument by demonstration in that video is particularly convincing. Instead, I think's its easier to note that that the _center_ of a circle of radius r travels 2 * pi * r distance over one rotation. In the problem, the center of the smaller circle has to travel further than the circumference of the bigger circle - i…
I don't think that argument holds water. Imagine rolling it on the inside of the circle – the center of the circle traces a path with only twice its radius, yet it still rolls 4 times around.
I don’t think that’s right. It if the radius of the inner circle is one third the outer, it only rotates twice rolling around the inside, which makes sense as it’s center traces a circle that’s the the difference of the radii.
Imagine the limiting case, as the inner circle approaches the size of the outer circle - the inner circle completes much less than one rotation per lap around the inside edge of the outer circle, and ‘seizes’ (if we’re imagining these as gears), completing zero rotations per lap when the circles are the same size. However, rolling around the outside, a circle of the same size completes two rotations.
In general the problem is like the old Spirograph toy (which I had to break out to convince myself)
Re: Conterintuitive facts in mathematics, CS, and physics
#159Earlier quoted context omitted.
That one got me good, so my future at College Board is as bright as yours. However, I don't think the argument by demonstration in that video is particularly convincing. Instead, I think's its easier to note that that the _center_ of a circle of radius r travels 2 * pi * r distance over one rotation. In the problem, the center of the smaller circle has to travel further than the circumference of the bigger circle - i…
Without the math: Radius is always proportional to circumference, so a circle twice the size is twice as big around. Take the case of two identical circles. To move a point on the first circle from 12 o'clock back to 12 o'clock, it only goes halfway around the other circle, which you can prove to yourself by imagining you've wrapped a string around the circle and marked it at 12 and 6 o'clock. If you unwrap half of t…
It’s interesting to read others reasoning on this. I find it hard to follow and much harder to generalize without some notation (and a diagram which this comment box struggles to reproduce)
Re: Conterintuitive facts in mathematics, CS, and physics
#160Earlier quoted context omitted.
Oh wow, learning a lot today. I was taught in the US that pounds are a measure of force, and that is how it was always used in physics problems.
As a European I learned in metric. When I first learned pounds, it was as the imperial system's equivalent of grams and a conversion factor was given. Force in physics class was taught in Newtons (kg*m/s^2).
Pounds as a mass unit are perverse enough. Things like pound force and psi (pounds per square inch) were used only to make fun of old mechanics papers and textbooks. Also, btu. It is quite amazing actually that someone would see the SI and think “no, too simple; I’ll keep my pounds, ounces, inches, and feet”.
Anyway, yes, the proper unit of force is the Newton.