# Calculating Large Exponents **URL:** \ **Category:** Factual Questions\ **Created:** [May 21, 2002, 7:58pm UTC](https://boards.straightdope.com/t/calculating-large-exponents/110163 "2002-05-21T19:58:04Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![cblackhand](https://avatars.discourse-cdn.com/v4/letter/c/ecc23a/32.png) [@cblackhand](https://boards.straightdope.com/u/cblackhand)\ **Post date:** [May 21, 2002, 7:58pm UTC](https://boards.straightdope.com/t/calculating-large-exponents/110163/1 "2002-05-21T19:58:04Z")
Some coworkers and I have run in to a problem we cannot solve and so I turn to the brilliance of the Teeming Millions. It came up off this [webpage](http://www.abarnett.demon.co.uk/theories.html#BITMAP). > [@](#): > > All knowledge in the universe can easily be displayed on your computer screen. How? I hear you cry (as well you might). It is quite simple, and requires nothing more than a computer than can count (which is what they are best at). Imagine, if you will, and 8x8 grid of pixels. This grid (or bitmap) contains 64 pixels, each of which may be 1 or 0 (on or off, black or white). The total number of combinations of on/off pixels is 2^64 (which is an extremely large number). This simple 8x8 grid can easily hold such images as all the letters of the alphabet, the digits 0-9, pictures of space invaders and PacMan, different shades of grey and so on. To display all the possible combinations simply requires the computer to treat the 8x8 grid as a 64 bit binary number (1x64 instead of 8x8). If the computer starts at zero and counts up to 2^64, displaying the binary number in the format of an 8x8 bitmap, it will show all the possible combinations of pixels that there are (including the above-mentioned letters and Pacmen). > If we expand this concept a little, it is easy to see that an entire computer screen may be treated as a single binary number. If your display is 1280 x 1024 pixels (quite a common size) then it could be treated as a single 1310720 bit binary number. It can store all the possible numbers from 0 to 2^1310720-1 (which is more than the number of particles in the universe). > If you set your computer (admittedly, it would have to be quite a fast one) counting, from 0 up to 2^1310720-1, it would display all the knowledge in the universe. Every frame from every movie, every page of every book, the DNA of every creature that has ever lived, the face of every person who has ever lived, the complete source of every computer program ever written, the entire life history of every single living organism, a picture of every square inch of every planet in the universe. Everything, ever would appear on your screen. There would be an awful lot of meaningless junk, but in amongst it will be all the knowledge in the universe. > And that’s just with a single bitplane. If you treat a 24-bit display as a single binary number, you’ll get the same information, but in colour! Assuming this could be done how long would it take to show everything? We ran in to the roadblock when we tried to calculate how many frames there were total in a non-exponential form. We were looking at the grayscale version, so there are 2[sup]1310719[/sup] frames, but that is not too useful for our purposes. We also assumed that the monitor this was displaying on was refreshing 100 times per second. 2[sup]1310719[/sup]/100 gives us the total time in seconds, but how do you calculate that? None of the digital means we tried could work it. Please help assuage our curiosity. Thanks in advance, John ---
**Author:** ![Karen\_Too](https://avatars.discourse-cdn.com/v4/letter/k/ce7236/32.png) [@Karen\_Too](https://boards.straightdope.com/u/Karen_Too)\ **Post date:** [May 21, 2002, 9:00pm UTC](https://boards.straightdope.com/t/calculating-large-exponents/110163/2 "2002-05-21T21:00:11Z")
Use logarithms, specifically, start with base 2: 2^y = A log\_2(2^y) = y = log\_2(A) y = log\_10(A)/log\_10(2) y\*log\_10(2) = log\_10(A) -k- ---
**Author:** ![tracer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tracer/32/20578_2.png) [@tracer](https://boards.straightdope.com/u/tracer)\ **Post date:** [May 21, 2002, 9:18pm UTC](https://boards.straightdope.com/t/calculating-large-exponents/110163/3 "2002-05-21T21:18:51Z")
So … log[sub]10[/sub]2 = 0.301, and here, y = 1310719. So y \* log[sub]10[/sub]2 = 394565.7. Therefore, A = **5 x 10[sup]394565[/sup] seconds** (This would be 1.6 x 10[sup]394558[/sup] years.) ---
**Author:** ![tracer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tracer/32/20578_2.png) [@tracer](https://boards.straightdope.com/u/tracer)\ **Post date:** [May 21, 2002, 9:23pm UTC](https://boards.straightdope.com/t/calculating-large-exponents/110163/4 "2002-05-21T21:23:28Z")
log[sub]10[/sub]2 = 0.301 y = 1310719 So, y \* log[sub]10[/sub]2 = 394565.7. A = 10[sup]394565.7[/sup] = 5 x 10[sup]394565[/sup]. Divide that by 100, and we get **5 x 10[sup]394563[/sup] seconds.** (This would be 1.6 x 10[sup]394556[/sup] years, which is a little longer than most of us are willing to wait around for.) ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [May 21, 2002, 9:26pm UTC](https://boards.straightdope.com/t/calculating-large-exponents/110163/5 "2002-05-21T21:26:34Z")
At least it would be faster than using monkeys and typewriters? ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [May 21, 2002, 9:30pm UTC](https://boards.straightdope.com/t/calculating-large-exponents/110163/6 "2002-05-21T21:30:38Z")
First of all, the leap from greater than the number of particles in the universe to all knowledge is just plain idiotic. (But see the note at the end.) Secondly, 2^131079 is about 10^394565.7349, so subtact 2 (dividing by 100) from the exp. gives 10^394563.7349 seconds. Now as we all know “there are (about) Pi seconds in a nano-century”. So subtract about 9.5 (losing accuracy here) to get roughly 10^394554.2 centuries. Since the universe is about 10^8.2 centuries old, the two numbers are noways comparable. Back to a small but meaninful number: Chaintin defined a number, [Omega](http://www.cs.auckland.ac.nz/CDMTCS/docs/god_dice.html), with a property that is "embodies an enormous amount of wisdom in a very small space … inasmuch as its first few thousands digits, which could be written on a small piece of paper, contain the answers to more mathematical questions than could be written down in the entire universe. " (Charles Bennett) Wolfram’s new book is suppose to also cover a similar issue. (Called away before final submit., so a little redundancy here.) ---
**Author:** ![cblackhand](https://avatars.discourse-cdn.com/v4/letter/c/ecc23a/32.png) [@cblackhand](https://boards.straightdope.com/u/cblackhand)\ **Post date:** [May 21, 2002, 10:02pm UTC](https://boards.straightdope.com/t/calculating-large-exponents/110163/7 "2002-05-21T22:02:24Z")
Thanks everyone.