Number 9, number 9, number 9.
The Beatles’ “Revolution 9” has the above loop, and their version of Rock Band is being released today. The movie 9 comes out today, too.
When a number has a lot of nines in it, like .99999999999999999, many computer systems can run into rounding problems. Fortunately, Mathematica can handle both exact and numeric forms. Here are exact forms of various
numbers whose numeric forms have lots of nines.
Can your system figure these numbers out? Here are the Mathematica input forms for them:
Numbers such as these occur in the study of almost integers. When trigonometric functions are added, then the number of nines can greatly increase. For example, 2017 21/5/π ≈ 737.50000000208, and thus sin(2017 21/5) ≈ –0.99999999999999997857. Pisot numbers can also be fantastically close to an integer.
Here are the numerical approximations for the numbers above.
1. 0.99999999999999999999999999999992878288974707564089
2. 0.99999999999999999999999999999992888272478918067295
3. 0.99999999999999999999999999999999990016495789496794
4. 110.99999996188658332
5. 0.99999996813007188185
6. 0.99999999871766046865
7. 5.9999999561918933296
8. 49.999999106159879944
9. 0.99999994563238375162
As the Beatles might say, “Take this, brother; may it serve you well…. Number 9, number 9, number 9.”
This is a great post.
Posted by DanSeptember 9, 2009 at 10:03 amAmazing.
Posted by bdzaSeptember 9, 2009 at 12:16 pmHowever, I cannot understand a word in it :_)
On a less mathematically mundane level (in other words real ‘Maat’) :
Posted by Justin GrantSeptember 9, 2009 at 5:09 pmhttp://jng.imagine27.com/articles/2009-09-09-090909_090909.html
Check out the Wikipedia article on harmonic series (http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)), section on the random harmonic series. I counted 39 “9″s. I’d love to read the AMM paper–he preprint doesn’t give the digits though.
Posted by Jason ZuckerSeptember 10, 2009 at 8:18 amI think you missed the best one: E^Pi-Pi !
Posted by Sander HuismanSeptember 10, 2009 at 12:27 pmAch! I remember that number 9 song! It was bizarre, number nine is the only thing that the guy says the entire time… For a while, I thought I would go mad. So, why did he do it?
Posted by IBYSeptember 10, 2009 at 11:21 pmBrilliant Blog.
Two simple “procedural” contributions:
Table[N[1 - 10^-n, n], {n, 20}]
(with some mysterious rounding at n=15 and 16)
and
Table[StringJoin[Table["9", {n}]], {n, 10}]
Posted by JaimeNovember 18, 2009 at 9:53 am