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Occasionally you'll read about something, and the author remarks that
"this has more than 10^100 possible combinations, which is more than the
number of atoms in the universe", or something similar.
Of course, 10^80 and 10^90 don't *sound* all that much different. They
*are* in fact extremely different (specifically, one is a thousand times
bigger!), but they don't look all that different. Something like 10^496
is rather difficult to grasp mentally.
Does anybody know of a list anywhere that gives examples of really large
numbers? I'm thinking of things like the number of grains of sand in a
cubic meter, the brain cells in a human brain, or the number of
subatomic particles in the visible universe. I for one have no idea even
approximately "how big" these numbers are.
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Invisible <voi### [at] dev null> wrote:
> Occasionally you'll read about something, and the author remarks that
> "this has more than 10^100 possible combinations, which is more than the
> number of atoms in the universe", or something similar.
>
> Of course, 10^80 and 10^90 don't *sound* all that much different. They
> *are* in fact extremely different (specifically, one is a thousand times
> bigger!), but they don't look all that different. Something like 10^496
> is rather difficult to grasp mentally.
>
> Does anybody know of a list anywhere that gives examples of really large
> numbers? I'm thinking of things like the number of grains of sand in a
> cubic meter, the brain cells in a human brain, or the number of
> subatomic particles in the visible universe. I for one have no idea even
> approximately "how big" these numbers are.
>
> --
> http://blog.orphi.me.uk/
> http://www.zazzle.com/MathematicalOrchid*
For the observable universe see:
http://en.wikipedia.org/wiki/Observable_universe
As a lower limit ~3x10^79 hydrogen atoms.
Isaac
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Invisible wrote:
> Of course, 10^80 and 10^90 don't *sound* all that much different. They
> *are* in fact extremely different (specifically, one is a thousand times
> bigger!), but they don't look all that different. Something like 10^496
Oh dear, no. One is 10 billion times bigger than the other.
> Does anybody know of a list anywhere that gives examples of really large
> numbers? I'm thinking of things like the number of grains of sand in a
> cubic meter, the brain cells in a human brain, or the number of
> subatomic particles in the visible universe. I for one have no idea even
> approximately "how big" these numbers are.
Didn't look that up, but the first Skewes number was once believed to
be the largest number ever used in a proof (or for anywhere useful?):
http://en.wikipedia.org/wiki/Skewes_number
--
Fax me no questions, I'll Fax you no lies!
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Mueen Nawaz wrote:
> Didn't look that up, but the first Skewes number was once believed
> to be the largest number ever used in a proof (or for anywhere useful?):
>
> http://en.wikipedia.org/wiki/Skewes_number
I believe that Graham's number is (significantly!) larger still and
first appeared as a bound in a proof. Hilariously, the lower bound was 6.
http://en.wikipedia.org/wiki/Graham%27s_number
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Invisible <voi### [at] devnull> wrote:
> Occasionally you'll read about something, and the author remarks that
> "this has more than 10^100 possible combinations, which is more than the
> number of atoms in the universe", or something similar.
> Of course, 10^80 and 10^90 don't *sound* all that much different. They
> *are* in fact extremely different (specifically, one is a thousand times
> bigger!), but they don't look all that different. Something like 10^496
> is rather difficult to grasp mentally.
In general, the human mind tends to think linearly and cannot easily
grasp the concept of exponential growth, no matter how much it's explained.
Heck, even people who have studied technical subjects filled with math
often have hard time grasping the concept of exponential growth.
This is sometimes used to present thinking problems with surprising
results. The most classic one is the problem of the chessboard and the
grains of wheat (or in some versions rice), as supposedly some man
presented to some king (although this is almost certainly just an invented
story).
In other words: The man wanted as reward one grain on the first square
of the chessboard, and for each successive square double the previous
(ie. 2 on the second square, 4 on the third, 8 on the fourth and so on).
This sounded reasonable to the king, so he accepted. Only when his men
started actually counting how many grains of wheat that would make, did
they realize the impossibility of the request. (Just the grains at square
45 or such are more numerous than the yearly production of wheat of the
entire world.)
Another funny example, which you can use on someone: Assume you have
a really, really large piece of cardboard which is 1 mm thick. Also assume
that you can fold it in half as many times as you want (thus doubling its
thickness each time you fold it). How many times do you have to fold it
before the thickness reaches the Moon?
Most people who have never heard of the concept of exponential growth
will usually give a guess which is at least some thousands. However, the
real answer is a surprisingly small number.
As for number series which grow very fast, I like the following one,
because it's easy to understand and state, and grows incredibly fast:
- Let's denote a series of exponentials with (!^n).
n(!^1) just means n!
n(!^2) means (n!)! (the exponential of the exponential of n)
n(!^3) means ((n!)!)!
etc.
- The number series is: f(n) = n(!^n)
This series grows *very* fast. The result of f(5) (represented as a regular
decimal number) is probably larger than the combined hard drive space of all
the hard drives in the entire world could store. It's probably so large that
even if each atom in our planet could be used to represent a bit of storage,
it wouldn't be enough to store the entire number.
--
- Warp
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"Invisible" <voi### [at] devnull> wrote in message
news:488de160$1@news.povray.org...
> Does anybody know of a list anywhere that gives examples of really large
> numbers? I'm thinking of things like the number of grains of sand in a
> cubic meter, the brain cells in a human brain, or the number of
> subatomic particles in the visible universe. I for one have no idea even
> approximately "how big" these numbers are.
Do you really find it easy to visualize the number of subatomic particles in
the visible universe? I don't see why thinking about that would be more
informative than, say, 1E80.
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somebody wrote:
> Do you really find it easy to visualize the number of subatomic particles in
> the visible universe? I don't see why thinking about that would be more
> informative than, say, 1E80.
Well, the number of grains of sand on the entire English coastline is
"obviously" a pretty damned big number. And the number of subatomic
particles in the universe is equally obviously *very* much larger.
Call it a failure of the simplistic human mind, but seeing a handful of
symbols on a page isn't very impressive. Likening it to something that
at least "feels real" makes it slightly easier to grasp.
For example, off the top of your head, how long is "10^14 seconds"? I
mean, is that like, months? Millenia? What?
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Warp wrote:
> In general, the human mind tends to think linearly and cannot easily
> grasp the concept of exponential growth, no matter how much it's explained.
> Heck, even people who have studied technical subjects filled with math
> often have hard time grasping the concept of exponential growth.
Yeah, pretty much.
Also, many people don't seem to comprehend that "five billion" and "five
thousand billion" are quite different numbers - people seem to go "oh
yeah, something with 'billion' in it. I guess it's really big then?"
In a way, I think this is where metric measurements come in handy. Most
people are able to "get" that KB is small change, MB is moderately big,
and GB is very large (and TB is utterly huge). On the other hand, if you
said "oh yeah, I have three thousand million bytes of RAM", people would
just kinda blank that, I suspect.
> This is sometimes used to present thinking problems with surprising
> results. The most classic one is the problem of the chessboard and the
> grains of wheat (or in some versions rice), as supposedly some man
> presented to some king (although this is almost certainly just an invented
> story).
>
> In other words: The man wanted as reward one grain on the first square
> of the chessboard, and for each successive square double the previous
> (ie. 2 on the second square, 4 on the third, 8 on the fourth and so on).
> This sounded reasonable to the king, so he accepted. Only when his men
> started actually counting how many grains of wheat that would make, did
> they realize the impossibility of the request. (Just the grains at square
> 45 or such are more numerous than the yearly production of wheat of the
> entire world.)
I heard that the total would be 2^64-1 grains which is "more than the
toal number of grains that has ever existed on Earth" - a figure far
exceeding merely the yearly production of wheat.
I have absolutely *no idea* whether this description is actually
accurate or not - which is why I'm after a table of big numbers! ;-)
> Another funny example, which you can use on someone: Assume you have
> a really, really large piece of cardboard which is 1 mm thick. Also assume
> that you can fold it in half as many times as you want (thus doubling its
> thickness each time you fold it). How many times do you have to fold it
> before the thickness reaches the Moon?
Weirdly, you can only fold a piece of paper about 7 times. I have
literally no idea why. Brainiac tried it with a huge roll of industrial
tissue paper (so it's both very large and extremely thin). To make the
7th fold, they actually drove a van over the thing, but still it didn't
"really" fold convincingly. I guess it's due to the large turn radius or
something...
> As for number series which grow very fast, I like the following one,
> because it's easy to understand and state, and grows incredibly fast:
>
> - Let's denote a series of exponentials with (!^n).
> n(!^1) just means n!
> n(!^2) means (n!)! (the exponential of the exponential of n)
> n(!^3) means ((n!)!)!
> etc.
>
> - The number series is: f(n) = n(!^n)
>
> This series grows *very* fast. The result of f(5) (represented as a regular
> decimal number) is probably larger than the combined hard drive space of all
> the hard drives in the entire world could store. It's probably so large that
> even if each atom in our planet could be used to represent a bit of storage,
> it wouldn't be enough to store the entire number.
Heck, apparently n! is [VERY approximately] proportional to n^n, which
gives you some idea just how damned fast it grows. (Faster than any
normal exponential function.)
A degree N polynomial grows faster than any possible degree N-1 polynomial.
An exponential function grows faster than any polynomial of finite degree.
A factorial function grows faster than any possible exponential
function. [And we've already established just how ****ing fast that grows!]
Your n(!^2) is guaranteed to grow even faster still.
Finally, n(!^n) grows faster than any other function yet mentioned.
Still, take heart. The busy beaver function grows so fast it's not even
a computable function... (!!)
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Orchid XP v8 <voi### [at] devnull> wrote:
> Weirdly, you can only fold a piece of paper about 7 times.
It's just a myth.
You can fold a regular-sized paper, and even a very big paper (A0 or
even larger) 7 or perhaps 8 times, yes, but if you take really, really
huge sheet of paper, you can fold it more. In Mythbusters they took
a really enormous sheet of paper and, iirc, folded it 11 times.
It's just a question of the size and thickness of the paper. There's
no magical physical limit of 7 folds.
> Still, take heart. The busy beaver function grows so fast it's not even
> a computable function... (!!)
I like the n(!^n) more because it's much easier to explain and understand,
even though it doesn't grow as fast.
--
- Warp
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Orchid XP v8 <voi### [at] devnull> wrote:
> Well, the number of grains of sand on the entire English coastline is
> "obviously" a pretty damned big number. And the number of subatomic
> particles in the universe is equally obviously *very* much larger.
The funny thing about the amount of particles in the universe is that,
if current theories are right, there's no way of knowing how big the
universe is and how much material there is. There's a thing called
cosmological horizon which makes it completely impossible for us to
observe the entire universe, no matter what the means.
That's where the term "observable universe" comes from: It's everything
inside the cosmological horizon, which is at least in theory possible to
be observed.
The real size of the universe is completely impossible to know. It
could be just slightly larger than the observable universe, or it could
be staggeringly larger. There's just no way of knowing.
--
- Warp
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>> Well, the number of grains of sand on the entire English coastline is
>> "obviously" a pretty damned big number. And the number of subatomic
>> particles in the universe is equally obviously *very* much larger.
>
> The funny thing about the amount of particles in the universe is that,
> if current theories are right, there's no way of knowing how big the
> universe is and how much material there is. There's a thing called
> cosmological horizon which makes it completely impossible for us to
> observe the entire universe, no matter what the means.
>
> That's where the term "observable universe" comes from: It's everything
> inside the cosmological horizon, which is at least in theory possible to
> be observed.
>
> The real size of the universe is completely impossible to know. It
> could be just slightly larger than the observable universe, or it could
> be staggeringly larger. There's just no way of knowing.
Well, when people say "number of atoms in the universe", they usually
mean "number of atoms in the *observable* universe" - since, as you
point out, the true size of the *entire* universe can never be known to
mankind.
However, even this (much?) smaller number cannot really be known with
much accuracy. One has to make assumptions about Dark Matter and Dark
Energy and so on and so forth. So it's really a wildly approximate estimate.
Even so, if you do a calculation and it says that the amount of storage
required comes anywhere near to our "estimated" number of particles in
the visible universe... you *know* you aren't going to find a computer
anywhere that can run this algorithm. (!)
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Kevin Wampler wrote:
> Mueen Nawaz wrote:
>> Didn't look that up, but the first Skewes number was once believed
>> to be the largest number ever used in a proof (or for anywhere useful?):
>>
>> http://en.wikipedia.org/wiki/Skewes_number
>
> I believe that Graham's number is (significantly!) larger still and
> first appeared as a bound in a proof. Hilariously, the lower bound was 6.
>
> http://en.wikipedia.org/wiki/Graham%27s_number
Wow. Mind-bogglingly wow.
Well, at least I wasn't wrong - Skewes number appeared first.
--
Fax me no questions, I'll Fax you no lies!
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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In case it amuses anybody, I have put together the following data from
Wikipedia:
1.000 * 10^0 sec = 1 second.
1.000 * 10^1 sec = 10 seconds.
6.000 * 10^1 sec = 1 minute.
6.000 * 10^2 sec = 10 minutes.
9.000 * 10^2 sec = 15 minutes = 0.25 hours.
1.800 * 10^3 sec = 30 minutes = 0.5 hours.
3.600 * 10^3 sec = 60 minutes = 1 hour.
2.880 * 10^4 sec = 8 hours.
4.320 * 10^4 sec = 12 hours = 0.5 days.
8.640 * 10^4 sec = 1 day.
6.048 * 10^5 sec = 7 days = 1 week.
8.640 * 10^5 sec = 10 days.
1.209 * 10^6 sec = 14 days.
2.592 * 10^6 sec = 30 days = Roughly 1 month.
8.640 * 10^6 sec = 100 days = Roughly 3-4 months.
1.577 * 10^7 sec = 0.5 years = 6 months.
2.366 * 10^7 sec = 9 months.
3.155 * 10^7 sec = 1 year.
3.155 * 10^8 sec = 10 years.
1.577 * 10^9 sec = 50 years.
3.155 * 10^9 sec = 100 years = 1 century.
4.102 * 10^9 sec = 130 years = Electric light, recorded round,
telephone, etc.
5.206 * 10^9 sec = 165 years = Orbit of Neptune.
5.806 * 10^9 sec = 184 years = Beethoven's 9th symphony.
6.910 * 10^9 sec = 219 years = French revolution.
7.321 * 10^9 sec = 232 years = USA formed.
1.243 * 10^10 sec = 394 years = Shakespear's last play.
1.625 * 10^10 sec = 515 years = Columbus finds America.
1.751 * 10^10 sec = 555 years = Oldest SURVIVING printing press.
2.161 * 10^10 sec = 685 years = The Devine Commedy.
3.155 * 10^10 sec = 1,000 years = 1 millenium.
4.831 * 10^10 sec = 1,531 years = Fall of Roman Empire.
4.891 * 10^10 sec = 1,550 years = First Anglo-Saxon invasion.
6.081 * 10^10 sec = 1,927 years = Pompeii razed by volcanoe.
6.336 * 10^10 sec = 2,008 years = Birth of Jesus Christ.
7.352 * 10^10 sec = 2,330 years = Death of Alexander the Great.
7.592 * 10^10 sec = 2,406 years = Death of Socrates.
7.889 * 10^10 sec = 2,500 years = Euclid's Elements.
8.779 * 10^10 sec = 2,782 years = Rome founded.
1.199 * 10^11 sec = 3,800 years = Alphabetic writing.
1.420 * 10^11 sec = 4,500 years = Domestic horse.
1.441 * 10^11 sec = 4,567 years = Great Pyramid, Giza.
1.704 * 10^11 sec = 5,400 years = Start Bronze Age.
1.803 * 10^11 sec = 5,715 years = Halflife carbon-14.
2.051 * 10^11 sec = 6,500 years = The wheel.
3.155 * 10^11 sec = 10,000 years = End last glacial age.
3.471 * 10^11 sec = 11,000 years = Jericho founded.
3.786 * 10^11 sec = 12,000 years = Start Neolithic.
6.626 * 10^11 sec = 21,000 years = End last Glacial Maximum.
9.151 * 10^11 sec = 29,000 years = Neanderthals died out.
3.155 * 10^12 sec = 100,000 years = Domestic dogs.
4.733 * 10^12 sec = 150,000 years = Mitochondrial Eve.
7.889 * 10^12 sec = 250,000 years = Homo sapiens.
2.492 * 10^13 sec = 790,000 years = Homo erectus first uses fire.
7.889 * 10^13 sec = 2.5 million years = Homo habilis.
1.262 * 10^14 sec = 4 million years = End last Ice Age.
1.072 * 10^15 sec = 34 million years = First cats.
2.051 * 10^15 sec = 65 million years = Start of Tertiary. End of dinosaurs.
6.153 * 10^15 sec = 195 million years = Start of Jurassic. First mammals.
7.889 * 10^15 sec = 250 million years = Galactic year. (1 revolution
around Milky Way.)
7.933 * 10^15 sec = 251.4 million years = Start of Triassic. Permian
extinction. Age of the dinosaurs.
1.072 * 10^16 sec = 340 million years = Start of Carboniferous.
1.672 * 10^16 sec = 530 million years = Cambrian explosion. First
vertabrates.
1.704 * 10^16 sec = 540 million years = Start of Cambrian.
2.220 * 10^16 sec = 703.8 million years = Halflife uranium-235.
4.029 * 10^16 sec = 1,227 million years = Halflife potassium-40.
1.293 * 10^17 sec = 4.1 billion years = Oldest life on Earth.
1.409 * 10^17 sec = 4.486 billion years = Halflife uranium-238.
1.420 * 10^17 sec = 4.5 billion years = Age of the Earth.
4 * 10^17 sec = 13 billion years = Age of the Universe.
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Mueen Nawaz wrote:
> Well, at least I wasn't wrong - Skewes number appeared first.
I sort of like Skewes number as well, as it provides a useful
counterexample to provide to someone who thinks that checking the first
several billion examples of a conjecture makes it virtually certain to
be true.
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On 28-Jul-08 20:16, Warp wrote:
> Orchid XP v8 <voi### [at] devnull> wrote:
>> Well, the number of grains of sand on the entire English coastline is
>> "obviously" a pretty damned big number. And the number of subatomic
>> particles in the universe is equally obviously *very* much larger.
>
> The funny thing about the amount of particles in the universe is that,
> if current theories are right, there's no way of knowing how big the
> universe is and how much material there is. There's a thing called
> cosmological horizon which makes it completely impossible for us to
> observe the entire universe, no matter what the means.
>
> That's where the term "observable universe" comes from: It's everything
> inside the cosmological horizon, which is at least in theory possible to
> be observed.
>
> The real size of the universe is completely impossible to know. It
> could be just slightly larger than the observable universe, or it could
> be staggeringly larger. There's just no way of knowing.
>
I have apparently missed a lot since my physics study. I was under the
impression that the size of the universe is of the order of a sphere
with a radius of the age of the universe times the speed of light. Could
you give a pointer to those current theories that you mentioned?
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Kevin Wampler <wampler+pov### [at] uwashingtonedu> wrote:
> Mueen Nawaz wrote:
> > Well, at least I wasn't wrong - Skewes number appeared first.
> I sort of like Skewes number as well, as it provides a useful
> counterexample to provide to someone who thinks that checking the first
> several billion examples of a conjecture makes it virtually certain to
> be true.
Hmm, I wonder if you aren't confusing it with the Polya conjecture...
(which is famous for having a rather big counter-example).
--
- Warp
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andrel <a_l### [at] hotmailcom> wrote:
> I have apparently missed a lot since my physics study. I was under the
> impression that the size of the universe is of the order of a sphere
> with a radius of the age of the universe times the speed of light. Could
> you give a pointer to those current theories that you mentioned?
Glad you asked.
It is, in fact, a rather common misconception that the theory of
relativity limits the speed at which the universe can expand (even
some scientists and cosmology papers hold this misconception).
However, the theory of relativity does not limit the speed at which
the universe can expand. The distance between two points in the universe
can grow faster than c without it breaking relativity. The reason why
people get confused is that they tend to think that if the distance
between two points increases at a rate which is larger than c, that means
that the points are *moving* away from each other faster than c, thus
breaking relativity. However, the points are not moving. The space
geometry between them is changing (in very simplistic terms, new space
appears between them). This is summarized, for example, here:
http://en.wikipedia.org/wiki/Metric_expansion_of_space
"The metric expansion leads naturally to recession speeds which exceed
the "speed of light" c and to distances which exceed c times the age
of the universe, which is a frequent source of confusion among
amateurs and even professional physicists.[1] The speed c has no
special significance at cosmological scales."
No information of any type whatsoever can be transferred by any means
between two points which are recessing faster than c. This is exactly
what causes the so-called cosmological horizon (stub article at
http://en.wikipedia.org/wiki/Cosmological_horizon )
In fact, assuming that the borders of the universe had always grown
at a constant rate of c is against observation. Moreover, it has been
conjectured that the universe suffered an exponential inflation period
at its first moments, which would explain many observed phenomena. This
is an interesting article about the subject:
http://en.wikipedia.org/wiki/Cosmic_inflation
--
- Warp
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"Orchid XP v8" <voi### [at] devnull> wrote in message
news:488e046e$1@news.povray.org...
> somebody wrote:
>
> > Do you really find it easy to visualize the number of subatomic
particles in
> > the visible universe? I don't see why thinking about that would be more
> > informative than, say, 1E80.
>
> Well, the number of grains of sand on the entire English coastline is
> "obviously" a pretty damned big number. And the number of subatomic
> particles in the universe is equally obviously *very* much larger.
>
> Call it a failure of the simplistic human mind, but seeing a handful of
> symbols on a page isn't very impressive. Likening it to something that
> at least "feels real" makes it slightly easier to grasp.
>
> For example, off the top of your head, how long is "10^14 seconds"?
It's 10^14 seconds.
> I mean, is that like, months? Millenia? What?
1 year is close to PI*10^7 seconds (something easy to remember), so it's
close to PI*10^7 years.
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Another interesting (and nerdier) example:
Assume we have a 3GHz processor, and that it can increment a 32-bit
register at each clock cycle. How long does it take to go through all
the values of that register?
Now assume that it's a 64-bit register instead. How long does it
take now?
--
- Warp
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Warp wrote:
> Hmm, I wonder if you aren't confusing it with the Polya conjecture...
> (which is famous for having a rather big counter-example).
Nope, although the Polya conjecture is also an excellent example for
such situations. I somewhat prefer Skewe's number for such things as I
can somehow remember the exact form of theorem from which it arises more
easily.
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On 28-Jul-08 23:27, Warp wrote:
> andrel <a_l### [at] hotmailcom> wrote:
>> I have apparently missed a lot since my physics study. I was under the
>> impression that the size of the universe is of the order of a sphere
>> with a radius of the age of the universe times the speed of light. Could
>> you give a pointer to those current theories that you mentioned?
>
> Glad you asked.
>
> It is, in fact, a rather common misconception that the theory of
> relativity limits the speed at which the universe can expand
is it?
> (even
> some scientists and cosmology papers hold this misconception).
yup, this physicist for instance.
> However, the theory of relativity does not limit the speed at which
> the universe can expand. The distance between two points in the universe
> can grow faster than c without it breaking relativity.
yup, but that has no additional implication for the speed at which the
universe can expand.
> The reason why
> people get confused is that they tend to think that if the distance
> between two points increases at a rate which is larger than c, that means
> that the points are *moving* away from each other faster than c, thus
> breaking relativity. However, the points are not moving. The space
> geometry between them is changing (in very simplistic terms, new space
> appears between them). This is summarized, for example, here:
>
> http://en.wikipedia.org/wiki/Metric_expansion_of_space
yes, but as I said above that has no implication for the size of the
universe.
>
> "The metric expansion leads naturally to recession speeds which exceed
> the "speed of light" c and to distances which exceed c times the age
> of the universe, which is a frequent source of confusion among
> amateurs and even professional physicists.[1] The speed c has no
> special significance at cosmological scales."
>
> No information of any type whatsoever can be transferred by any means
> between two points which are recessing faster than c. This is exactly
> what causes the so-called cosmological horizon (stub article at
> http://en.wikipedia.org/wiki/Cosmological_horizon )
>
> In fact, assuming that the borders of the universe had always grown
> at a constant rate of c is against observation. Moreover, it has been
> conjectured that the universe suffered an exponential inflation period
> at its first moments, which would explain many observed phenomena. This
> is an interesting article about the subject:
>
> http://en.wikipedia.org/wiki/Cosmic_inflation
And what exactly does this all prove? I haven't seen anything in those
links that I did not know (but I admit I did not read everything) and
nothing that even remotely supports your 'The real size of the universe
is completely impossible to know. It could be just slightly larger than
the observable universe, or it could be staggeringly larger. There's
just no way of knowing.' but I might have missed it.
Unless you are in a roundabout way referring to the problem that you can
not define the 'now' for which you are computing the size.
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On 28-Jul-08 23:33, somebody wrote:
> "Orchid XP v8" <voi### [at] devnull> wrote in message
> news:488e046e$1@news.povray.org...
>> somebody wrote:
>>
>>> Do you really find it easy to visualize the number of subatomic
> particles in
>>> the visible universe? I don't see why thinking about that would be more
>>> informative than, say, 1E80.
>> Well, the number of grains of sand on the entire English coastline is
>> "obviously" a pretty damned big number. And the number of subatomic
>> particles in the universe is equally obviously *very* much larger.
>>
>> Call it a failure of the simplistic human mind, but seeing a handful of
>> symbols on a page isn't very impressive. Likening it to something that
>> at least "feels real" makes it slightly easier to grasp.
>>
>> For example, off the top of your head, how long is "10^14 seconds"?
>
> It's 10^14 seconds.
>
>> I mean, is that like, months? Millenia? What?
>
> 1 year is close to PI*10^7 seconds (something easy to remember), so it's
> close to PI*10^7 years.
<nitpick> that'll be 1/PI*10^7 or (as PI^2 is almost 10) PI*10^6 </nitpick>
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"Kevin Wampler" <wampler+pov### [at] uwashingtonedu> wrote in message
news:488e4494$1@news.povray.org...
> Warp wrote:
>> Hmm, I wonder if you aren't confusing it with the Polya conjecture...
>> (which is famous for having a rather big counter-example).
>
>
> Nope,
Heh, I knew you were going to say "Nope" before I looked at your reply.
:)
Time for bed said Zebedee...
~Steve~
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St. wrote:
> Heh, I knew you were going to say "Nope" before I looked at your reply.
> :)
Clearly I have become far too tomato.
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andrel <a_l### [at] hotmailcom> wrote:
> And what exactly does this all prove? I haven't seen anything in those
> links that I did not know (but I admit I did not read everything) and
> nothing that even remotely supports your 'The real size of the universe
> is completely impossible to know. It could be just slightly larger than
> the observable universe, or it could be staggeringly larger. There's
> just no way of knowing.' but I might have missed it.
Uh? I said that the current widely agreed consensus is that the universe
not only can expand faster than c (which you don't seem to disagree with),
but most probably has done so (because that would explain many observed
phenomena). I gave links to wikipedia pages where you could find references
to more material.
Of course there's no absolute *proof* of this. By the very definition
of cosmological horizon it's *impossible* to have an absolute proof of
this (ie. that the universe is larger than the observable universe).
However, currently science most agrees that this is very likely.
Your way of writing seems to imply something like "you have not given
me any proof about this, and thus I don't believe you". In other words,
you still state that the size of the universe is at most the size of
a sphere with a radius of the age of the universe itmes the speed of
light (although you don't seem to deny that the universe *can* expand
faster than c).
Well, where's your proof? Or any serious references, for that matter.
At least I gave you *something*.
--
- Warp
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Invisible <voi### [at] devnull> wrote:
> Does anybody know of a list anywhere that gives examples of really large
> numbers? I'm thinking of things like the number of grains of sand in a
> cubic meter, the brain cells in a human brain, or the number of
> subatomic particles in the visible universe. I for one have no idea even
> approximately "how big" these numbers are.
I'd recommend Project Euler ( projecteuler.net ). It's a couple hundred math
problems for the not-particularly-mathematically-inclined whose answers are all
moderately large integers, e.g. the sum of the digits in 100!. (That's
factorial, not emphasis.) You create an account and can't see the answer...
until you have the answer. Most require programming, but a good share require
nothing more than brute force. Realizing how quickly problems become
intractable with brute force though will give you a real appreciation for how
quickly numbers grow. Fun too.
- Ricky
P.S. Keep your distance from #160. I lost at least a couple evenings to that
one. Never figured it out.
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Invisible wrote:
> the brain cells in a human brain,
>
Human brain was thought to be about 10^11 neurons.
Hit google books search and look for "On Number Numbness" by Hofstadter,
right up your alley I think.
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> For example, off the top of your head, how long is "10^14 seconds"? I
> mean, is that like, months? Millenia? What?
GIYF
http://www.google.com/search?q=1e14s+in+millenia
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Warp wrote:
> Another interesting (and nerdier) example:
>
> Assume we have a 3GHz processor, and that it can increment a 32-bit
> register at each clock cycle. How long does it take to go through all
> the values of that register?
Ooo... 2^32 = 4 billion. 3 GHz = 3,000,000,000 incriments per second. So
we have... a little over 1 second.
> Now assume that it's a 64-bit register instead. How long does it
> take now?
2^64 = (2^32)^2, so if 2^32 takes 1 second, 2^64 should take 4 billion
seconds (which is about 136 years). So... a ****ing long time then.
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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And lo on Mon, 28 Jul 2008 17:44:55 +0100, Warp <war### [at] tagpovrayorg> did
spake, saying:
> Another funny example, which you can use on someone: Assume you have
> a really, really large piece of cardboard which is 1 mm thick. Also
> assume
> that you can fold it in half as many times as you want (thus doubling its
> thickness each time you fold it). How many times do you have to fold it
> before the thickness reaches the Moon?
31 times to roughly match the length of the UK. 40 times will easily reach
the Moon and about 58 times to get very close to the Sun. Yeah I was bored.
--
Phil Cook
--
I once tried to be apathetic, but I just couldn't be bothered
http://flipc.blogspot.com
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And lo on Mon, 28 Jul 2008 23:20:44 +0100, andrel
<a_l### [at] hotmailcom> did spake, saying:
> On 28-Jul-08 23:27, Warp wrote:
>> andrel <a_l### [at] hotmailcom> wrote:
>
> And what exactly does this all prove? I haven't seen anything in those
> links that I did not know (but I admit I did not read everything) and
> nothing that even remotely supports your 'The real size of the universe
> is completely impossible to know. It could be just slightly larger than
> the observable universe, or it could be staggeringly larger. There's
> just no way of knowing.' but I might have missed it.
> Unless you are in a roundabout way referring to the problem that you can
> not define the 'now' for which you are computing the size.
Imagine standing in a circle of light and all around you is darkness. If
the light increases in diameter and the floor increases in size (i.e. not
revealing new areas of floor) and the increase is such that you cannot
reach the circumference by any means; does the floor you're standing on
extend beyond the light?
--
Phil Cook
--
I once tried to be apathetic, but I just couldn't be bothered
http://flipc.blogspot.com
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On Mon, 28 Jul 2008 20:20:59 EDT, "triple_r" <nomail@nomail> wrote:
>moderately large integers, e.g. the sum of the digits in 100!.
Reminds me of how I was quite surprised at the number of unique sorts that can be
obtained from a deck of cards, 52!, or 8.0658175170943878571660636856404e+67.
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On 29-Jul-08 1:26, Warp wrote:
> andrel <a_l### [at] hotmailcom> wrote:
>> And what exactly does this all prove? I haven't seen anything in those
>> links that I did not know (but I admit I did not read everything) and
>> nothing that even remotely supports your 'The real size of the universe
>> is completely impossible to know. It could be just slightly larger than
>> the observable universe, or it could be staggeringly larger. There's
>> just no way of knowing.' but I might have missed it.
>
> Uh? I said that the current widely agreed consensus is that the universe
> not only can expand faster than c (which you don't seem to disagree with),
Oh, but I do. Two points may move from one another faster than c, making
it impossible for one point ever to see the other one. That does not
imply, however, that the universe itself expands faster that c. I think
that is also the point that
http://en.wikipedia.org/wiki/Metric_expansion_of_space is trying to make.
> but most probably has done so (because that would explain many observed
> phenomena). I gave links to wikipedia pages where you could find references
> to more material.
No you gave a website that discusses the 'observable universe' which is
a totally different beast than the size of the universe at some moment
in time.
> Of course there's no absolute *proof* of this. By the very definition
> of cosmological horizon it's *impossible* to have an absolute proof of
> this (ie. that the universe is larger than the observable universe).
> However, currently science most agrees that this is very likely.
What is in the page you referred to seems to me to be a standard
relativistic approach of the concept of what the observable universe is.
'Currently' would then seem to mean 50+ years or so.
> Your way of writing seems to imply something like "you have not given
> me any proof about this, and thus I don't believe you". In other words,
> you still state that the size of the universe is at most the size of
> a sphere with a radius of the age of the universe itmes the speed of
> light (although you don't seem to deny that the universe *can* expand
> faster than c).
I think that the size of the universe (at least from some time, say, a
million years, after the big bang) is expanding with at most c. I have
not followed cosmology really closely over the last 20 years or so, so I
am not really familiar with any recent theories on the early years of
the universe. Yet, the couple of hundred thousand years that span that
era are dwarfed by the nearly 14 billion years when normal physics
applies. What I do know is that I studied special and general relativity
and some cosmology when I was at the university and that your claim that
the real size of the universe may be 'staggering larger' than the
observable universe does not seem to fit what I remember. Although I
moved to applied physics for my masters, I have still some links with
the physics and astrophysics community and I tend to think that I would
have noticed a theory that would imply that.
Please also note that even if someone comes up with a new cosmological
theory that does not mean that from now on that is the accepted theory.
Not even when it is reported on discovery channel. Science journalists
have a habit of suggesting that (and not only for cosmology), but
science does not work that way.
> Well, where's your proof? Or any serious references, for that matter.
> At least I gave you *something*.
Yes, a page that was about something else and if anything was
contradicting your claim. ;)
BTW the concept of the universe having a size at one point in time is
rather useless, indeed because of relativistic reasons. Observable
universe is a more useful concept, I grant you that one.
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andrel wrote:
> No you gave a website that discusses the 'observable universe' which is
> a totally different beast than the size of the universe at some moment
> in time.
Two fallacies: (1) "observable universe" technically is the same as
"universe". (2) "moment in time" is a meaningless term, even for things
fairly close together, let alone separated in a spacelike way.
In other words, define "universe" first. This gets you into Clinton
areas, of trying to define what "is" is.
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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Invisible wrote:
> Does anybody know of a list anywhere that gives examples of really large
> numbers?
I don't know about "really big", but here's a start for moderately large.
http://www.kokogiak.com/megapenny/
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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On 29-Jul-08 22:09, Darren New wrote:
> andrel wrote:
>> No you gave a website that discusses the 'observable universe' which
>> is a totally different beast than the size of the universe at some
>> moment in time.
>
> Two fallacies: (1) "observable universe" technically is the same as
> "universe".
why? For many people God is part of this universe, yet not part of the
observable universe. (I know you did not mean that).
> (2) "moment in time" is a meaningless term, even for things
> fairly close together, let alone separated in a spacelike way.
yes. I am aware of that. Here informally meant to mean every point in
space time that has the same value for the time component. So rather
undefined indeed.
> In other words, define "universe" first. This gets you into Clinton
> areas, of trying to define what "is" is.
Hmm, I did not know he once tried to define the universe. I know that
his definition of sex was peculiar, so I assume you refer to that.
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andrel wrote:
> why? For many people God is part of this universe, yet not part of the
> observable universe. (I know you did not mean that).
Because "universe" means "everything there is". Define "is" in a way
that includes things that do not and never will have any chance of
affecting any sort of causality in your observations.
>> (2) "moment in time" is a meaningless term, even for things fairly
>> close together, let alone separated in a spacelike way.
>
> yes. I am aware of that. Here informally meant to mean every point in
> space time that has the same value for the time component. So rather
> undefined indeed.
It's the informality that kills you, tho. There's no such thing as "the
same time component". The same point in spacetime has many values for
the time component. You can have two things that happen "at the same
time", and I experience A happening before B, and you experience B
happening before A, and Andrew experiences them both happening at the
same time. There isn't any universal clock you can use to determine the
"time component" for a place in space, any more than there's any
universal clock you can use to determine the Y component of a parabola.
>> In other words, define "universe" first. This gets you into Clinton
>> areas, of trying to define what "is" is.
>
> Hmm, I did not know he once tried to define the universe. I know that
> his definition of sex was peculiar, so I assume you refer to that.
No, I was referring to his arguing over what "is" is. Does something
completely outside your causality exist? Is there an actual definition
you could come up with that would make "the universe" bigger than "the
observable universe"?
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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andrel <a_l### [at] hotmailcom> wrote:
> What I do know is that I studied special and general relativity
> and some cosmology when I was at the university and that your claim that
> the real size of the universe may be 'staggering larger' than the
> observable universe does not seem to fit what I remember.
You might find this article interesting:
http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03
A quote:
"Notice that, according to Hubble's law, the universe does not expand
at a single speed. Some galaxies recede from us at 1,000 kilometers
per second, others (those twice as distant) at 2,000 km/s, and so
on. In fact, Hubble's law predicts that galaxies beyond a certain
distance, known as the Hubble distance, recede faster than the speed
of light. For the measured value of the Hubble constant, this distance
is about 14 billion light-years."
And btw, I didn't say the universe *is* staggeringly larger than the
observable part of it (observable by us, that is). What I said is that
it's perfectly *possible* for it to be enormously larger. We simply have
no physical means of knowing. However, AFAIK, current consensus is that
it's probable that the size of the universe is significantly larger than
the observable part, because that's one good explanation of some observed
phenomena related to the big bang theory.
--
- Warp
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On 29-Jul-08 23:35, Warp wrote:
> andrel <a_l### [at] hotmailcom> wrote:
>> What I do know is that I studied special and general relativity
>> and some cosmology when I was at the university and that your claim that
>> the real size of the universe may be 'staggering larger' than the
>> observable universe does not seem to fit what I remember.
>
> You might find this article interesting:
>
> http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03
>
> A quote:
>
> "Notice that, according to Hubble's law, the universe does not expand
> at a single speed. Some galaxies recede from us at 1,000 kilometers
> per second, others (those twice as distant) at 2,000 km/s, and so
> on. In fact, Hubble's law predicts that galaxies beyond a certain
> distance, known as the Hubble distance, recede faster than the speed
> of light. For the measured value of the Hubble constant, this distance
> is about 14 billion light-years."
That is what I said also. So given the time of day, I skip that one for
tonight.
> And btw, I didn't say the universe *is* staggeringly larger than the
> observable part of it (observable by us, that is). What I said is that
> it's perfectly *possible* for it to be enormously larger. We simply have
> no physical means of knowing. However, AFAIK, current consensus is that
> it's probable that the size of the universe is significantly larger than
> the observable part, because that's one good explanation of some observed
> phenomena related to the big bang theory.
AFAIK that is not the consensus. I.e. if significant means a couple of
times. If it means a P<.05 that is probably true.
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On 29-Jul-08 23:26, Darren New wrote:
> andrel wrote:
>> why? For many people God is part of this universe, yet not part of the
>> observable universe. (I know you did not mean that).
>
> Because "universe" means "everything there is". Define "is" in a way
> that includes things that do not and never will have any chance of
> affecting any sort of causality in your observations.
There may be things that you know existed or must have existed once but
now are outside of your observation forever.
>>> (2) "moment in time" is a meaningless term, even for things fairly
>>> close together, let alone separated in a spacelike way.
>>
>> yes. I am aware of that. Here informally meant to mean every point in
>> space time that has the same value for the time component. So rather
>> undefined indeed.
>
> It's the informality that kills you, tho.
I think I acknowledged that already.
> There's no such thing as "the
> same time component". The same point in spacetime has many values for
> the time component. You can have two things that happen "at the same
> time", and I experience A happening before B, and you experience B
> happening before A, and Andrew experiences them both happening at the
> same time. There isn't any universal clock you can use to determine the
> "time component" for a place in space, any more than there's any
> universal clock you can use to determine the Y component of a parabola.
>
>>> In other words, define "universe" first. This gets you into Clinton
>>> areas, of trying to define what "is" is.
>>
>> Hmm, I did not know he once tried to define the universe. I know that
>> his definition of sex was peculiar, so I assume you refer to that.
>
> No, I was referring to his arguing over what "is" is.
Ok, never heard that one before.
> Does something
> completely outside your causality exist? Is there an actual definition
> you could come up with that would make "the universe" bigger than "the
> observable universe"?
As said, I think you can.
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andrel <a_l### [at] hotmailcom> wrote:
> AFAIK that is not the consensus. I.e. if significant means a couple of
> times. If it means a P<.05 that is probably true.
If, according to you, the universe is expanding at velocity c (what does
that even mean?), and given that current measurements seem to indicate that
the expansion is actually accelerating, how do you explain this?
Two questions:
1) What do you mean by "expanding at a velocity of at most c"? Where is
this c velocity, and compared to what?
2) Since you have clearly agreed that the expansion of the universe is
not bound to speed limitations between objects, then what is the
phenomenon which limits the expansion of the universe to c?
In other words, you seem to say "yes, it is possible for two points in
the universe to recede faster than c from each other, but that's not
happening anywhere in the universe". Why not?
--
- Warp
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andrel wrote:
> There may be things that you know existed or must have existed once but
> now are outside of your observation forever.
I'm pretty sure those count as part of the "observable universe". They
have a causal influence on your observations.
>> Does something completely outside your causality exist? Is there an
>> actual definition you could come up with that would make "the
>> universe" bigger than "the observable universe"?
>
> As said, I think you can.
I'd like to hear it. :-)
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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Warp wrote:
> on. In fact, Hubble's law predicts that galaxies beyond a certain
> distance, known as the Hubble distance, recede faster than the speed
> of light. For the measured value of the Hubble constant, this distance
> is about 14 billion light-years."
Here's an interesting question, tho: Consider the galaxies that are
receding at (1-epsilon)*c from us. Galaxies[1] that will cross the
boundary from timelike to spacelike distances in (say) the next year.
It can't accelerate faster than light, right? Is it going to go slower
and slower, like an object falling into a black hole will, from our
point of view?
Hmmm, thinking on it, if the expansion is caused by expanding space,
maybe time doesn't seem to slow down? Do you not get relativistic time
dilation from expanding space? If not, why do photons red-shift? If so,
why would we see any galaxy ever "cross" the speed of light as it recedes?
I did see an interesting speculation once that the 3-degree background
radiation was actually all the galaxies piled up against the lightspeed
barrier, but I have no idea how much sense that makes.
[1] Or planets, or specs of dust, or whatever's small enough for you to
accept this happens.
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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Darren New wrote:
> I don't know about "really big", but here's a start for moderately large.
>
> http://www.kokogiak.com/megapenny/
Ooo... interesting.
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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