All the Movies

I thought it might be fun to compute the total number of possible movies, based on frame rate, resolution, and the number of possible colors on a standard high-definition digital blu-ray or something. So with some input from people who know more than I do about those things, I came up with the following numbers:

2203 = 10648000 colors/pixel
24 frames/second
1920*1080=2073600 pixels/frame

And I want to calculate all of the possible movies with runnings times less than or equal to exactly four hours.

(4 hours = 14400 sec) * (24 frames/sec) = 345600 frames

Based on the numbers for colors/pixel and pixels/frame, the number of possible frames should be

106480002073600

So for any particular number of frames n, we should have exactly

(106480002073600)n

possible movies. So for all possible movies less than or equal to four hours, we have to sum up these terms for all n from 0 to 345600. According to the appropriate summation formula, this is equal to exactly

(106480002073600)n+1

106480002073600-1

And by plugging in the appropriate value, we get, finally,

(106480002073600)345601	=
106480002073600-1
=
10648000716638233600
106480002073600-1

If we want to use logarithms to estimate the size of this number, we can safely ignore the relatively inconsequential bottom half of the fraction and compute

log10(10648000716638233600) =

716638233600(log1010648000) = 5036008956987

So we're talking about a number that has over five trillion digits. That's so many movies that if you watched a movie every day for the rest of the month, you still wouldn't be done!