OOTS #1264 - The Discussion Thread

[Peelee]

I can imagine a square of 10 people by 10 people (with 9 more people stacked up, sitting on each of their heads) pretty well... but I could only make wild guesses if you asked me to estimate whether a non-arrayed crowd was 1000 or 10000.

Where I really trip myself up, though, is imagining dimensions of infinity.
There are infinite points in a line, right? It can't be any greater, because infinity +1 is meaningless.
But aren't there also infinite lines containing infinite points in a plane?
And infinite planes containing infinite lines containing infinite points in space?
And infinite spaces containing infinite planes containing infinite lines containing infinite points in time? (^_^)°

I missed this before. One of the more basic intros to infinity is the idea of countable Infinities and uncountable Infinities. Countable infinities are, as it says on the tin, countable. Think of the natural set of numbers. One, two, three, and so forth. Four, even. It's an infinite set. No matter how high you get, there's another number higher than that. But you can count to it - if I said, for example, what's the next number after 2,849,572,837,549,285,704, you would say "Why Peelee, that is an unreasonably large number and you are quite silly for counting that high - you must be orders of magnitude older than the known universe to have counted to that! But the next number would be 2,849,572,837,549,285,705, of course!" And you would be correct, because it is trivially easy to count to that. If we just don't bother with that pesky concept of time, at least. Anyway, point is, any number in the infinite set of natural numbers, you can count to. Heck, they're even called the counting numbers. Convenient, that!

However, that's not the only set of numbers. I don't mean the set of whole numbers, which is the exact same thing as the natural numbers but with 0 added in. I'm not going to pull a Spinal Tap and go "well, it's one more, isn't it?" Nah, I'm going to be much more sneaky than that! Rational numbers! Once we get to rational real numbers, hoo boy, does that infinity get just way bigger. It's no longer countable. See, we start at 0, but if I ask you what the next number is, you can't just go to 1, oh my no. You would never deign to commit such a faux pas, I know you, you're much more elegant. You would say, "well, Peelee, since were dealing with decimals now, 0.1 would be between 0 and 1, and would thus come first. But 0.01 is even closer to 0. And 0.001 is even closer to 0, that would come before all three of those." And soon enough you would be stuck in a never-ending pattern of constantly realizing there is a smaller and smaller number between 0 and whatever tiny fraction you last pulled out. You can't count to the next number becuase there is an infinite drop between 0 and the "next" number in the sequence. And just when you think that's big, just remember that 1 is still in there! And even 2! And all the rest. Except instead of a nice, clean, simple progression between them, there is an infinite amount of numbers in between each natural number.

"The audacity!", you say, quite reasonably. "Countable infinities are so much easier to deal with. Why, it would seem as if infinity actually existed as a concrete concept in the world, the infinite set of natural numbers would be the only way to do it!" And I would beam with pride at just how correct you are. "INDEED!", I would shout with glee, "and in fact, that set even has a special name! It is the Aleph naught set, using the Hebrew letter Aleph and the subscript 0 (א)₀)


And then you would exclaim in disbelief, almost in a rage, "Why Peelee, did you seriously go to all this effort, and did I really read this entire thing, all for the sole payoff that I now understand a ****ing 2-second background visual joke from Futurama?!"

[Fyraltari]

Rational numbers! Once we get to rational numbers, hoo boy, does that infinity get just way bigger. It's no longer countable.

That's not true. Rational numbers are countable, it's just a bit tricky.

You see, by definition, rational numbers are just a ratio of two relative numbers, so you can represent them as points of a graph, where the x-axis is the numerator and the y-axis is the denominator.

Now, you can simply count each rational number by going in a spiral motion starting at the origin point of the graph (and being careful to "jump" over points that are actually numbers you've already gone over like how "2/4" is really "1/2")

So the list goes like this:

0: (0;0) => 0; 1: (1;1) => 1; 2: (-1;1) => -1; 3: (2;-1) => -2; 4: (2;1) => 2; 5: (1;2) => 1/2; 6: (-1;2) => -1/2; 7: (3;-2) => -3/2; 8: (3;-1) => -3; 9: (3;1) => 3; ...

On so on and on in a spiral. Each rational number is associated with only one natural number and vice-versa, so there are, in fact just as many rationals as there are naturals.

And the number after 0.1 (that is 1/10) is -0.1 (-1/10). This also work without taking the egative numbers into account, you just limit yourself to one quarter of a graph, but the motion is harder to explain than "spiral" it's a back-and-forth between the axis.
Anyway, in that case, the list goes:
0: (0;0) => 0; 1: (1;1) => 1; 2: (2;1) => 2; 3: (1;2) => 1/2; 4: (1;3) => 1/3; 5: (2;3) => 2/3; 6: (3;2) => 3/2; 7: (3;1) => 3; 8: (4;1) => 4; 9: (4;3) => 4/3; ... And so on. Again, to every natural one and only one rational and vice-versa.

In this case the number after 0.1 (1;10) is 0.3 (3;10). (0.2 already came up at (1;5).)

[Peelee]

That's not true. Rational numbers are countable, it's just a bit tricky.

You see, by definition, rational numbers are just a ratio of two relative numbers, so you can represent them as points of a graph, where the x-axis is the numerator and the y-axis is the denominator.

Now, you can simply count each rational number by going in a spiral motion starting at the origin point of the graph (and being careful to "jump" over points that are actually numbers you've already gone over like how "2/4" is really "1/2")

So the list goes like this:

0: (0;0) => 0; 1: (1;1) => 1; 2: (-1;1) => -1; 3: (2;-1) => -2; 4: (2;1) => 2; 5: (1;2) => 1/2; 6: (-1;2) => -1/2; 7: (3;-2) => -3/2; 8: (3;-1) => -3; 9: (3;1) => 3; ...

On so on and on in a spiral. Each rational number is associated with only one natural number and vice-versa, so there are, in fact just as many rationals as there are naturals.

And the number after 0.1 (that is 1/10) is -0.1 (-1/10). This also work without taking the egative numbers into account, you just limit yourself to one quarter of a graph, but the motion is harder to explain than "spiral" it's a back-and-forth between the axis.
Anyway, in that case, the list goes:
0: (0;0) => 0; 1: (1;1) => 1; 2: (2;1) => 2; 3: (1;2) => 1/2; 4: (1;3) => 1/3; 5: (2;3) => 2/3; 6: (3;2) => 3/2; 7: (3;1) => 3; 8: (4;1) => 4; 9: (4;3) => 4/3; ... And so on. Again, to every natural one and only one rational and vice-versa.

In this case the number after 0.1 (1;10) is 0.3 (3;10). (0.2 already came up at (1;5).)

I thought I was missing something but couldn't quite nail down what it was. Changed to real numbers, thanks!

[Fyraltari]

interesting, I did not know that

honestly I like the SI but I'm still pissed at mega (M) mili (m) and micro (µ) (I literally had to google that one, it doesn't even look like an m!)

That's the Greek letter Mu, that's where we got m from.

[Vikenlugaid]

yes, but I've been lucky with my education and as such I know we also have a billion, it's just that it's a million million rather then a thousand million
but seeing as how english media is a big thing I need to keep mind that our billion is not their billion but rather our milliard

the only thing I don't really know is who exactly is using the short form or the long form (the only rule of thumb I've come up with is that if it's in english it's probably the short form, I just wasn't really sure about the UK since halfeye asked about which billion is meant and he's from the UK)

edit: ah, I see where the disrepancy happened: personally I've never seen milliard used anywhere in english but as explanation of the difference between long form and short form

Here in Spain a million means 1.000.000.000.000.

[brian 333]

Here in Spain a million means 1.000.000.000.000.

It is waaaay easier to become a millionaire in the U.S.

Way to lower the bar!

[bunsen_h]

Ah, now you're just trying to cheat. I see this game.

You can imagine that because you can imagine 10 of a thing. Minor change to the parameters, exact same structure but now it's 1000 units on each axis, and suddenly the problem re-appears plain as day. Our feeble human brains can't grasp the concept of it. The number is just too big. And a million? Forget about it. We're so bad at conceptualizing big numbers we by and large are overwhelmed when we are first introduced to decent breakdowns about the true difference between a million and a billion.

1000 units on each axis would still be manageable by subdividing into identical groups. The procedure fails if you can't manipulate the objects to be counted into an ordered array. If you have, say, 36 marbles arranged in a 6x6 grid, or possibly even a 4x9 grid, it's easy to grasp how many there are in an instant. If you have around the same number "arranged" as a pile in your cupped hands, you probably wouldn't be able to figure out how many there are without counting them. The simple ordered pattern makes the crucial difference.

[Vikenlugaid]

It is waaaay easier to become a millionaire in the U.S.

Way to lower the bar!

I meant "a billion" sorry 🤣

[Fyraltari]

As far as I can tell, all European languages but English use the long form. This is just the ananas thing all over again.

[brian 333]

I meant "a billion" sorry 🤣

It is also easier to become a billionaire in the U.S.

Team USA really knows how to lower the bar!

[Vikenlugaid]

It is also easier to become a billionaire in the U.S.

Team USA really knows how to lower the bar!

It can be far easier lowering the value of the currency.

[Peelee]

1000 units on each axis would still be manageable by subdividing into identical groups. The procedure fails if you can't manipulate the objects to be counted into an ordered array.

You have the ordered array. A cube one thousand units to a side.

Really, you're exemplifying my point. You can't imagine a thousand easily, but you can break it down into manageable sizes and then imagine that way. You can't imagine a thousand of a thing, but you can imagine ten of a thing, so you break it down into tens. You can't imagine 36 of a thing, so you break it down into sixes.

But those tricks don't always work. You can't imagine a thousand miles, and you can't shuffle that down into a ten-mile cube. Hell, even ten miles is hard to fathom because it's already a condensed version of 52,800 feet. A foot is simple to grasp, it's about yea long, you can fauge ten feet or twenty feet or if you're heavily trained like a military sniper you can measure a few hundred feet. But fifty thousand? Mighty tricky, that. We can calculate on paper how to travel the distance to Mars, but conceptualizing the distance in our heads? Forget about it. We may know that some people in the world are worth 200 billion dollars, but that's just gobbledygook when we try to imagine exactly how much money that is.

[brian 333]

On flat terrain in my youth I could do a mara miles in half a day. Ten years ago I could walk ten miles in a day but my hips were sore for a week. I know exactly how far ten miles are! Farther than my hips can go.

But to translate that into a hundred miles?

I stood on a ridge in New Mexico and saw the top of a red hill on the horizon. Hours of driving later I was in Arizona, still driving in the general direction of that hill. I'd have said that hill was only ten miles away. It's just East of the Grand Canyon. I think it's called Red Hill. Check it out.

Perspective is what units of measure are all about. If we could imagine 1000 cc we wouldn't need a liter. If we could imagine 1000 meters we wouldn't need a kilometer.

[bunsen_h]

But those tricks don't always work. You can't imagine a thousand miles, and you can't shuffle that down into a ten-mile cube. Hell, even ten miles is hard to fathom because it's already a condensed version of 52,800 feet. A foot is simple to grasp, it's about yea long, you can fauge ten feet or twenty feet or if you're heavily trained like a military sniper you can measure a few hundred feet. But fifty thousand? Mighty tricky, that. We can calculate on paper how to travel the distance to Mars, but conceptualizing the distance in our heads? Forget about it. We may know that some people in the world are worth 200 billion dollars, but that's just gobbledygook when we try to imagine exactly how much money that is.

Distance is a somewhat different kettle of fish, as it's continuous rather than discrete. One doesn't usually consider a mile in terms of it consisting of some number of foot-long units, which themselves consist of numbers of still-smaller units, and so on. "I can't imagine a centimetre, since that's 65,000,000 times the length of a carbon-carbon single bond!" (See also: previous discussions of aleph numbers.) But I agree with you that in general, humans don't have a good grasp of large numbers of things *unless* they're clumped into manageable blocks.

I've been told that some people in the world are "worth" billions of dollars. I concede that they own assets which have been assessed with such values. I'm not sure I'd concede that those people are "worth" such amounts.

[Peelee]

I've been told that some people in the world are "worth" billions of dollars. I concede that they own assets which have been assessed with such values. I'm not sure I'd concede that those people are "worth" such amounts.

Ha! Great to hear excellent wordplay from the sciences instead of the humanities. I love how you put that.

[Rrmcklin]

I knew americans used the short form but does the UK as well?

personally I've never seen milliard used anywhere but as explanation of the difference between long form and short form

The short form is standard in basically all English dialects. Point of order, that's something taken from the French, but apparently that's gone out of style (and did so awhile ago).

[a_flemish_guy]

The short form is standard in basically all English dialects. Point of order, that's something taken from the French, but apparently that's gone out of style (and did so awhile ago).

must have gone out of style a really long time ago, we have an achaic expression of frustration that's bassicly the french pronounciation of milliard which has itself gone through at least 3 bastardisations, you don't do that in a few generations

[Peelee]

must have gone out of style a really long time ago, we have an achaic expression of frustration that's bassicly the french pronounciation of milliard which has itself gone through at least 3 bastardisations, you don't do that in a few generations

I'm American, don't tempt me.

[hamishspence]

I knew americans used the short form but does the UK as well?

personally I've never seen milliard used anywhere but as explanation of the difference between long form and short form

I remember it mostly from the Piet Hein rhyme:


“Nature, it seems, is the popular name
for milliards and milliards and milliards
of particles playing their infinite game
of billiards and billiards and billiards.”

[bunsen_h]

I remember it mostly from the Piet Hein rhyme:

“Nature, it seems, is the popular name
for milliards and milliards and milliards
of particles playing their infinite game
of billiards and billiards and billiards.”

Decades ago now, I stumbled across one of his Grooks books in a used-book store, priced at something like 50 cents. I still vividly remember making a point of being very nonchalant as I took it to the cashier/owner, so she wouldn't notice and reassess it.

[dave_smith354]

must have gone out of style a really long time ago, we have an achaic expression of frustration that's bassicly the french pronounciation of milliard which has itself gone through at least 3 bastardisations, you don't do that in a few generations

Honestly, I think it was fairly recent. I'm "only" 44, I remember in my youth a lot of printed material and quite a few actual people who took the use of long form as a point of national pride (especially in contrast to America). On the other hand, news media and common usage had switched to short form before I was born (maybe as early as the 1950's).

Nowadays it's only an issue for people who need to co-operate across borders (Scientists at post-graduate level, major engineering companies, diplomats etc).

[Peelee]

The Mod on the Silver Mountain: Closed for Necromancy