People often think that they can think of large numbers. A couple of weeks ago, a discussion came up in the newsgroup sci.math, where I gave the example d(120)(9), where d(m)(n) denotes m-repeated composition, and d(n)=A(n,n,n), where A(k,l,m) is the 3-argument Ackermann function.
d(n) corresponds to the Ackermann Number A(n). To get just a tiny idea of the humongous magnitude of that number, consider just the first few terms, using Knuth's Up-Arrow Notation:
d(9) = A(9,9,9) = 9^^^^^^^^^9 = 9^(9)9. To estimate the unimaginable magnitude of this number, let's use some heuristics:
9^^9 = 9^9^9^9^9^9^9^9^9. That's just with 2 arrows, tetration. Let's ignore the margin of error between 9 and 10 and try to estimate.
10^10 is 1 with 10 zeroes or 10000000000.
10^10^10 is 1 with 10^10 = 10000000000 zeroes or a 10000000000-digit number. Maple is already choking with this number. We are already far past the number of electrons in the universe (~10^60) by many orders, and the total number of chess board configurations (~10^120) also by many orders.
10^10^10^10 is a 10^10^10-digit number. If we used a glyph which was 1 Angstrom thick, and we stacked the glyphs side by side, writing this number down we would need:
10^10^10(g)*10^(-10)(m/g) = 10^(10^10-10) = 10^9999999990 meters. Unless i am making a typo, and if memory serves right the diameter of the known universe is approximately 26 billion light years (give or take a couple of billion), hence:
26000000000(ly)*300000000(m/sec)*60(sec/min)*60(min/h)*24(h/d)*365(d/y) = 245980800000000000000000000 m ~ 10^27 m.
This means that our number, 10^10^10^10, would need approximately 10^9999999990/10^27 = 10^9999999966 universes the size of our own, stacked side by side as spheres to accommodate it, with each glyph being one Angstrom thick, with the number piercing through all universes diametrically.
The last number, 10^10^10^10, was close to 9^9^9^9 = 9^^4 = 9^(2)4. Now try to imagine the magnitude of:
9^9^9^9^9^9^9^9^9 = 9^^9 = 9^(2)9.
After you manage to "see" the last number, convince yourself that it is very far below 9^(3)9 = 9^^^9 = 9^^9^^9^^9^^9^^9^^9^^9^^9.
Now leap mentally (via hand-waving and coffee) and go to:
9^(9)9 = 9^^^^^^^^^9 = 9^^^^^^^^9^^^^^^^^9^^^^^^^^9^^^^^^^^9^^^^^^^^9^^^^^^^^9^^^^^^^^9^^^^^^^^9.
That's just d(9) = d(1)(9) or the ninth Ackermann Number, A(9). Then consider, d(2)(9)=A(d(9),d(9),d(9)) = d(9)^(d(9))d(9) = d(9)^^^...^^^d(9), with d(9) up-arrows. That's the A(9)'th Ackermann Number.
Finally, if your brain allows, try to imagine the magnitude of the final number: d(120)(9) = d(d(...d(9)...))) (120 parentheses for the indicated space).
If you think Ackermann Numbers are large, wait till you see Conway's Chained Arrow Notation. The correspondence between Conway's arrow notation and Knuth's Up-Arrow notation is relatively easy:
a®b®1=a^b=ab (exponentiation)
a®b®2=a^^b=ba (tetration)
a®b®3=a^^^b (pentation)
...
Larger numbers inside the arrows and more arrows correspond to much larger final numbers. For simplicity we will use the correspondence:
a®b®c®d...=[a,b,c,d,...].
This means, for example, that:
a®(b®c)®d=[a,[b,c],d], etc.
Let's consider the number [9,9,9,2]. Following the rules on the Wikipedia page:
[9,9,9,2]
[9,9,[9,9,[9,9,[9,9,[9,9,[9,9,[9,9,[9,9,[9,9],1],1],1],1],1],1],1],1] (Conway Rule #1)
[9,9,[9,9,[9,9,9,9,[9,9,[9,9,[9,9,[9,9,[9, 9]]]]]]]]] (Conway Rule #2)
[9,9,[9,9,[9, 9,[9,9,[9,9,[9,9,[9,9,[9,9,387420489]]]]]]]] (Conway Rule #3)
The innermost number, [9,9,387420489] is 9^(387420489)9 = 9^...(387420489 arrows)...^9.
This is already MUCH larger than d(9)=9^(9)9=A(9). This means:
[9,9,387420489] > d(9) Þ
[9,9,[9,9,387420489]]=9^([9,9,387420489])9 > 9^(d(9))9
Writing it out explicitly using Ackermann Numbers:
(9^((9^((9^((9^((9^((9^((9^(9^(387420489)9)9))9))9))9))9))9))9)=
9^...^9, with (9^((9^((9^((9^((9^((9^(9^(387420489)9)9))9))9))9))9))9) arrows!!!
This means that d(120)(9) is of similar order to [9,9,120,2]. Thus [9,9,121,2] > d(120)(9).
Now try to imagine the size of numbers like:
[999,999,999,999,999,999,999,999,999,999,999,999] or even with more arrows!
Here is a Maple version 9 classic worksheet which implements Conway's arrow notation using lists, with examples from Wikipedia.