Playing with numbers...: zorac

Playing with numbers...

I was idly playing with numbers in my head the other day, something to occupy my mind while I trudged back from Sainsbury's with a load of shopping. In particular, I was looking at number sequences (you know, what number comes next - 1, 3, 5, 7, ? or 1, 4, 9, 16, ?) and came up with the following (related, but calculated in opposite ways):

1 ... 4 ... 19,683 ... ?
1 ... 4 ... 7,625,597,484,987 ... ?

The answer to the first series is a mere
340,282,366,920,938,463,463,374,607,431,768,211,456
The second, as you can probably guess, is larger. Much larger. In fact, it's exactly
2^{26,815,615,859,885,194,199,148,049,996,411,692,254,958,731,641,184,786,755,447,122,887,443,528,060,147,093,953,603,748,596,333,806,855,380,063,716,372,972,101,707,507,765,623,893,139,892,867,298,012,168,192}
or approximately
5 x 10^{8,072,304,726,028,225,379,282,369,632,412,842,737,810,037,345,818,396,835,407,833,195,922,269,447,488,939,454,873,506,883,657,604,175,736,107,203,603,983,535,115,431,016,379,958,411,831,793,261,615,652,260}
Even more approximately, that's
5 x 10^{8 x 10¹⁵³}
Time to play with some big numbers:
5 x 10^{googol x 8 x 10⁵³}
5 x googolplex^{8,000 x 10⁵⁰}
5 x googolplex^{8,000 √googol}

However I put it, that is one very big number, almost impossible to visualise - Asimov's essay Skewered! details just how hard to visualise big numbers can be. Final challenge, if you want to prove that you actually solved the puzzle: explain how the series was derived.

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zorac

September 5 2003, 14:16:15 UTC 17 years ago

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I can't argue with that. Trying to calculate those numbers in one's head is however difficult. Particularly when they come out differently in different directions - as explained below, both of the series are the same, just with the parentheses in different places. Using smaller numbers (and no more sups)

((2^2)^2)^2 = (4^2)^2 = 16^2 = 256
2^(2^(2^2)) = 2^(2^4) = 2^16 = 65,536

With 4s in place of the 2s, you get the original final answers.

imc

September 5 2003, 14:51:27 UTC 17 years ago

But of course ((4^4)^4)^4 is the same as 4^(4*4*4) so your first series condenses to that of n^(n^(n-1)).

September 5 2003, 16:08:50 UTC 17 years ago

Ooh! That's neat. Should've figured that one myself.

Unfortunately, the second series isn't no neatly condensing (unless there's an operater which is to ^ as Σ is to + and (IIRC) Π is to *). Best I can do is functional:

f(n) = g(n, n);
g(x, 0) = 1;
g(x, y + 1) = x ^ g(x, y)

jiggery_pokery

September 5 2003, 16:17:06 UTC 17 years ago

There's Arrow Notation which I think is what we're after and which uses uses a ^ symbol with a | symbol under it. I cannot think how to make this character on a PC, which is annoying, because it used to be very easy on a Spectrum. (SYMBOL SHIFT and 6, I think...?) (Checks - not particularly close. SYMBOL SHIFT and H.)

There's almost certainly a Unicode glyph for it.

September 5 2003, 16:42:00 UTC 17 years ago

You're a star!

n↑↑n

September 5 2003, 17:03:02 UTC 17 years ago

But the Spectrum's arrow character is the `^' character (both in ASCII value and in function) - it just looks different.

What's more, the Arrow Notation works perfectly well using `^' instead of an arrow, because the single-arrow operator is the same as the power operator.

I think the most common use for Arrow Notation is in describing Graham's Number (which you can google for yourself as I can't be bothered to at this time of night).

By the way, on an unrelated note, I've just noticed that all the dates on this comment page have ` ' (sic) in them - that surely can't be what was wanted!

September 5 2003, 17:17:03 UTC 17 years ago

Thanks for the arrow notation tips and pointers to more big shiny numbers.

The   glitch was a temporary thing while I was fiddling with S2 to try and work around a table rendering glitch with entries whereby wrapping a <nobr> around the while date/time was causing the title to get broken onto two lines even when it wasn't neccessary. I now know that HTML entities may not be embedded in LJ date/time formats...

September 7 2003, 11:08:19 UTC 17 years ago

^{It's^{my^{journal^{I^{can^{do^{what^{I^want!}}}}}}}}

September 7 2003, 11:57:10 UTC 17 years ago

_{You_{are_{welcome_{to_{do_{whatever_{you_{want,_{just_{don't_{come_{crying_{to_{me_{complaining_{that_{I've_{spoiled_{your_fun.}}}}}}}}}}}}}}}}}}}

;-)

September 7 2003, 12:36:53 UTC 17 years ago

Nifty! IE, however, renders all levels of <lsu[pb]>s at the same size - does spoil the fun a little...

17 years ago

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