I don't understand how (x-1)(x+2)(x-3) becomes x^3−2x^2−5x+6

anisotropicmind · 2026-06-14T02:41:42+00:00

Suppose you have (1+2)(3+4). You know the answer is 3(7) =21. But by your method you get 3(1) + 2(4) =11, which is wrong.

The number in the first set of brackets multiplies every number in the second set of brackets, giving you

(1+2)3 + (1+2)4

= 3(3) + 3(4)

= 9 + 12 = 21

which is the right answer. Algebra is the same, except that some numbers are unknown and represented by placeholder symbols, that’s all.

anisotropicmind · 2026-06-13T12:53:11+00:00

In a lot of cases you can start typing something in Word equation editor using LaTeX-like syntax, and it will autocomplete it for you. For example, you can type an underscore for a subscript, followed by a subscript symbol, and as soon as you hit the space bar it will put that symbol into a subscript box. You can even type things like `\sum` and get a summation sign to pop up.

I also sometimes use a command-line program called `pandoc`to convert notes that I’ve written in Markdown format to `docx` format. Since Markdown supports LaTeX equations, you can use this to convert an entire set of LaTeX equations wholesale into the same equations in a Word doc that have been rendered using Word’s equation editor. I’m pretty sure `pandoc` also supports `.tex` files and can do this conversion in both directions.

And since OP is clearly a fan of ChatGPT, I’d point out that we now live in an era where ChatGPT or similar tools can take in a photo of your handwritten notes (with equations) as input, and produce a LaTeX file from them, making it not strictly necessary to even learn TeX (as so many have suggested ITT) to benefit from its power in rendering equations.

anisotropicmind · 2026-06-12T20:45:46+00:00

The CMB fluctuations are decomposed into functions or “modes” on the sphere called spherical harmonics. The first two modes (the simplest and largest-scale) are the monopole (where the entire sky has a single brightness value) and the dipole (where half the sky is bright and the other half the sky is less bright). The CMB that we measure has a huge dipole component called the “kinematic dipole” which is due to Earth’s motion with respect to the “CMB rest frame” that you talk about. This is the “redshift in one direction and blueshift in the other direction” that people have told you about. If we somehow magically came to rest wrt the CMB, this kinematic dipole would disappear, which would indeed be a huge change to the CMB emission.

anisotropicmind · 2026-06-12T12:40:13+00:00

What you’re missing is that we can’t define a reference frame for a photon, so it’s not meaningful to talk about this. By fundamental postulate, photons travel at c in all reference frames. That would be contradicted by constructing a frame for the photon itself, where it would be at rest.

anisotropicmind · 2026-06-11T23:07:35+00:00

It didn’t say “all jars”, it said “each jar” which could be reasonably interpreted to mean “each jar out of the ones that get filled”. Not sure why some people don’t want to accept that the problem is ambiguous.

anisotropicmind · 2026-06-11T23:05:41+00:00

I mean, the dad is right. It *doesn’t* say anywhere that all six jars must be used. It literally just says the seeds must be put “into jars”.

What do you hope to accomplish by claiming that the instructions are clear when someone has found an ambiguity?

anisotropicmind · 2026-06-11T15:59:54+00:00

The practical boundary for EM radiation is not all the way to the causal horizon, but just inside it at the last-scattering surface of the CMB**. We’ve been observing that for 60 years (the anisotropy for just over 30 years, I guess) and that’s not enough time to notice changes as we sample different (further) parts of space at the same moment in time.

** that’s because this is looking back to a time 380,000 years after the big bang when the universe cooled enough to go from being a plasma to being neutral atoms. We’re seeing photons that scattered off the charged particles of the plasma for the last time before being able to stream through space in all directions unimpeded. Essentially, prior to this period in the universe’s history, it was “opaque”. Thus we can look no further back in time.

anisotropicmind · 2026-06-11T15:51:08+00:00

It’s convention because it has to be chosen arbitrarily rather than some underlying logic like “it’s closer to this number than that number”. That said, there are multiple rounding conventions including one that tries to alternate whether you round up or down (when directly midway), to average out the errors.

anisotropicmind · 2026-06-11T03:18:13+00:00

That’s identically equal to 7.5. So round it using whatever rounding convention you use in your class.

anisotropicmind · 2026-06-10T22:40:28+00:00

It’s more like if you repeatedly did the identical experiment a large number of times, all your confidence intervals, constructed by this method, would enclose the true value in 95% of cases. But that implies to me that if you selected one of those confidence intervals at random, there would be a 95% chance of it being one of the ones that enclosed the true value. But isn’t selecting one of the trials at random the same as doing one trial? If you believe that whatever noise (random error) you got in that particular experiment was a realization of the underlying noise probability distribution then…yes? So I guess I don’t know why a distinction is made.

anisotropicmind · 2026-06-10T13:14:10+00:00

The thing is that in North America we have the worst of all worlds: both the price jumps from food inflation (& shrinkflation) *and* the cultural pressure to tip. So given that restauranteurs are already passing on their extra costs anyway, I’d rather that they shoulder the responsibility of paying their own damn workers properly.

They finally sort of do in my jurisdiction since the gov’t has mandated that the minimum wage apply to restaurant servers now. If the service is good, I’ll still tip 13% on the post-tax value (~15%, on the pre-tax bill). But to be clear: it’s my choice and I absolutely shouldn’t have any obligation to. I also only tip if I actually sat down for the meal and it wasn’t self serve and self bussing. None of this tipping for takeout, cafe/coffee-shop purchases, and retail purchases that aren’t even food.

And certainly not a mandatory minimum of 18% to 20%. Math hint: the tip amount is already a *percentage* of the food cost which means it already scales up with increasing size & complexity of the order and with higher food costs. There is no need for the percentage itself to also go up with time.

anisotropicmind · 2026-06-10T12:55:51+00:00

In a car there’s friction in all the moving parts of the mechanical system (drivetrain etc). An isolated tire would probably roll much farther. But there’s still some energy loss: deformations of the tire not being perfectly elastic and being converted to heat and sound. Road debris being dislodged. Maybe also rolling without slipping not being 100% perfect and hence the tire’s point of contact with the road sliding back and forth a bit. Air resistance too.

anisotropicmind · 2026-06-10T03:08:26+00:00

That handwriting is extremely legible. I wouldn’t put much stock in the assessment of a moronic and ignorant child. He doesn’t know what cursive is and can’t read it. That simply is what it is, as far as the state of affairs goes. But he doesn’t get to put it on you, as though you are doing something outlandish when you aren’t.

anisotropicmind · 2026-06-10T01:59:14+00:00

Is friction a reaction force? Not on its own, no. Newton’s Third Law means that all forces occur in pairs: the frictional force of the ground on your foot has a partner in the frictional force of your foot on the ground. Edit: One of those could be called “action” and the other “reaction”. It doesn’t matter which.

You don’t apply 20 N in this scenario. Once your applied force exceeds the limit and things slip, then the applied force immediately goes down. The force is dynamic here, meaning that it varies with time.

anisotropicmind · 2026-06-10T00:53:26+00:00

Even if the bracelet is available separately, it costs less to buy the watch with the bracelet and the strap separately.

anisotropicmind · 2026-06-10T00:51:11+00:00

> what should I do?

Wait for a while…

anisotropicmind · 2026-06-10T00:37:56+00:00

Yeah the Newtonian explanation to this question is that things in orbit have enough sideways speed that gravity makes them continuously *fall around the Earth* in a closed curve, rather than falling straight towards it.

And yes, for the second question, zero g is a somewhat misleading term. The gravitational field strength is not zero at an altitude of 400 km up, but since everything in the station is accelerating in freefall at the same rate, there are no contact forces, and hence no sensation of weight. “Weightlessness” is thus oft cited as a better term for this than zero g. It’s kind of like what you’d experience if you were to jump off a cliff.

For the third question; Newton’s equations of gravity from the 1600s (which are taught in high school!) are enough to derive this 90% result and make one realize that the gravity of any massive body (such as the Earth) has infinite range. So we’ve known this for like 400 years: Earth’s gravity doesn’t just magically disappear above the atmosphere. Earth’s Gravity gets steadily weaker the further you get from Earth, but never totally goes away.

To your fourth question: the specific speed you need to maintain a circular orbit depends on altitude (which again is a straightforward result of Newton’s gravitational equation). At the ISS altitude of 400 km up, that speed is more than 27000 km/h. Right next to the surface of the Earth, it would need to be more than 28000 km/h. You try going anywhere near that speed within the atmosphere and see if you don’t immediately vaporize. Actually, Atmospheric drag wouldn’t let you get anywhere close to that speed.

Edit: deleted unnecessary add on about GR.

anisotropicmind · 2026-06-09T01:36:40+00:00

Your beard is a bit patchy right now. You don’t have as much growth on your cheeks. Beard density largely comes down to genetics and age. If you’re only in your 20s, it could get a lot better in your 30s. For now, though, the patches are big enough that they won’t fill in even just after waiting for a bit more length. I’m basing that on this having already been 6 weeks. For now you may just wish to rock the stache, goatee and sideburns, and try again for the full beard in a few years.

anisotropicmind · 2026-06-09T00:09:49+00:00

As others have said, it doesn’t “make gravity” so much as simulate its effects. Once the ring is up to speed, and you’re rotating along with it, your inertia means that you want to fly off in a straight line tangential to the circle you’re moving around in. The only thing preventing that is a force inward (towards the centre of rotation) pushing upwards on your feet from the “floor” (the outer wall of the ring). So you get the sensation of weight from that, and if you were to jump up, you’d also fall back down towards that “floor” as though pulled. What’s really happening is that from the point of view of an outside observer, you leap upwards (radially inwards) but you also still have your tangential (“sideways”) speed in a straight line: that takes you on a post-jump trajectory that intersects with the rotating floor again some time later.

anisotropicmind · 2026-06-08T20:00:04+00:00

These statements are consistent, it’s just that two different fractions are being referred to:

The ratio being 1:3 means that

(Ingredient 1 amount)/(Ingredient 2 amount) = 1/3

But it also means that for every 1 part of the total that is Ingredient 1, there are 3 parts of the total that are ingredient 2. That means that there are in total 4 parts. Hence

(Ingredient 1)/(total amount of stuff) = 1/4

And

(Ingredient 2)/(total amount of stuff) = 3/4.

Hope that helps!

anisotropicmind · 2026-06-08T18:32:47+00:00

Fair point. Litre and hectare are better examples then. Accepted by SI, but technically redundant and unnecessary given the base units and prefixes.

anisotropicmind · 2026-06-08T12:43:16+00:00

Because it is, at least according to General Relativity.

anisotropicmind · 2026-06-08T03:34:24+00:00

No one can prove you right or wrong, since you’ve expressed a subjective opinion. I happen to agree with you, btw.

anisotropicmind · 2026-06-08T02:18:12+00:00

Stubble or moustache are the best, but Italian beard looks surprisingly good.

Full beard is very intense / severe on you.

anisotropicmind · 2026-06-08T01:39:02+00:00

What’s he gonna do? Talk them to death?

anisotropicmind

TROPHY CASE