A Short Tutorial On Big Numbers

People often think that they can think of large numbers[1]. A couple of weeks ago, a discussion came up in the newsgroup sci.math, where the author 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 a few familiar number magnitudes, using Knuth's Up-Arrow Notation.

Horizon: 380000000=3.8*10^8

This is the distance from the Earth to the Moon in meters. It took Apollo 11 around a week to travel this distance. It takes light which travels at c=3*10^8 m/sec a little more than a second to cover it.

Horizon: AU=150000000000=1.5*10^11

This is the distance between the Earth and the Sun in meters, more commonly known as an AU or astronomical unit. It takes light approximately 8.2 minutes to cover it.

Horizon: 115*AU=1.7*10^13

This is the distance the most remote human-made object (Voyager 1) has travelled so far.

Horizon: 9460800000000000=9.46*10^15

This is a light-year, or the distance light covers in one year in meters.

Horizon: 4.3*9.46*10^15=4*10^16

This is the distance to the closest star after the Sun at 4.3 light-years, Proxima Centauri.

[... huge leap...]

Horizon: 1.23*10^26

This is the approximate radius of the known (observable) universe in meters. Light covers this distance in 13 billion years.

Horizon: 10^58

This is the approximate number of electrons in the known universe.

Horizon: 2*10^71

This is the approximate number of all chessboard configurations[2].

Now let's remember the number mentioned by the author.

d(9) = A(9,9,9) = 9^^^^^^^^^9 = 9^(9)9. To estimate the unimaginable magnitude of this number, let's use some heuristics:

Horizon: 9^(2)4

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 10000000001-digit number. Maple is already choking with this number. We are already far past the number of electrons in the universe by many orders, and the total number of chessboard configurations also by many orders.

10^10^10^10 is a 10^10^10+1-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 approximately:

10^10^10(g)*10^(-10)(m/g) = 10^(10^10-10) = 10^9999999990 meters. The diameter of the known universe is approximately 26 billion light years (see above), 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.

Horizon: 9^(2)9

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.

Horizon: 9^(3)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.

Horizon: 9^(9)9=d(9)

Now leap mentally (via hand-waving and coffee) and go to the unimaginable:

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).

Horizon: d(d(9))=d(2)(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.

Horizon: d(120)(9)

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).

Horizon: [9,9,121,2]

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[2]. 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) =>
[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).

Horizon: [9,9,122,2] and beyond

Now try to imagine the size of numbers like,

[9,9,999,2], [9,9,999,9] or something like [999,999,999,999,999,999,999,999,999].

You will see that your mind will most likely fail you here. Such numbers are almost certainly beyond comprehension relative to unity or smaller numbers[3][4].

Notes/References

  1. You can define your mind's horizon to be the largest number you can perceive clearly on the headers in bold, relative to unity, without resorting to tricks. Your horizon will be approximately equal to the number near the text where your mind chokes. See article Is The Mind Infinite?.
  2. To get a sense of how humongous some the numbers in this page are, you can download a Maple 18 document which allows some symbolic calculations (for up to 3 arrows (a->b->b->d) with Conway's notation) here.
  3. For example, see document Every Chess Configuration.
  4. For an in depth view of large numbers, consult Robert Munafo's Large Numbers (Page 4).

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