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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.
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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