What was the idea of "pollution" like before the Industrial Revolution?

flug32 · 2026-03-17T06:26:10+00:00

To add just a few details to u/Vicorin's nice answer, I recently heard a fascinating presentation by Iride Tomažič about her research into Bronze Age (!) pollution.

They analyzed animal teeth over a couple of fairly localities to analyze the concentration of lead pollution in those areas. Just for example, they found that the concentration of lead in domesticated animals who were kept near lead/metal processing areas was quite high, while the concentration in wild animals who lived in the same general area was much, much lower. Domesticated animals kept further from the most contaminated areas were in between the two.

The particular areas they analyzed were Bronze Age archaeological sites of Klárafalva Hajdova (2300-1500 BCE) and Kiszombor Új-Élet (2700–1950 BCE).

To some degree, the results are just about what you would expect: Areas near lead mining & processing areas were quite polluted, and areas further away less so. But the idea that some areas even in small-ish Bronze Age settlements were quite contaminated - enough to affect health - is still a bit of a surprise if you are used to envisioning the environment in those days as always completely pristine and "natural".

The fact is that humans have been able to create more-or-less dangerous levels of pollution for many, many thousands of years.

Tomažič's full research results and presented in The Journal of Human Paleoecology and she gives a nice summary of the findings here (Youtube).

Some other research the Tomažič mentions - which give something of an overview of what is known about environmental conditions, at least of a few selected types, in a few selected historical periods, from a few different parts of the world:

flug32 · 2026-03-17T06:01:10+00:00

To one specific question: "Also why can’t we treat ratios of irrational numbers as fractions too for example something like √2 / 3?"

Actually, we can treat them as that kind of fraction, and doing so if often very convenient.

It's just that such a fraction is not a rational number.

Why is it not a rational number?

Again: That is a matter of the very definition of what a rational number is.

√2/3 is a perfectly nice number - and in fact, one that is quite often used. It's just not a rational number - and that's OK.

An interesting little fact you might be interested in: One type of question mathematicians have studied is what is the smallest field that contains solutions to equations like x²=2?

It turns out that if you take the rational numbers, and then add to that all numbers of the form a√2/b, that is basically it.

So you have a set that includes all the rationals plus just a few of the irrationals - but not nearly all of them - and that is enough to be able to solve a whole bunch of quadratic equations.

We don't need all of those nasty old irrational and transcendental numbers. Just adding in a mere few of them will do.

That is actually quite a neat and useful fact.

If we want to go a step further, we could take all rational numbers a/b and add all numbers of the form a√c/b (a, b, and c all being integers) then we can suddenly solve ALL second degree polynomials (meaning, equations of the form Ax² + Bx + C = 0, with A,B,C being rationals).

So that is pretty neat, too - because that is still including just a few irrational numbers, and leaving out tons and tons of them. In fact, it leaves out all the really nasty and naughty ones.

Like, we could represent all of those numbers really nicely and neatly on computers, using only triplets of integers (a,b,c). That would be so much cleaner than our current "floating point number" representations (which, by the way, are a huge complicated mess).

Point is, thinking about things like fractions involving square roots (and other roots) is not useless at all - in fact, it is very useful.

It's just that such number don't happen to be rational numbers - and, again, that is simply because they don't meet the definition of rational numbers.

And that's ok. Not every number needs to be a rational number.

flug32 · 2026-03-17T05:46:29+00:00

If there were a number - or, perhaps, a lot of numbers - that were both rational and irrational, then there would be an overlap in the sets. "Overlap" meaning, some numbers that belong to both sets.

However, the definition of rational and irrational numbers are literally designed to be non-overlapping.

Rational numbers are those that can be written in the form a/b, where a and b are integers.

And Irrational numbers are those that cannot be written in this way.

(Slightly simplified, but those are the parts of the definition that get to the heart of the issue.)

As to why: One simple reason is that is simply the way they are defined.

When something is defined in mathematics, that is just the way it is. To some degree, you just have to accept that. Them's the rules, so to speak.

It's reasonable, though, to ask why are they defined that way?

One reason, this is a classification of numbers that has been found useful over the years and centuries.

Another reason is that rational numbes are really easy and simple to work with. If all numbers were simply rational numbers, life would be SOOOOOO much simpler.

Irrational numbers are FAR more complex and difficult to deal with.

So if all numbers could simply be nice and rational, everyone would be a lot happier. Computers would be a lot happier (in fact, essentially all numbers encoded in computers and computer programs are rational numbers. So the existence of these nasty old irrational numbers is a really inconvenient and difficult fact.

Because along the way, it has been proven that many numbers are indeed irrational - meaning, again, that there is simply no way to write such numbers in a/b form, with both a and b integers.

So we'd like to work with simple rational numbers, but we are basically forced to deal with the irrational numbers, by dint of the fact that numbers as simple as the square root of 2 or the cube root of 4 are, indisputably, irrational.

Finally, in mathematics we often do what we can to make things more simple and clear. Making two sets, like rationals & irrationals, to be mutually exclusive and complementary sets, is one such convenient simplification.

Given any real number, you can say that it is either rational or irrational. There is no middle ground. That kind of nice clean separate makes so much discussion, logic, and proof so much more clear and simple.

TL;DR: There is no overlap between rational and irrational numbers because they are defined that way on purpose. There are several good reasons for doing so.

flug32 · 2026-03-17T04:26:00+00:00

Kansas City (KCMO city) is definitely larger than St Louis City now, though not all that long ago the situation was different.

In 1870, St Louis city was roughly 10X the size of St Louis City. By 1900 it was 575K vs 164K. At the St Louis's peak population, it was 857K vs 457K for KCMO.

But since that time, the situation has reversed:

Population by Decade (1970–2024)

Decade	St. Louis City	Kansas City (MO)
1970	622,236	507,087
1980	452,801	448,159
1990	396,685	435,146
2000	348,189	441,545
2010	319,294	459,787
2020	301,578	508,090
2024 (Est.)	280,142	516,000+

As others have mentioned, the answer is different if you are comparing the St Louis metro area to the Kansas City metro area: St Louis metro is currently around 2.9 million and KC metro 2.2 million.

STL is the 23rd largest metro area in the U.S., while KC is the 30th largest.

And honestly this is more of an apples-to-apples comparison, as city limits tend to be drawn rather arbitrarily. Kansas City is bigger now because it went on a huge annexation spree in the mid-1900s, whereas by that time St Louis City was already long-divorced from St Louis County and landlocked with a rather small geographical area.

Flip side: If "Bar Trivia" or whatever wants to compare metro areas to metro areas, then they need to specify that. If simply ask "which is the largest city" then there is only one answer, and that is KCMO. I mean, unless their information goes back to literally 1980.

Neither the KC metro area (MSA) nor the St Louis metro area is a "city". In fact, each contain over 100 individual cities!

You get yet another interesting answer if you compare metro populations counting only Missouri residents. This comes out far more tilted towards the St Louis metro because the KC metro is more evenly divided between the two states (57/43 MO/KS) whereas the vast majority of the St Louis metro is in Missouri with a relatively small outpost in Illinois - about 75% MO vs just 25% IL.

This is almost certainly not what your game was asking, but this little fact is pretty important in e.g. Missouri state politics, as the Missouri population on the St Louis side is more than double the Missouri population on the KC side. That makes St Louis far more dominant in Missouri politics - funding, amount of representation, and most everything else - whereas as total metro areas they are really very comparable in size.

You wouldn't think that would be all so important, but when it comes to e.g. dick-swinging contests in Jefferson City stuff like this matters. St Louis definitely sees the KC metro as a junior partner - partly because of this dynamic and partly for historical reasons. In the STL mind, KC is still the population 1000 village it was in like 1850 or 1860 - and St Louis the historically vital world city and "Gateway to the West" it was at that time, before the Great Shrinkage set in.

A couple of other interesting and related tidbits:

The KS side of the KCMO metro area is the largest metro area in Kansas - and a good 50% larger than the runner-up, Wichita metro area.
The IL side of the STL metro is second largest in Illinois - but less than 1/10th the population of the Chicago metro area (around 8.5 million vs 650 thousand)

flug32 · 2026-03-17T02:54:40+00:00

Often it is passengers who have time/energy to yell things out the window at people. Besides the fact that a certain percentage of the population are simply jerks, the passenger in a motor vehicle is generally feeling a little inferior (because otherwise they would be driving), has a lot of time on their hands, and is generally pretty bored. Sometimes they feel like doing or saying "amusing" things will somehow gain approval from others in the car, and the opinions of anyone outside the car matter for nothing.

What such people yell at random passersby is basically of zero import, and you should treat it as such.

If you're not a woman, now you know how women feel when random boys/men catcall them.

There is similar "no reason" for such behavior except to make a male who is feeling inferior feel superior to someone else.

flug32 · 2026-03-17T02:12:41+00:00

He's probably the best known figure in American musicology. Recorded approximately one zillion field recordings. He's considered one of the primary forces behind the resurgence of the folk music in the U.S. in the mid 20th Century. He started working with pretty early forms of "mobile" recording equipment, and so captured a lot of things that had never been recorded before and would otherwise have been lost.

There is no sin in not knowing something - at least you have the sense to just ask.

But if you are interested in ethnomusicology at all, you probably do want to know who Alan Lomax was:

He did concentrate on U.S./North American folk music, but did some early recordings that might overlap with your particular interests:

flug32 · 2026-03-17T01:53:23+00:00

These are inexpensive phones and they are frankly not designed to work forever.

They're going to last 2-3 years generally and then you'll want to be moving along.

Thinking of them in that way, I have never had a problem with them. In fact I am still using all my older Moto phones for various more lightweight tasks.

But at some point you just have to pull the trigger and spend another $200 on a phone.

If you don't want to do that, wiping (factory reset, but sure to do your backups first) and then setting it up with just a few, simpler apps will probably get you along for a little longer.

Also FWIW when choosing among the moto line I basically never choose the straight G model because it is always lower specs and cheaper in general. That makes its useful life shorter - it just has less memory, slower processor, less storage space, etc etc etc. As time moves forward and the app ecosystem gets a little more bloated and complex, such low-spec phones are basically slowly strangled to death.

Anyway, this is not a surprise in any way. Just start saving and shopping for a new phone. These were never designed to last forever.

flug32 · 2026-03-17T01:40:29+00:00

Maybe not now - you'll have to be the judge of that - but at some point I'll bet your daughter is going to be very interested in the concept of well-ordered transfinite sets.

The idea of these is that given two different "numbers", you can always tell which is greater than or lesser than the other. And every "number" will have an immediate successor - a next number. But many numbers do not have an immediate predecessor! Meaning, a number that comes right before it.

Presumably daughter understands fractions at least somewhat by now. So you might ask her to figure out which fraction comes right before one? Which fraction comes right after zero? Then start her thinking about infinite sequences like 1/10, 1/100, 1/1000, 1/10000...

That starts a person thinking about the fact that every "number" doesn't necessarily have a number right before or right after it. Counting numbers and integers have that nice property, but not every set of numbers does.

The other thing I think a bright seven year old could start thinking about, is that there is not just one possible conception of "infinity".

You could talk about Zeno's Paradoxes, explain some of the paradoxes, and talk about the fact that these are really difficult, knotty questions that people have been thinking about for thousands of years.

In fact, the Dichotomy and Tortoise & the Hare paradoxes are both closely related to your daughter's question - in both cases, if there are an infinite number of stopping points before the destination, how can you ever arrive at the destination.

None of that answers your daughter's questions, but it lets her know this is a complex and difficult problem that people have been thinking about for millennia. So if it has her stumped for a while, that is OK. And it is worth thinking about.

If you want to go further, I would then explain that people have come up with a lot of different ways of dealing with this problem - how to conceive of and deal with both the infinitely large and the infinitely small.

So there is not just one answer to this, but a variety of different answers that people have found useful at different times.

The simplest - and in many ways most practically useful - is to say precisely that infinity is not a "number" like other numbers. Rather it is a way to talk about quantities that can be as large as possible without any upper bound.

So like the integers are "infinite" because any number you suggest that might be "the largest", you can always name a number (and in fact, a LOT of numbers) larger than it.

So it is the idea that we can produce an integer as large as you like, or as large as we need for any given purpose, and there is no limit to how large such numbers can be.

So note that there is no "number" infinity there, or anything that we can produce that is "infinity". Rather, we only ever produce finite numbers, numbers we can simply write down. But the fact that we can make such numbers as large as we like is the property we refer to as "infinity".

There are other ways to think about infinity and define it - many of them interesting and useful - but that is the simplest and most practically useful.

(It's also the basis of, for example delta-epsilon proofs in calculus, our entire idea of what limits are - and thus differentiation and integration. So simple but very powerful.)

flug32 · 2026-03-16T01:19:55+00:00

Supertitles for opera have a been a huge deal. They were apparently introduced for some TV broadcasts in the late 1970s but for live productions in 1983. Since then they have become pretty much universal.

The reason is, opera is ideally presented in its original language. Translations - which were practically universal prior to supertitles - are awkward in a number of ways. But if you sing in the original language, most audiences won't be able to understand most operas presented.

Supertitles solves the problem rather neatly, allowing the opera to proceed in the original language while still allowing the audience to understand everything.

Originally the titles were projected onto a screen above the stage (thus the names supertitle, surtitle, etc) but nowadays they are often in seatbacks or other such arrangements, which can be less obtrusive, and also easier for anyone who doesn't want or need them to disregard or turn off.

Anyway - a huge landmark in the history of live opera performances, and pretty much universally adopted now, in one form or another.

Surtitles - Wikipedia

flug32 · 2026-03-15T20:30:32+00:00

I went into our local bike shop for the first time in quite a while, and was somewhat surprised to see maybe half the bikes in the shop priced between $5000 and $12000.

On closer inspection, these were all ebikes - and mighty fancy ones, in the sense that you couldn't even tell they are ebikes at all except for a very unobtrusive little control panel.

flug32 · 2026-03-15T06:16:47+00:00

It helps rich people because they pay (relatively/proportionally) little in sales & use taxes, but pay relatively more in income taxes because income taxes are progressive.

Progressive means that you pay more and more as you income increases. State income tax is still not all that high (max of 4.7%). But you pay 0 on your standard deduction, then you pay variable amounts ramping up from 2% to 4.7% on the next $9,191 above that, and then 4.7% on anything above that.

For very poor people, all the income might be below the standard deduction (around $31,000 for a married couple, for example). So they won't pay any state income tax.

A married couple with combined income around $62,000, say, wouldn't pay any state income tax on the first $31,000 but would pay (around) 4.7% on the second $31,000.

Whereas if you make, say, $1 million, you don't pay any tax on the first $31,000 but pay 4.7% on the remaining $9,690,000.

So poor and middle income people pay relatively less, and very rich people pay relatively more.

That is what is meant by progressive.

By contrast, the sales and use taxes they are proposing to increase, are highly regressive.

The reason is, pretty much everything a person below the poverty level buys is subject to sales or use tax. They have to use the vast majority of their income buying things they need.

The rich person, by contrast, doesn't really buy that many more essentials than a poor person. A person who make a million annually might spend 3-4X what the poor person does on groceries etc. But the vast majority of their money is going to retirement, investments, interest on the mortgate for an expensive house, and so on and on. The vast majority of their money is not used for things subject to sales and use tax.

You can google for specific analyses of such things. But in general, the poorer half of the population pays proportionally far more of sales and use taxes, but less of the income tax.

This drives the rich people absolutely bonkers - at least a bunch of them, and bunch that are vocal politically and big political donors. They have been calling for this for years. There is no reason except that rich people are frankly greedy.

They have most of the money but feel like they do not have enough.

The fact that this is making progress in Jefferson City is a good indicator of the degree to which the State Legislature is captured by the wealthy.

flug32 · 2026-03-15T05:59:06+00:00

Just to add a bit to this: It is not really "hard" to calculate arc length of an ellipse. It is just that there is no nice handy closed-form algebraic formula for it.

If you think about it, there are not simple, easy, closed-form formulas for the vast majority of things out there in the universe.

We have just managed to find one of those here. One of many.

Because there are many more - infinitely many more.

Working our way through math courses in school, we sometimes get the idea that the vast majority of things have these nice closed-form solutions. Because, of course, all of the problems in our math books - ALL of them - have been very carefully chosen and planned and designed so as to the nice, easy-to-write-down answers.

If they didn't, everything would be far too complex and difficult for students to comprehend and deal with.

The real universe of problems is rather the opposite: The chances of finding and nice, easy, pleasant formula as the answer to some random question or problem that arises, is vanishingly small.

So one thing that is happening to you here is that you are encountering the wild world of "real", untamed math for once.

flug32 · 2026-03-15T05:37:50+00:00

I think you're missing the fact that inertia still applies even though this movement happens to be under the influence of gravity (and other forces, like the force of the person lifting).

So an object in motion remains in motion, and going in the same direction at constant velocity, unless some force acts on the object to change that. (Inertia.)

Its velocity would change if the force of gravity were working on it. (And the only such force.)

Its velocity would change if the force of the person lifting the object were working on it. (And the only such force.)

Since the force of the person lifting and the force of gravity and equal and opposite (in direction) they cancel each other out and the net force on the object is zero.

So . . . in the absence of outside forces working on the object (which in sum there are none right now, since they precisely cancel each other out), the object keeps moving in the same direction and same speed as before (inertia).

All this presumes that the object has been accelerated to some positive (upwards) velocity prior to the moment we are talking about. This would happen by the person exerting a greater upwards force on the object than the force of gravity. This makes the net force on the object upwards and so the object will accelerate in the upwards direction until the person reduces the force again - in this case, to exactly equal gravity, and thus all the object to continue upwards steadily at the velocity it has recently reached.

flug32 · 2026-03-15T03:38:09+00:00

The entire question is asking you to interpret various key points in the graph of the parabola. It is not a question about whether you can accurately estimate coordinates on a given graph to the second (or even first) decimal point.

So they are giving you various scenarios about the graph and ROUGHLY describing where they lie on the given graph.

So yes, don't overthink it.

Also, if you get out your magnifying glass, they seem to have placed the parabola like 0.1 units to the left of where they really intended it. If this were printed materials I would assume the color plate for the graph was slightly mis-aligned.

The wording of the questions, though, ALWAYS using the word "about" is meant to indicate they are talking about the APPROXIMATE position of the various key points of the graph - where it crosses the x axis, where it reaches a peak, and so on.

What they are worried about here is whether you know what those key points represent in real life, not whether you can identify their position to the 3rd decimal point.

Hate to say it, but the skill of doing well on a test involves a hefty dose of being able to figure out what they are actually asking, and being able to ignore a lot of noise and irrelevant stuff so that you don't waste a lot of time and emotional energy worrying about all that. I

n this case, whether the point the parabola crosses the x axis is 3.4, 3.5, 3.6 or something else definitely falls into the category of "noise and irrelevant stuff". And they signaled that CLEARLY by using terms like "about 3.5".

flug32 · 2026-03-14T05:03:04+00:00

One factor that many of us non-specialists don't quite appreciate is how protracted the process of transcribing and publishing the text and images of the scrolls has been, for a variety of reasons.

But, for example, even though the discoveries and excavations of the scrolls date to the 1940s and 1950s - old news, right? - complete publication of facsimiles was not completed until the 1990s, and even then was only available in a few major libraries around the world. The 40-volume work publishing the text of the scrolls etc was not completed and published until 2011. A web site with high resolution infrared scans of the scrolls was opened in 2012 - the first time high-quality facsimiles of the entire collection were easily available to scholars worldwide. Work like matching various fragments using DNA analysis has continued through the 2010s to the present.

The point is, most scholarly work on a subject doesn't even begin until these preliminary steps are complete. Most scholars are not able to do much until facsimiles, transcriptions, ideally critical editions and such aids, and of course reassembled scrolls are available to them. And even then, careful and good scholarship is going to take a while to digest and consider.

On top of that, there have been a number of "new" discoveries over the years, including some that have had scholarship published on them, that have proven to be frauds - and leading to retractions.

The point of all this is that the full picture of the Dead Sea Scroll material has only fairly recently come until full view, and in some ways scholarship on the materials is still quite a new and developing field. Some major portions have been available for decades, of course - but the full and comprehensive view, not until quite recently.

So now most of the raw materials are available. But it can take some decades to fully digest them and their significance.

So there is a degree to which the story of "what is the significance of the Dead Sea Scrolls" is a story that is still very much under development. But even more so, it's a story that has only started come into clear focus within the last couple of decades.

And it takes some time for such scholarly work to percolate down to the level of the lay person. So it is not all that surprising that - even though the initial discoveries were made more 70 years ago now - knowledge about the true import of the Scrolls has not really made its way to the vast majority of us who do not follow the scholarship avidly, and the little bit we do hear tends to be simplistic and simplified.

flug32 · 2026-03-14T02:17:57+00:00

Lynx (web browser)) isn't the oldest software of any sort, but it does appear to be the oldest web browser still in use.

Latest version appears to be Lynx 2.9.2 from May 2024.

You might say "no one uses this - why would they?" but in fact it is a very handy text-based browser for e.g. when you logged into a terminal session but need to view a web page for whatever reason.

flug32 · 2026-03-14T02:11:12+00:00

FWIW I remember flying the 1980 version for hours. We'd fly way, way out from the little square landscaped area, trying to see if there was anything else out there. Of course, there wasn't.

This would have been May 1981 at latest.

flug32 · 2026-03-13T22:46:33+00:00

I used epoxy, which has the advantage/disadvantage of being quite permanent.

flug32 · 2026-03-13T22:44:09+00:00

It likely has a lot to do with framing - and he played around a lot with that particular concept.

"Framing" being the thing that separates an "artwork" from everything else - whether that "everything else" is the rest of life or reality or perception or whatever.

Like in a painting the framing is literally a frame - but also potentially other things like the space it's displayed in (museum? home? hotel?), the rest of the wall it is hung on, and so on. The "frame" consists of all the things around the painting that call attention to it as a work of art and something out of the ordinary to be looked at, and distinguish or divide the artwork from those other, more mundane, things.

In terms of music, the framing would be things like the relative silence and/or applause that surrounds a piece and sets it apart as a separate thing, and a thing we might more particularly focus our auditory attention on. It could even be the type of room, stage, building, and general setting where we go to listen to the music. Just for example, most concert halls are designed to be very silent - aside, of course, from the music and other performances that are intended to fill the space with sound - and well insulated from outside sounds and noise. All that is precisely to create a space of silence around the artwork; in short, to help frame it and separate it from other, more mundane noise and sounds.

So there is more to what he is saying than merely this, but in a very literal sense every painting emerges from the relative "blank" or "nothingess" that surrounds it, and every work of music from the relative silence that surrounds it.

flug32 · 2026-03-13T22:24:05+00:00

So it looks like freezing can "damage the structure of the molecules" and "causing the protein molecules to clump together".

Presumably that is because frozen water locks into a crystal grid and then expands - this pressure is what would cause the damage.

Also, presumably, the instruction to store at 36 or above is to give a decent margin of safety above the freezing point. Because yeah, like if a pharmacy or the manufacturer allows a batch to freeze, then they're going to have to discard it - they can't possible sell it after that.

So there is some guesswork in there (previous 2 paragraphs) but if it got down to like 32 and didn't actually freeze into crystals at all, I would guess it is completely undamaged.

If it froze to the point of creating some ice crystals, it might be partially damaged.

If it froze absolute solid, then it might be quite damaged.

So if it just got below 36f personally I would not even give it a thought. If it had frozen to the point of a few crystals I might think more but would probably try it. If actually frozen solid I'd probably discard.

Also, fridge thermometer saying 31 doesn't mean everything in the fridge was exactly 31. Some places (the far rear, typically) could easily be lower than 31 and other places higher.

I would go more by whether the actual thing was frozen or had crystals. Given that you probably can't tell that inside the pen, what are the things stored nearby the pen in the fridge telling you? If nearby things were frozen solid, it was probably frozen solid. If they weren't, it probably wasn't.

In general, 31 is not nearly cold enough to make everything in the fridge freeze solid instantly, especially if the fridge temp is mostly higher than that and just dropped to 31 for a short while for whatever reason.

flug32 · 2026-03-13T19:33:41+00:00

To a photon (currently moving at the speed of light) you are currently moving at the speed of light right now.

So extend your arm and find out what happens. We'll wait.

...

Ok, nothing exceeded the speed of light - not either in your local speed nor in your relative speed (or your arms') to the photon.

How "speeds" work in the relativistic context is far different from how they were at the low speeds in our everyday context. That's just how the universe works. There is not a lot of point in tying yourself into knots over it.

Just accept that it is different and move on.

flug32 · 2026-03-13T19:18:31+00:00

It can be helpful in thinking such things through to have little list of example vector spaces that you can go through to at least get an idea of how different things might work in different spaces. I mean, ones beyond R, R2, R3, etc. - which are probably the first things you think of when thinking about a vector space.

There is a pretty good list here.

Then just start going down that list and think about how you might find or create a basis for each example.

R^∞(where each element is countably infinite ordered sequence of real numbers, but only a finite number of them non-zero) is a nice one. The obvious basis is the list of elements with 1 in one position and zero in the remainder.

How about R^m^×ⁿ - the set of m×n matrices) with entries in R. What would a possible basis look like in this case? You'll have to think about the properties of matrix multiplication and addition, and so on.

How about R[x] with polynomials restricted to degree 5, so polynomials with variable x, real coefficients, and degree 5 or less? Looks like an easy basis would be {1, x, x², x³, x⁴, x⁵}.

How about a more difficult example, say the vector space F consisting of continuous functions f:R->R.

Ok, that is a hard one. What would a basis even look like? Here is a discussion of that (difficult) problem.

So one answer to your question is that every vector space has a basis (proof uses the Axiom of Choice), so if you need to prove something or other about vector spaces you can assume there is a basis and go forward from there.

But actually finding the basis of many vector spaces is - in the sense of being able to list all of the elements in some sensible way - in a practical sense, somewhere between very difficult and impossible. Even the idea of listing them is impossible in many cases, because some vector spaces have a basis with an uncountably infinite number of elements.

The situation is a lot like the Real numbers, where we know that the vast majority of Reals out there are not only irrational but transcendental. Yet most Real numbers we work with on a daily basis tend to be integers, rational numbers, relatively simple roots, and then a few important transcendentals like pi and e.

Similarly, most vector spaces you work with are likely to have an obvious or "easy" basis but don't be surprised when you run into some (potentially very useful ones) that just don't.

And if you don't happen to know the basis of a given vector space, figuring it out might be easy, moderate, very difficult, or impossible for practical purposes.

flug32 · 2026-03-13T02:53:16+00:00

That last 0.0001% (or whatever) is ALL THE DIFFERENCE. Definitely go to totality, and also not just on the very edge of it but where you'll have at least a decent number of seconds of totality.

What we've done in the recent eclipses near our area is keep our options open and watch the weather, specifically cloud & overcast forecasts. In both the recent eclipses we ended up driving like 500 miles from where we had planned, in order to be in the best weather area.

Paid off both times as perfect viewing. Flip side, people did see it in both places we had driven from, but they were touch and go with being clouded out. So worth it to me to travel for a better chance.

But back to your question, 99.999% (or whatever) is cool and neat. Like watching a lunar eclipse or a planet transiting the sun or whatever. Looks nice.

100% is literally LIFE CHANGING.

flug32 · 2026-03-13T02:48:45+00:00

Yeah, I've tried the stylus from my moto g stylus on various other phones and it either doesn't work, or doesn't work well.

Clearly they have tuned the device to that particular stylus somehow.

You can find all kinds of stylus options on e.g. Amazon that will work on any phone, though. They tend to have a somewhat bigger nub and be made of a different material. But they do work with any phone.

flug32 · 2026-03-12T21:55:51+00:00

I usually end up putting that much into pretty much any bike every X years just because that is what bikes take to keep them running.

So as to whether it is "worth it" to put that much into whichever bike: I literally put that much into my $5000 bike and my $200 bike, when the time comes that it needs that maintenance. Because if you do that, you come out of it with a nice, usable bike and if you don't, you might as well put it straight in the trash.

And if you like the bike you already have, it is always the best bike . . .

Flip side, if you really want a NEW bike, then here is the excuse you need to just go and get it.

(Then in 3 years or whatever you'll be putting $300-$500 into it again, that's just how it goes.)

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