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 Subject: Mathematics, Large Numbers Category: Computers > Algorithms Asked by: fehler-ga List Price: \$5.00 Posted: 13 May 2006 03:53 PDT Expires: 12 Jun 2006 03:53 PDT Question ID: 728368
 ```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...'``` 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``` 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```
 Subject: Re: Mathematics, Large Numbers Answered By: eiffel-ga on 13 May 2006 14:44 PDT
 ```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``` 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``` 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)".```
 Subject: Re: Mathematics, Large Numbers From: kottekoe-ga on 13 May 2006 07:46 PDT
 ```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.```
 Subject: Re: Mathematics, Large Numbers From: kottekoe-ga on 13 May 2006 10:15 PDT
 ```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."```
 Subject: Re: Mathematics, Large Numbers From: kottekoe-ga on 13 May 2006 10:23 PDT
 ```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.```
 Subject: Re: Mathematics, Large Numbers From: kottekoe-ga on 13 May 2006 21:26 PDT
 ```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.```
 Subject: Re: Mathematics, Large Numbers From: kottekoe-ga on 13 May 2006 21:37 PDT
 ```One last way to write this number: In hexadecimal, it is given by: 1000....000 - with 58,720,256 zeros```
 Subject: Re: Mathematics, Large Numbers From: eiffel-ga on 14 May 2006 05:08 PDT
 ```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.```
 Subject: Re: Mathematics, Large Numbers From: myoarin-ga on 14 May 2006 08:25 PDT
 ```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.```
 Subject: Re: Mathematics, Large Numbers From: matchett808-ga on 11 Jul 2006 07:27 PDT
 ```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```