# All the Movies

2009-10-17

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!