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Old 2008-10-02, 02:18   #859
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Quote:
Originally Posted by Orgasmic Troll View Post
I think M45 and M46 are outside of our ability to really grasp the notion of how big it is.
How about: M46 is 1 less than 1/(probability of flipping a fair coin 43,112,609 times and getting all heads)?

(assuming that we can grasp the notion of how big 43,112,609 is)

Last fiddled with by jinydu on 2008-10-02 at 02:19
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Old 2008-10-02, 02:37   #860
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This reminds me to watch (again) Rosencrantz & Guildenstern Are Dead
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Old 2008-10-02, 13:42   #861
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Quote:
Originally Posted by jinydu View Post
How about: M46 is 1 less than 1/(probability of flipping a fair coin 43,112,609 times and getting all heads)?

(assuming that we can grasp the notion of how big 43,112,609 is)
Do you have an intuitive notion of how long it would take to flip a coin 43 million times? (btw, if you start doing any arithmetic in your head, that's not intuitive)
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Old 2008-10-02, 15:03   #862
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Quote:
Originally Posted by Orgasmic Troll View Post
I think M45 and M46 are outside of our ability to really grasp the notion of how big it is.
Sound energy is something people can grasp.

We know that M46 is ~ 130 million dB bigger than, say, the number of fingers on your hand.

Code:
130M dB = 10 log10 ((10^13M)/5)
The explosion of Krakatoa is usually quoted as the loudest thing witnessed on earth, and had a sound pressure of 20,000 kPa, or about 170 dB louder than the sound of rustling leaves.

Code:
170 dB = 20 log10 (20e3 / 60e-6)
So M46 is bigger than the number of fingers on your hand by approximately the same amount that 10^6.5 million exploding Krakatoa's are louder than the sound of rustling leaves.

Code:
 
130M dB ~= 20 log10 (10^6.5e6 * (20e3/60e-6))
Errrm, ok, nevermind... still can't really grasp that.

p.s. does anything in nature follow a log log relationship? That's what we need here to make the right analogy.
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Old 2008-10-02, 19:47   #863
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Quote:
Originally Posted by bsquared View Post
p.s. does anything in nature follow a log log relationship?
Sure: the natural logarithm of trunk diameter.
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Old 2008-10-02, 19:56   #864
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Quote:
Originally Posted by cheesehead View Post
Sure: the natural logarithm of trunk diameter.
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Old 2008-10-02, 20:03   #865
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This is actually an example from math, not from the physical world, but...

How about \sum{\frac{1}{p_i}}, where the sum is taken over all primes less than or equal to 2^43112609 - 1? I think the result is approximately 17.5 (but I'm not sure because of the o(1) term; I'm not sure if M46 is "large enough" to make it less than 0.1).

Last fiddled with by jinydu on 2008-10-02 at 20:08
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Old 2008-10-04, 16:12   #866
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\sum_{p<n}\frac1p\approx\log\log n+M

with Mertens' constant M = 0.2614972...

Quote:
Originally Posted by jinydu View Post
How about \sum{\frac{1}{p_i}}, where the sum is taken over all primes less than or equal to 2^43112609 - 1? I think the result is approximately 17.5 (but I'm not sure because of the o(1) term; I'm not sure if M46 is "large enough" to make it less than 0.1).
The error should be very small. A simple geometric model of the errors at 10^k for k in {3, 4, 5, 6, 7, 8} (a good fit, R = 0.99987) suggests all* the error will be in your choice of M. The error around 1e8 is only about 4 millionths...

I expect your result is 17.47431035675907928519109475..., accurate to all displayed decimals.

* Unless you use more than, say, ten million digits of M.
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Old 2008-10-04, 19:30   #867
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Default At Long Last!

http://www.perfsci.com/wall-art.asp#primes
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Old 2008-10-04, 21:15   #868
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I've always liked their "Click for larger view" gag.
I strongly suspect that they reused the same picture, too!
That's cheating!
Attached Thumbnails
Click image for larger version

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Last fiddled with by Batalov on 2008-10-04 at 21:16
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Old 2008-10-04, 21:38   #869
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Actually, the "larger" image is twice the size. Still, it's so blurry that it might as well be the first size (not that I'm expecting to be able to see the digits or something, but come on...).
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