What is:

(128*128)^16,700,000

as an integer?

I need all this expressed as a number with all the digits listed not
as 'X to the power of...'

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Request for Question Clarification by eiffel-ga on 13 May 2006 05:26 PDT

Dear fehler-ga,

Unfortunately the answer box is not big enough to take all seventy
million three hundred and eighty thousand eight hundred and thirteen
digits, nor will I live long enough to type them all in.

How would you like to proceed? Would you accept an approximation using
scientific notation?

Regards,
eiffel-ga

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Clarification of Question by fehler-ga on 13 May 2006 09:20 PDT

Dear All,

Many thanks, I suspected it would be an insanely large number. ;-)
Obviously the gazillion digits won't work. To give you some
background, and an idea of what we're looking for, it's for a project
called 'Imbecil' which generates random icons. It's here:

http://www.fallt.com/imbecil

http://recons.tructed.info/graphics/

The first version generated a random 32 x 32 pixel, 1 bit icon (black
and white), i.e. one of a possible:

17976931348623159077293051907890247336179769789423065727343008115773267580550096
31327084773224075360211201138798713933576587897688144166224928474306394741243777
67893424865485276302219601246094119453082952085005768838150682342462881473913110
540827237163350510684586298239947245938479716304835356329624224137216

The second version generates a random 128 x 128 pixel, 32 bit icon (full colour).

We're trying to express how many icons are possible - (128 * 128) ^
16,700,000 - hence the need for the extremely large number. If you can
think of a better way of expressing this huge number - an
approximation using scientific notation - or something else that
suggests just how vast the possible array of icons is we'd accept this
as an answer. We'd also credit you, should you wish, as a contributor
to this work of art.

Many thanks,

fehler-ga

Hi fehler-ga,

The number of different possible icons is an enormous number,
comprising more than seventy million digits. However, we do know that
the number ends in a four (because 128 x 128 equals 16384, and if you
multiply any number ending in 4 by itself an even number of times the
answer will also end in 4).

Here's another way to think of just how big this number is:

Suppose every person in the world produced one million icons every
second, and had been doing this for the entire estimated history of
the universe (say 20 billion years). The number of icons produced
would be then less than ten to the power 31. That's not even one
billionth of one percent of the possible icons. In fact it doesn't
even scratch the surface of the journey towards one billionth of one
percent of the possible icons.

Let's look at it another way. If you were to write this number down,
at 10 digits per inch, you would need a piece of paper more than a
hundred miles long to contain it.

The number of particles in the universe is well below one googol (a
number with 101 digits), yet the number of different icons is a number
with more than seventy million digits!

Incidentally, I get 70,380,813 digits whereas kottekoe-ga gets
70,706,234. I'm pretty sure that's because Google's calculator is
using logarithms internally when it evaluates 14*(2^24) and therefore
generating additional rounding error. If I type the following into
Google:

   14 * 16700000 * (log 2)

then I get 70,380,813. This agrees with the KCalc calculator, which
gives an approximate answer of 9.68810425007 times ten to the
70,380,812 when I enter the  expression (128*128)^16,700,000.

I trust you find this information interesting and useful for your work of art.

You may also be interested in the following question by ioawnat-ga,
who is printing every possible version of a book and is seeking a
universe big enough to hold a box that will contain them all:

"Google Answers - Large Number Computation"
http://answers.google.com/answers/threadview?id=725701

Regards,
eiffel-ga

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Request for Answer Clarification by fehler-ga on 14 May 2006 09:50 PDT

Dear All,

Thank you all so much for your enthusiastic contributions. Can I
clarify one thing... Is the number 2^24 (16,777,216) the same as 32
bit?

Lastly, if any of you would like to be added to the credits for this
work of art, please email me with your names:

chris (at) fallt (dot) com

Thanks again!

Chris

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Clarification of Answer by eiffel-ga on 14 May 2006 12:35 PDT

Hi fehler-ga,

When a graphics format has "32 bits per pixel", it generally contains
8 bits of information for each primary colour - red, green, blue. This
gives 24 bits per pixel of colour information. The remaining eight
bits per pixel may be unused, or they may be used for some other kind
of information (typically for transparency data).

Therefore, each pixel within your icon has one out of 2^24
(16,777,216) possible colours. If all 32 bits carried colour
information, there would instead be 2^32 (4,294,967,296) possible
colours for each pixel.

Thanks for the kind offer of credit - you may credit my work as
"eiffel-ga (Google Answers Researcher)".

I would be willing to honor your request for a mere penny a digit.
Please send me a down payment of $500,000 and I'll get started. I will
give a substantial discount if you let me express the answer in
binary, octal, or hexadecimal.

Why not go with Eiffel's orginal statement. You can calculate the
number of digits yourself with Google's calculator. I believe you
really mean to say 24 bits of color, not 32. Then the number in
question is:

N = (128*128)^(2^24) = (2^14)^(2^24) 

Taking the base 10 logarithm and rounding up will give us the number of digits:

log(N) = (2^24)*log(2^14) = 14*(2^24)*log(2)

Type the last expression into Google, round to the next highest
integer and you get:

70,706,234 digits

So you could say:

"The number of possibilities is so large that it requires more than 70
million decimal digits to express it in numbers."

And of course, you could add that this number is vastly larger than a
googol (10^100). Also, if you made each icon smaller than a proton,
you could fill the whole universe with icons, reduce that universe to
the size of a proton, fill the universe again with those, and repeat
this procedure hundreds of thousands of times without running out of
icons.

Eiffel, the difference in our calculations is because you used 16.7
million, which is what the questioner asked. I used 2^24, which is the
correct number of different web colors.

2^24 = 16,777,216

The only way I can think about numbers this big is the way I gave in
my answer with repeated shrinking and refilling of the universe.

One last way to write this number:

In hexadecimal, it is given by:

1000....000 - with 58,720,256 zeros

kottekoe, thanks for pointing out the source of the discrepancy in the
number of digits. Your "recursive universe" notion is a good one -
perhaps if you give it a catchy name and promite it, it will become
accepted as a standard way of thinking about bigger-than-big numbers.

I like the hexadecimal version because it is exact (as wound a binary
one be, of course), but not everyone is as comfortable with hex as
computer people are.

This all way above me, but it seems that the number of visually
differentiable icons would be only a small fraction of the theoretical
total.

a simple way round it is to generate 128^167000 or wotever it is 128
times break it down in2 a number of operations is the gazzilion digit
randomization is too much