The online cartoon xkcd is one of my favorite things. It is published three times per week on M-W-F. One episode, number 1047, gave approximations for physical events. One of the items mentioned was the number of seconds in a year. It has a fairly exact value of
60 seconds x 60 minutes x 24 hours x 365.25 days per year = 31557600 seconds per year
The approximate value given on http://m.xkcd.com/1047/ for the number of seconds in a year was 75^4 = 31640625
31640625 is 0.263% higher than 31557600
Today I was playing with differences of factorials on my TI-89 and found a closer approximation.
317! ÷ 314! = 31554200 which is 0.0108% lower than 31557600
Update: After reading Rob's comments I decided to see how long the 0.0108% of a year was in hours, and eXpress that value as a factorial division.
31557600 - 31554200 = 3400 seconds
One hour equals 60 x 60 = 3600 seconds per hour
So 16! ÷ 13! = 3360 which leaves 40 seconds
40! ÷ 39! = 40
But that last step is just cheating to get to an eXact number, because
X! ÷ (X-1)! = X
So one year equals (317! ÷ 314!) + (16! ÷ 13!) + (40! ÷ 39!) seconds
But then I noticed that 17! ÷ 14! = 4080 which is fairly close to 3400, and keeps the equation pretty with translational symmetry, the repeating of the 17 and 14's.
So calculating an approximate pretty value 317! ÷ 314! + 17! ÷ 14! = 31558280 which is 0.0022% larger than a real year
Pretty is important!!!
60 seconds x 60 minutes x 24 hours x 365.25 days per year = 31557600 seconds per year
The approximate value given on http://m.xkcd.com/1047/ for the number of seconds in a year was 75^4 = 31640625
31640625 is 0.263% higher than 31557600
Today I was playing with differences of factorials on my TI-89 and found a closer approximation.
317! ÷ 314! = 31554200 which is 0.0108% lower than 31557600
Update: After reading Rob's comments I decided to see how long the 0.0108% of a year was in hours, and eXpress that value as a factorial division.
31557600 - 31554200 = 3400 seconds
One hour equals 60 x 60 = 3600 seconds per hour
So 16! ÷ 13! = 3360 which leaves 40 seconds
40! ÷ 39! = 40
But that last step is just cheating to get to an eXact number, because
X! ÷ (X-1)! = X
So one year equals (317! ÷ 314!) + (16! ÷ 13!) + (40! ÷ 39!) seconds
But then I noticed that 17! ÷ 14! = 4080 which is fairly close to 3400, and keeps the equation pretty with translational symmetry, the repeating of the 17 and 14's.
So calculating an approximate pretty value 317! ÷ 314! + 17! ÷ 14! = 31558280 which is 0.0022% larger than a real year
Pretty is important!!!
I would love to know how the spam bots read a post like this.
ReplyDeleteMia: Spam bots stop maps.
DeleteI prefer the lower approximation as I think that gives me some seconds to spare where I can sit back and chill. I'm just glad you did an average leap year although in all the confusion I don't now know if I only have some spare seconds on leap years. I have my own solution to Spam Bots
ReplyDeletehttp://www.youtube.com/watch?v=anwy2MPT5RE
I made an update to the blog post to get a prettier equation. My original gap was about an hour.
Delete