From: Tony Lezard <tony@mantis.co.uk>
Newsgroups: sci.math,rec.puzzles
Subject: Re: Naming Large Numbers
Date: Thu, 17 Dec 92 12:11:03 GMT
Followup-to: rec.puzzles
Distribution: world
Organization: Mantis Consultants, Cambridge. UK.

amorgan@Xenon.Stanford.EDU (Crunchy Frog) writes:

> Let me invent a notation.  a&b means
> 
> a^a^a^a ...... ^a
> \               /
>      b times
> 
> I submit that writing 10^100 & 10^100 is slightly tricky in standard
> notation.  I believe Don Knuth invented something called arrow
> notation to describe really *huge* numbers like Skewes number.

Skewes' number isn't all that bad.

For small n, integral(2 to n) dx/ln x overestimates pi(n), the number
of primes <= n. In 1933 Skewes proved that it becomes an underestimate
somewhere before n reaches 10^(10^(10^34))), assuming the truth of the
Riemann Hypothesis. This had been the largest number used in a serious
theorem until Mr. Graham came along...

> Consider another case.  Let triangle x be x^x.  Let square x be
> triangle triangle ..... x.  Let circle x be square square ..... x
> \   x times     /                            \   x times     /
> 
> circle 2 is a *large* number.  It can't be written with standard
> notation.  circle 10 is beyond the mind-pummling huge and moves into
> the brain squooshingly gigantic.  Let us then pause to consider
> circle 10^100 & circle 10^100 which I modestly suggest be called
> frog's number.
> 
> What I have always found... humbling is the word I want I guess
> is that however large these numbers may be, they aren't a patch on
> infinity.  Even frog's number is *nothing* compared with some
> of the *really* big numbers out there.

Fun puzzle: devise (or read about) notations for huge numbers. Then
construct some biggies in different notations and see if you can work
out which is bigger. For example, is Graham's number bigger than frog's
number as defined above?

To help you decide, here is the recipe for Graham's number, taken from
the Penguin Dictionary of Curious and Interesting Numbers. It used
Knuth's "arrow" notation,which I also describe:

3^3 means 3 cubed, as usual.
3^^3 means 3^(3^3) or 3^27, or 7625597484987, which is still graspable.
3^^^3 means 3^^(3^^3) or 3^^7625597484987 which is
3^(7625597484987^7625597484987) which is getting a bit serious now.

3^^^^3 = 3^^^(3^^^3) is of course so huge that you can't even imagine
the height of the tower of exponents!

Now, consider 3^^^^^.....^^^3 where the number of arrows between the 3's
is 3^^^^3. Got that in your head? Right. Now construct 3^^^^....^^3 in
which the number of arrow is that previous 3^^...^^^3 number.

You get the idea. Graham's number is what you get when you continue
this process until you are *63* steps away from 3^^^^3. Yow!

Well if you ask me Graham's number takes the cake. Now, what about
Ackermann(100,100)? (Ackermann's function was defined in rec.puzzles
recently) or Ackermann(1000000,1000000)? More generally, what is the
least n such that Ackermann(n,n) exceeds Graham's number?

All less than an insignificant drop in the ocean compared to even the
smallest of infinities!

Followups set to rec.puzzles.

--
Tony Lezard IS tony@mantis.co.uk OR tony%mantis.co.uk@uknet.ac.uk OR things
like tony%uk.co.mantis@uk.ac.nsfnet-relay OR (last resort) arl10@phx.cam.ac.uk
      "The two most common things on Earth are hydrogen and stupidity."
              -- Geoff Miller (geoffm@purplehaze.Corp.Sun.COM)