Absent For A Bit ....

I am away for a little while working on a few or more episodes for The Adventures of My Space Alien Alter Ego story ... will return (to Earth) soon!

Notice: Blogger has screwed up and lost a bunch of photos out of my blog! They are replaced with a gray silhouette of a human head. I will eventually get them replaced with the correct photo, it may take a while to find and fix everything. So some of my stories don't make much sense without their photos, sorry for the inconvenience.

esbb

2013-02-03

In Seconds Place

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!!!

4 comments:

Mia said...

I would love to know how the spam bots read a post like this.

esbboston said...

Mia: Spam bots stop maps.

Rob Z Tobor said...

I 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

http://www.youtube.com/watch?v=anwy2MPT5RE

esbboston said...

I made an update to the blog post to get a prettier equation. My original gap was about an hour.

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