The number googol is 10^100. As the number of atoms in the universe is estimated at something like 10^79, how is this number ever practical?
Not to even mention googolplex, does the number googol have any real use?
In computing possible ways of arranging n objects like a deck of cards, the number n! comes up. For large n big numbers are not unusual.
For a fun read on really big numbers get hold of Isaac Asimov’s essay on Skewes Number. This was thought to be the largest number ever encountered in a mathematical question at the time the essay was written.
I’ve forgotten the name of the essay but maybe someone will be along to give us a hand.
Aha! The Azimov essay is Skewered! and it can be found in his book Of Matters Great and Small
Skewes’ Number arose in a proof concerning prime numbers and equals:
10 to the 10 to the 10 to the 10 to the 34.
(in this notation a googol would be 10 to the 10 to the 2)
I can’t figure out the coding to write it any differently.
How about 10[sup]34,000[/sup]
Not even close, friedo. The number you just wrote has only 34,001 digits. 10 to the 34 has 35 digits. 10 to the 10 to the 34 has 10,000,000,000,000,000,000,000,000,000,000,001 digits. 10 to the 10 to the 10 to the 34 has ((10 to the 10 to the 34) + 1) digits. 10 to the 10 to the 10 to the 10 to the 34 has ((10 to the 10 to the 10 to the 34) + 1) digits.
There are actually two Skewe’s numbers, the first (which assumes that the Reimann hypothesis is true) is as noted above
10^10^10^34.
The second is substantially larger and is based on the assumption that the Reimann hypothesis is false
10^10^10^10^3
Both are mind bogglingly large
WTF?
when I saw 10^10^10^10^3 I was like that’s it? And then the vastness hit me. It almost seemed like a religous experance. I don’t know where to even begin to rationalize it’s size now.
Whereas 10^10^10^34 was a total yawner for you? A bit of pocket change? 
I would scoff (amiably) at anyone who claims to have rationalized even the lowly googol, let alone the Skewes numbers and those beyond. Even the fact that a googol is roughly the number of subatomic particles in our universe isn’t helpful. After all, we can’t sense individual subatomic particles, as well as sense all of them collectively — all 10[sup]80[/sup] of them — and thereby get a feel for the size of a googol. Anything larger still is even more hopelessly beyond our intuition.
In my own musings on huge numbers, I’ve decided that the limit of my grasp is about 10[sup]31[/sup], give or take, and even that modest mental achievement is debatable. I base it though on the largest physical ratio I can visualize: the volume of the Earth expressed in cubic millimeters. This would be, for example, the approximate number of sand grains in the Earth, if it were made entirely of sand. (Or since it’s the holidays, substitute chocolate sprinkles.)
Obviously we can still perform computations with all these large numbers, and compare their relative sizes, and discover their many properties. But I really doubt any person can rationalize them, if I took your meaning right. They’re just too far outside the realm of human perception and experience. Googolplex and the Skewes numbers are like stars in the sky: varying greatly in distance from us, yet all appearing to be on the surface of a giant sphere, as if they were equally remote.
There could be a biological reason for this “blindness”: natural selection never provided any pressure for us to comprehend the super huge or the super small. After all there’s little practical advantage, living in the wild, in being able to sense an Angstrom’s worth of length or a billion years of time. Our senses and mental images are limited to things of human scope, plus or minus a few orders of magnitude in each direction.
Much of all this is opinion of course.
I disagree with your nitpick. An exponent raised to another exponent certainly becomes the product of the two exponents.
(a[sup]b[/sup])[sup]c[/sup] = a[sup]bc[/sup]
Your explanation assumed that the exponentiation is right-associative, as a^(b[sup]c[/sup]). But that’s not good enough to say he was wrong, because the notation used in 10^10^10^10^34 doesn’t provide a clue either way. And exponentiation is not associative in both directions equally.
In a lot of computer languages, such notation would be right-associative. But as most people to solve that expression, and they’re going to compute from left to right.
It may be moot, since the number was subsequently shown to be two numbers, both distinct from the original reference. But whatever the actual number is (are), the question remains…does the spirit of Skewes’ number as written in the post imply left- or right-associative exponentiation?
No mathematician in the world would ever assume that the notation meant anything other than a^b[SUP]c[/SUP]. I can’t imagine why anybody in computers would think differently.
The number is indeed
10[sup]10[sup]10[sup]34[/sup][/sup][/sup]
and is right-associative, if I understand your terminology correctly, meaning that first you compute 10[sup]34[/sup] and then 10 to the number that results, and then 10 to the number that results from that.
It may be a mistake to think that because something can’t exist in our universe, it can’t be practical. Consider the circle: no exact example exists anywhere, but it’s a very useful concept.
Exponentiation is always right associative. Always.
10^10^10^34 is equivalent to either
(((10^10)^10)^10)^34
or
10^(10^(10^(10^34)))
I think the semi-consensus in this thread is that it’s the second, as it is in the calc program (to pick an example of an algebraic-notation computer language).
There is honest confusion here, and I think we need an authoritative resolution.
Agreed Expano
I believe the convention for evaluating exponents is “right associative”. (A term I never heard before). As a rule of thumb, I’ve found when dealing with exponents raised to powers, etc, whatever yields the larger number is the correct evaluation.
An old math puzzle states ‘what is the largest number you can make with 3 digits’?
The answer is 9[sup]9[sup]9[/sup][/sup]
You could evaluate this number as 9[sup]9[/sup] which equals
387,420,489.
Then raisng this to the ninth power we get:
1.066 x 10[sup]77[/sup] (A relatively puny number for this discussion).
Now going in the other direction, 9[sup]9[/sup] still equals:
387,420,489.
Then we get 9 and raise it to the power of that number:
9[sup]387,420,489[/sup]
Now there’s a number more worthy of this thread.
As far as anything more than a googol being meaningless? Well, here is something in reality much larger than a googol. (Granted the figures are somewhat speculative). But, if we were to express the volume of the observable Universe in terms of electron volumes, it would be about 4 x 10[sup]122[/sup] electron volumes.
Source is http://www.1728.com/convert.htm
(yes, it is my site)
As our late, good friend Isaac Asimov said in his essay T Formation (I think) there is a fascination with large numbers. He said that he didn’t think anyone could explain it but “Believe me, it exists.”
I was still a bit shaken up when I wrote that. 10^10^10^34 was astounding also when I truely considered it’s size. 10^10^10^10^3 was the one I first realized what it meant. I was just lucky I was not eating when I read that or I might have choked.
I won’t even try to rationalize what a skews (either true or false version) worth of stuff looks like. The long version is 3,000 digits long if I did my math right. It’s hard to explain. It’s not how many 10^10^10^3 would be it’s just it’s extance. Then further thought leads to what about 10^10^10^10^10^3? and then numbers even larger. I discovered 10^10^10^10^10^10^3 makes the windows calculator screw up by the way. Trying to rationalize time gave me problems but I would never thought they could be riveled with 10 chareactors
So is a googleplex a useless number?
What is a googolplex? Compare it to a googol please.
-RED MATRIX
Also, i’m remembering something in Sky & Telescope that the visible universe is one dimention. This “sphere” of dimention has the probability to repeat itself, in a pattern, where only 1 atomic particle is different… yeilding a second universe… hmmm…
A googolplex is 10 raised to the power of a googol or 10[sup]10[sup]100[/sup][/sup] or 10[sup]10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000[/sup]. It is unimaginably huger than a googol.
Sets of things can be much larger than the things themselves.
The salesman problem is a classic. The salesman has to visit 100 cities around the U.S. What is the shortest route to take? There is not an algorithm that will give the best answer in all cases. Approximations may be calculated, but a true answer would involve trying every path from every point to every point. There is an enormous number of these paths, so many that even fast computers would take longer to do them all than would be worth it.
It’s not hard to come up with theoretical problems in which the numbers you would be working with are huge. Weather patterns, climate changes, nuclear explosions, the evolution of solar systems or of galaxies for that matter, all those things that supercomputers were invented to batter into submission.
Then there are the more abstract problems in mathematics of the sort in which Skewe’s number first popped up. That number is extremely small compared to later numbers that require being used. Despite abstractness, math gets used in unexpected ways all the time.
Large numbers have surprising ways of turning out to be very useful. Indeed, new forms of symbol representation have been devised to handle numbers that make Skewe’s number a teensy parvenu wannabe by comparison.
Googleplex may or may not turn out to be useful, but for sure numbers unimaginably huger will be.