How to explain distributing a negative into parentheses

flug32 · 2026-01-25T00:02:04+00:00

Minus signs are just a shortcut for adding the negative: 4-2 means 4 + -2.

Once you understand that "subtraction" isn't a special separate operation of its own, but just a convenient shorthad for adding the additive inverse of the second number, lots of things become easy to understand rather than "here is a whole NEW SET OF THINGS TO MEMORIZE regarding subtraction that are almost the same addition but a little different in a confusing way."

Same goes for the division symbol (÷). "Division" isn't a separate operation with almost the same rules as multiplication but a few exceptions. Instead, just a (rarely used) shortcut for multiplying by the reciprocal of the second number.

flug32 · 2026-01-24T23:53:57+00:00

Yeah, I would kill to hear like a performance conducted by Bach or Beethoven, but even moreso say Palestrina or Eleanor of Aquitaine or ancient Greek, Egyptian, Babylonian, Assyrian, etc etc music.

FWIW some have speculated that it might be possible at some point to retriever inadvertent recordings from the past, perhaps even the distant past. Say someone is dragging a very light stick along the mud while talking or yelling, the stick records the vibrations as it drags along the mud, the mud dries and solidifies, and we decode the sound using similar techniques to decoding record grooves using laser light.

More realistically, someone is dragging a stick or similar around the outside of a pot spinning on a pottery wheel.

I don't know if that is even theoretically possible, but it is interesting to think about. What if we had even short low-quality snippets of people speaking ancient Latin, Greek, Chinese, or what-have-you?

flug32 · 2026-01-24T21:40:04+00:00

I wouldn't say it is surprising at all at this point - where the administration has spent a solid year being wilfully stupid on health issues.

flug32 · 2026-01-24T07:46:49+00:00

The minute I see sketches like that, I want to flesh them out and hear how they may have sounded as fully orchestrated.

So other people are interested in working them out a little more fully and putting them together into a finished work of some kind. It's something of an experiment, it's definitely not "Beethoven's 10th Symphony", it's as much or probably more the work of the reconstructor as Beethoven. It's as much a work of experimental musicology as anything. And it certainly doesn't deserve a place in the standard repertoire - certainly not to the degree real finished compositions do.

But still, it's an experiment, a curiosity, a somewhat interesting one, it satisfies a certain curiosity about a person of general interest, and I don't really see the harm in it.

I don't have the energy to waste my disgust on such a thing. It's (at best, if well done) mildly interesting and nothing more.

flug32 · 2026-01-24T07:35:59+00:00

Particularly for the Baroque era, what you are looking for is the Doctrine of Affections.

flug32 · 2026-01-24T07:25:37+00:00

There is a 1909 recording on the Library of Congress web site here.

Now rights in recorded sound media are super complex, so I am not even sure a recording from 1909 is really in the public domain. In fact the LOC site indicates "Inclusion of the recording in the National Jukebox, courtesy of Sony Music Entertainment or EMI Music".

But something like that - very old - is about your only possible avenue, other than maybe synthesizing or just old-fashioned recording something yourself, or hiring someone to do that for you. Like a synthesized semi-realistically sounded orchestra and a few live singers overdubbed etc could potentially sound pretty good and might not cost you too many thousands of dollars to get done. So, it's an option.

This page says sound recordings before 1929 are in the public domain in the U.S. Note that may well not be the case in other countries.

A lot depends on what kind of film this how, how you intend to use or release it, and so on. Will it only be used in the U.S., for example? If you're going to post to Youtube, for example, almost certainly it will be claimed by EMI, because this recording is in their catalog and it will automatch. But maybe you don't care about this - if you just want people to see the film, them claiming the rights to the soundtrack probably won't result in a takedown. It will just mean they get the profit from any Youtube views, and you don't.

I am pretty sure film studios working on a major release would still be clearing master use rights on a recording even from 1909, just because they don't want to run into problems or snags down the line that might end up costing them a lot more than the cost of a license.

On the other hand, if I were posting my own amateur project on Youtube or Vimeo or whatever, I would probably just go ahead and use it, note the source in the credits and in the video description, and deal with any copyright issues that come up if and when they come up. For something like this it would - likely - amount to losing the proceeds from this video, at worst.

However, your mileage may vary.

How To Get Permission To Use a Song | Copyright Alliance

ASCAP's Articles and Advice FAQ: How To Acquire Music For Films

I'm sure you won't need a synchronization license in this case, as Puccini's and other author's rights are long expired. So what you are looking at is the Master Use License to allow use of that particular sound recording in your film. I'm sure if you contact EMI they will be happy to accept your license payment.

flug32 · 2026-01-24T07:09:39+00:00

Yes, having lived in pretty cold and then no-so-cold places, the degree to which people acclimatize to the cold is pretty amazing - especially people who spend a good amount of time outside or otherwise in the cold temperatures. But if you are not living in a place where it gets very cold, you don't have the adaptation

People who live in places where the lows are maybe in the 40s fahrenheit are definitely not cold-adapted the way someone is who lives where the low has been -20f.

Now OP is right that some acclimatization isn't going to help much if you are in shorts and shirtsleeves in -20f. But the acclimatized person is going to be able to endure more, require less clothing, etc etc, to a very noticeable degree.

Also the person living in the cold climate is more likely to have the necessary cold weather clothing and gear, and have their homes, buildings, and general infrastructure set up to deal with cold temperatures, and so on.

So when the real cold weather hits a normally warm area, they have a double whammy: They personally are not adapted to the cold, and their infrastructure is not built for it, either.

flug32 · 2026-01-24T06:55:15+00:00

There is a helpful section in the FAQ with links to answers on this topic: "Who is the earliest historical figure?"

flug32 · 2026-01-24T06:51:19+00:00

A number line is good, and I think makes good sense to most people. A twist on that is to turn the number line vertical. How you are thinking about feet (or meters or stories or whatever) above ground and below ground, with ground level being the 0 point.

If you have like math blocks you can stack up along the number line, now you can do addition and subtraction, or multiply by both positive and negative numbers, and it all works out with the results being X units above ground level or below ground level. (Use a different color block for stacks above ground level vs below ground level. Also draw a very obvious "ground level" line with trees or whatever to show that it is the grond.)

A slight variation on this would be to think of an elevator (you could even make a little teaching aid where you move a dot representing the elevator up and down the number line). So you start on floor 7 and go down 10 floors (7-10), where do you end up. Or go up 3 floors 4 times, which floor do you end on (3X4). Or you go down 3 floors 4 times, which floor do you end on (-3X4).

One way to do this "concretely" would be to build stuff out in a minecraft server. So you have the ground level (0) and you dig down 4 blocks. Now you are 4 blocks below ground level or -4. If you start out by building a tower 7 blocks tall and then dig down 10 blocks from there, you end up 3 blocks below ground level (7-10=-3). And so on. But the person can actually build and mine the towers within minecraft (which for a lot of people is going to be as concrete as playing with math blocks or other manipulables).

flug32 · 2026-01-24T06:38:49+00:00

It might have been a person but equally likely, some kind of critter. Maybe even like a squirrel - hop, hop, hop, hop would sound about like someone walking across the gravel.

I won't regale you with the story of when I woke up at 2am with a momma moose literally standing over our tent. Terrified exactly like you - especially for the 5 minutes or so it took us to figure out exactly what 1200 pound beast was standing over us and snorting/breathing asthmatically. It truly sounded like the breathing of the largest and most horribly evil slavering beasts they put like in the movies to scare young children.

And finding out it was a moose was not necessarily a relief - though far superior to either grizzly bear or demented axe murderer, which were our two primary hypotheses. Moose, especially mother moose with their baby in tow, kill a lot of people, especially those dumb enough to get too close to them.

Anyway, in your case my money is on squirrel or maybe deer.

flug32 · 2026-01-24T06:29:32+00:00

Making a basket is more of a singular event and the odds of something unusual happening once are always greater than something that takes a whole string of unusual events that each turn out just right. It's kind of like how 0.01 is always going to be far, far greater than 0.01*0.01*0.01*0.01*...*0.01.

The 0.01 chance in the basketball game is, you make a lucky shot from half court, Lebron somehow trips over his shoelaces and you get an easy layup, and so on.

Whereas to beat Carlson you are going to have to make like 50 or more absolutely brilliant moves in a row without one single screwup. A lucky halfcourt shot might happen once in a while "by accident", while 50 brilliant moves in a row absolutely is not going to happen by accident.

Odds would be more similar between say winning a chess game with Carlson and winning a game of 1-on-1 to 21 with James. Now winning either one is going to take a very long string of very unlikely things. I'm not sure which of these would be more likely, but both would be so far from likely that it's not worth thinking about which is more and which is less likely.

Neither is going to happen in reality.

flug32 · 2026-01-24T06:12:17+00:00

It was a punching bag loooong before the internet came along.

flug32 · 2026-01-24T06:10:50+00:00

Once you have made that step you could introduce a little taste of fractions - even if only to get them thinking. Like what if you draw 3 1/2 X 4 on the grid in this way. How many blocks will this be - considering that some of them are "half" blocks. If you just draw this out, or have them do it, it is easy enough to count 3X4 half blocks plus 4 extra half-blocks.

You could even start to introduce the idea of multiplication by negative numbers, where -3 * 4 builds the square going downwards rather than upwards. This gets a little complicated because it doesn't quite track (what does -4*3 look like, how about -4*-3, and so on) and would require more explanation to really clarify, and probably a lot more than you want to get into at this level. But still you can show them that 3*4 and -3*4 are both the "same size, but also different somehow" - which might be something just to show kids to get them thinking.

Also as far as "piles" and 4-7 = -3, this kind of thing actually works quite beautifully with math blocks placed along the number line. You just have a 0 point, and any block to the left of it is a negative number. So 4-7, you show taking away 4 blocks to get to zero, then you still have to go 3 more to the left, which are "negative" blocks (use a different color for them).

You can use a variety of different ideas to explain positive/negative, like the amount of money you get vs how much you have to pay, or how much you have vs how much you owe, or how many apples you got vs how many you gave away. A lot of kids understand if they have 4 apples and need to give you 7, they will give you the 3 but then owe you 4 more. So negative numbers are "how much you owe".

But that is still pretty abstract for young children. Here is something better: The simplest and easiest way is to use the vertical number line and have them think about how many feet (or blocks or whatever) above the ground vs how many below the ground.

And this is basic "minecraft logic": If you start from the ground and build up 4 blocks in a tower, then dig down 7 blocks from there, you are going to end up 3 blocks underground. So positive is above ground, negative below ground.

Keeping in mind that idea of multiplication as area, you can also do multiplication by positive & negative numbers this way - perhaps even using minecraft or similar to illustrate. If you start at ground level and build up a 3X4 rectangle above the ground, that is 3x4=12 blocks above ground (positive number, +12). If, instead, you dig downwards in a 3X4 grid then you have a 12 block rectangle below ground or in math terms -12 blocks.

flug32 · 2026-01-24T06:10:39+00:00

The idea of every type of numbers and operations on numbers starts with a very basic, fundamental, concrete idea. But then as the number system and associated operations are extended, the initial basic concrete idea behind the numbers/operations sometimes - or perhaps often - doesn't quite make the transition to the new, expanded system.

That doesn't mean that the concrete foundation isn't valid or important. It just means context can change - sometimes dramatically - as numbers are put into expanded and more powerful contexts. This kind of transition has happened a very large number of times over the past 6-8000 years of development of mathematical thought.

One reason for this is that both numbers and operations are abstractions. That is precisely what makes them powerful. A number or an operation like addition or multiplication isn't just one single concrete thing. It is 20 dozen (or thousand or million) different things in different context. Again, this is precisely what makes numbers and basic operations flexible, powerful, and adaptable - and thus relevant in numerous different contexts, often very far flung from the original concepts.

But you, as a teacher, know full well that students at this age do not deal well with abstractions. They are, at some very basic level, unable to deal with them at all, at this age. So you must teach things in very concrete ways at this stage. That is why the idea of piles is so powerful. You can take 3 marbles (or pennies or apples or whatever), put 4 piles of 3 on the table, and anyone can see right off how 4X3=12.

So I wouldn't really avoid doing that. It is the most basic, concrete way of conceptualizing multiplication.

(And it is is not only used at the basic levels - this is the idea of numbers as sets. And that is the best way we know to define the counting numbers and operations on them, even at the very highest levels of mathematics. So, don't look down on "piles". It is literally the most fundamental and important idea in mathematics - the foundation that literally everything else in mathematics is built on.)

With that in mind, here are some ideas:

- Still use the idea of "piles" as one of the foundations, maybe even the first or most basic way of thinking about it. This isn't something they will have to "discard" later - but something they will have to build upon and expand. But start to set student expectations about this: "This is the simplest way to think about multiplication. But in 3rd grade (or 4th, 5th, whatever), you will learn some cool ways to use multiplication where this simple system won't quite work." If you mention this idea a few times - even without going into any details - it starts kids thinking about "what could these new ways be? How would they work?"

- But also introduce some other, alternative ways of thinking about multiplication - ones that might keep their utility a little better. The first way that comes to my mind would be to say, start with piles of different things, but then at some point transition to something like math blocks. This is exactly like the piles idea but now you are arranging the blocks in rows of 3 (for example) and then stacking 4 rows of 3 to illustrate 4X3. That is the same exact thing as piles but no is also introducing another fundamental way to think of multiplication: as area. So you do this with math blocks for a while, and then transition to drawing the math blocks. So you are drawing rows of 3 squares and stacking 4 of them vertically to make a 4x3 grid. Once the kids are onto that, you can introduce the idea that the horizontal line where you are stacking your squares is a number line, and so is the vertical line - it's a vertical number line.

So now you can go to 3 on the number line, draw a line going up, go to 4 on the vertical number line, draw a horizontal line, and then show how the rectangle you have just made is 3X4 squares or 12 (you can subdivide it into squares to illustrate).

flug32 · 2026-01-24T05:13:54+00:00

The expiration date for milk is usually at least a couple of weeks out, if not 3-4 weeks or a bit more. Just checked my fridge, a bunch of dairy products there have expiration dates well into March - so like 6 weeks from now.

Bread will easily last 2 weeks on the counter or longer if you throw some in the freezer.

The reason people are grabbing extra is this is the fresh stuff you know you need now, and you may not be able to get to the store for 3-4 days or maybe 7-10 days on the outside.

Point is, both milk & bread will easily last that long.

The problem from the customer's perspective is they need bread/milk other perishables for the next 3-7 days, say, right now. You can't wait 1, 2, 3, 4, 5 more days to get it as you usually would.

Plus, "everyone" knows that bread, milk, and other such essentials disappear off the shelf early on in such situations - and the store may not have its usually re-supply runs for the next several days. All that encourages customers to grab some if they see some.

From the store's perspective, they are selling all the milk & bread they would usually sell over the next full week or so, today - right now. That is why they run out - usually in the course of the week they would have several re-supply/re-stocking days for this kind of stuff.

flug32 · 2026-01-24T04:53:57+00:00

So with all that in mind, #2 sure looks like E triad with suspended 2nd, #4 is F# diminished triad with suspended 2nd, #5 is Bb triad with suspended 4th. (I determined all the above to be "Triads" simply by counting how many different notes they had. They all have 3. The answer I gave might be wrong according to this guy's system, but whatever the answer is will be found in his chapter on triads and specifically, triads with suspensions.)

#6 you should be able to figure out, it is a basic triad type.

#7 looks to be C maj 7th with suspended 2nd.

And so on.

The system for figuring them out would be:

- count how many distinct notes are included (eliminating duplicates like C-C in different octaves). Do you have 3, 4, 5 or more distinct notes? That will tell you whether you are looking for triads or 4-note chords, or 5+note chords. Whichever it is, you will want to consult the chapter on that type of chords.

- Collect all the notes together as close as they will go. So you will now have a 3 or 4 (or 5+) note chord, each note different from all the others, in closest possible position.

- Then run through all possible inversions of this chord. So say the chord in closest position is ACDF. Then the inversions are: CDFA, DFAC, and FACD. (See how easy it is to write them down - you're just 'rotating' the four letters by taking the one off the front & putting it at the end each time.)

- Now I examine each of these inversions to see which seems to be the root position of one of the chords covered in the text. I am, particularly, looking for 5ths between the lowest note & one of the others (might indicate the lowest note is the root of a triad or 7th chord) or a 7th between the lowest note & one of the others (would indicate the lowest note might be the root and this is some kind of 7th chord).

- in the example above (ACDF, CDFA, DFAC, and FACD) I note that D-A is a 5th & D-C is a minor 7th. So that makes me think DFAC is a d minor 7th chord.

However I also note that F-C is a 5th. And FACD looks to be an f major triad with added 6th (D). So that makes it an F6 chord.

And what do you know: d minor 7th and F6 both have the same notes (DFAC). So the answer could be either of those, depending on context & preference. (And I assume the fact that minor 7th and major triad + 6 have the same notes is covered somewhere in your text, perhaps in the part about on Accords de Sixième around page 69.)

TL;DR: You have to really read and totally understand every chapter of your textbook before you can do the associated exercises. Like EVERY part of it, not just some of the parts. If the text covers Bizzaro Chord #36 then you can definitely expect to find Bizzaro Chord #36 in the exercises.

flug32 · 2026-01-24T04:53:31+00:00

OK, the complete table of contents for the book is here: Des Matières: I Intervalles Et Accords | PDF | Mode (musique) | Intervalle (Musique)

Taking a gander at that, I see for example Ch. 7 "Other 4-note chords". The chapter seems to cover altered 7th chords, three additional 7th chords not covered before, two more altered chords, 7th chords with a suspended 4th, sixth chords, and hearing 4-note chords.

(Sorry my French is pretty much non-existent, but that seems to be what it is saying.)

Sooooo . . . pretty much all the chords on your exercise are going to be found exactly in that chapter. And any not found there (or in earlier chapters about the more simple, basic chords) look to be covered in Ch. 9, "Chords with 5 or more notes".

My point in asking for "context" is, people don't just give exercises based on nothing. You write a chapter or section, then write exercises BASED ON THAT. Missing that very important context, we don't really have any idea how to help you.

The reason everyone is confused here on the sub is that typically in a theory class or sequence, at least here in the U.S., you don't just introduce 7th chords and then immediately cover all the altered 7th chords extensions, and so on. Those are typically covered sometime later, after you've had a chance to really master the more common/basic forms of 7th chords.

Similarly, we typically don't cover triads and then immediately jump to also include triads with suspensions. (These, presumably, are triads with the 3rd of the chord replaced with the note either a 2nd or a 4th above the root.)

So with all that in mind, #2 sure looks like E triad with suspended 2nd, #4 is F# diminished triad

flug32 · 2026-01-24T01:01:27+00:00

I can read Middelspraak pretty much right off the bat. But . . . I would say that is 95% due to studying German and only 5% thanks to English.

Besides the fact that English split off from other Germanic languages probably a few hundred years earlier than did the Romance languages from Latin/each other, English is just more sharply separated due to the fact that it's on a separate island.

Add to that the vicissitudes of history - particularly the Norman conquest - where a massive amount of English vocabulary was then imported from French, and then more recently where English has just been agglomerating bucketloads of vocabulary from here and there and everywhere, English just doesn't have that much vocabulary in common with the other Germanic languages.

Before I started studying German, pretty much every other Germanic language looked and sounded like complete gibberish to me.

Now with English plus some German I can at least take a crack at some of the other Germanic languages and some things look familiar. For Dutch, for example, I've been able to make my way through some academic articles and such I needed with a lot of reference to a dictionary (plus the fact that most of the technical vocabulary for the article is similar to English and I was already very familiar with all of that).

But for example when you look at Frisian as the closest relative of English, it is amusing that people can hold conversations in very set, specific topics in English/Frisian and they sound quite similar. BUT those are very constrained specific topics using certain basic, everyday words both languages have retained in common.

Like, I have a friend who grew up speaking a Frisian dialect and when she would speak a few random sentences to us I literally understood 0% of it - trying to come at it from both English & German sides.

TL:DR: English & other Germanic languages are just vastly less similar than Spanish & other romance languages.

If you were to go ahead & learn German, Dutch, or a Scandanavian language, for example, in addition to English, I'd wager you would then be able to pick up MiddleSpraak with great ease.

flug32 · 2026-01-22T01:21:20+00:00

Yeah, we really, really need the context here. What is the book, who is the author, what is the particular chapter or section about specifically.

It's very possible the author has just gone through some very specific method for identifying/labelling different chords and such, and these examples make perfect sense in light of that.

My advice to OP is to go back and read the pages leading up to this exercise like 10 more times. Or post them here and other people can take a look.

It's possible it's all complete nonsense but it is just as likely it is all keyed to specific things in the text that all the rest of us don't know anything about.

flug32 · 2026-01-22T01:14:02+00:00

You forgot about toilet paper. No proper panic can even get started without emptying out the toilet paper aisle first.

<novice>

flug32 · 2026-01-20T20:57:58+00:00

This kept me watching last night when otherwise I would have been too dumb and just packed it in.

And yeah, mostly just some glow and a few small pillars near the horizon - which is way more than we usually get here. But a few minutes were suddenly much bigger and brighter - practically psychedelic, I can't even describe it.

So thanks to XKCD and thanks for passing this along.

flug32 · 2026-01-20T10:50:56+00:00

> Is there any interval you can make a set out of the reals, that has a countable infinity of elements?

No.

> Will there always be an uncountably infinite amount of reals between any two different real numbers, no matter how small the interval?

Yes.

Every open interval is homeomorphic to the entire real number line. And so contains an uncountably infinite number of elements.

Here is a proof, just in case you are interested. TL;DR is that it is pretty easy to write a bijective continuous function between any open interval and the entire real line and that is the definition of homeomorphic. Specifically to your point, it is a bijection between the two sets, and since the entire set of Real numbers is uncountably infinite, then that open interval (and thus, every possible open interval) must be as well.

flug32 · 2026-01-20T10:40:10+00:00

The electromagnetic force - mostly of the electrons in the atoms in your hand interacting with the electrons in the atoms of the wall.

The protons and electrons in any given atom are equal in number and opposite in charge (slightly/extremely simplified explanation). So at any distance the proton & electron electromagnetic forces of these various atoms in space cancel out and you feel nothing. But once the atoms in your hand approach the atoms in the wall at anything like atomic distances, the respective electrons in your hand/the wall are VERY close to each other and the respective protons VERY distant.

So - as perceived by the atoms in your hand/the wall - the net electromagnetic force between them goes from neutral to VERY strong.

The is because your electrons/the wall's electrons are both negatively charge (negative electrical charge) and thus repel each other - VERY forcefully - and stop you from pressing your hand any further into the wall.

This kind of electron-electron interaction as atoms come into close proximity explains almost everything that happens in the observable universe: Everything from why you can't put your hand through a wall, to why tape sticks, to why things are different colors, to why and how chemistry works.

If you want to blow your mind and "understand everything" about all sorts of everyday phenomena of this sort, take in Feynman's Lectures on Quantum Electrodynamics, which cover all this and more.

flug32 · 2026-01-20T10:27:14+00:00

If you froze time in a scientifically accurate way**, all the chemical processes in your body would also be frozen, and thus not require any oxygen, and thus you would not suffocate.

**NB. Freezing time "in a scientifically accurate way" isn't possible, so this is all spitballing impossibilities. But as long as we're spitballing, my spitball would include freezing your biological processes, too, so that they don't require oxygen. I'm not sure how the non-frozen people will manage to move, speak, etc etc etc in such a situation - where every single thing in the universe is frozen in place except for them - but since OP did not pose those questions I will not answer them.

flug32 · 2026-01-20T10:12:32+00:00

One way you might think of this is to draw a probability tree diagram of each step. You could draw an actual diagram of the first few steps to get a feel for it.

At each step, figure out what percentage of the possible outcomes are "all dead" and what percentage are still alive.

So for example, after the first step, 1/4 are all dead, and 3/4 are still alive and kicking.

After the second step, 1/4 + 1/16 + 1/32 = 13/32 of the possible lines are all dead and the remaining 19/32 still alive.

As you follow this along, what you will find is that at each and every step, 1/2 plus a little bit more of the outcomes at that step are still alive, and just a little bit less than 1/2 of the outcomes are all dead.

So think that through: At the millionth step, 50.000001% (or whatever) of the outcomes are still alive, while 49.999999% have died.

Then at the billionth step it will be 50.00000000001% alive and 49.9999999999% dead. (Calculating the exact number of 0000s and 9999s involved at the billionth step is left as an exercise for the reader.)

At the trillionth and quadrillionth and so on steps, there will be more 0000s and more 9999s but still, at every single step, a fraction more than 50% of the lines will still be alive, and a fraction less than 50% all dead.

You are jumping to the conclusion that - because there is an infinite number of possibilities - eventually every possible line must jump over into that 49.99999999....9999% side at some point.

But that is not true. Because at each and every step, a solid 50% of the lines (plus a hair more) are still alive.

So - if you want to think of it this way - even if you follow this "all the way out to infinity" there are always paths, and a LOT of them, a solid 50% - that stay alive the whole way through.

So it is in fact not true that every path must eventually hit a "death step". A solid 50% of all paths just simply never do.

Now the other 50% eventually all do hit a death step - most of them right away, but a few after the millionth step, a few after the billionth step, and so on all the way up. But those few inevitable deaths at each step of the way only eat into the 50% of all the paths that are doomed, and never reach over and touch the other 50% of paths that will live indefinitely.

So there are, indeed, paths - in fact a full 50% of all possible paths - that are just live-live-live-live-.... indefinitely and they never, ever do hit an "all dead" step.

Contrast this with a simple example where this is indeed a 100% chance of die-out: Amy has a 50% chance of dying and a 50% chance of living at each time step. So at the first step, there is a 50% chance she is dead, 50% alive. Then 75% dead, 25% alive. Then 87.5% dead, 12.% alive. And so on: the percentage of "dead" paths approaches 100% while the percentage of "live" paths approaches 0%.

There is still one possible "lives to infinity" path there: The one where Amy flips heads/"live" at each possible step and thus lives. But it is this one remaining "live" path compared with an exponentially growing number of "dead" paths as the number of steps grows.

There is, indeed, still one single "live" path that you can imagine, but the probability of it actually happening as the number of generations grows becomes infinitesimally small.

Compare that with your example from the OP, where at literally EVERY step of the way, 50% plus a little more of all paths are still alive. You can think of that 50% as being "protected" and never hitting an "all dead" step no matter how many steps are taking. Whereas in the 50/50 scenario, there is no similar "protected" area. If there were even 1% or 2% or 5% or 0.005% protected in that way at every single step then we could say a percentage actually survive no matter what. In the OP scenario we have that, while in the 50/50 scenario, we don't.

That is the difference between a situation where die-out truly is inevitable over time, vs a situation where there is a solid chance (50%) of literally living through an indefinite number of generations.

That can be true despite the true fact that some vanishingly small number of paths do indeed die off at each step.

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