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 2^{1/5}/*π* ≈ 737.50000000208, and thus sin(2017 2^{1/5}) ≈ –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.”

## 7 Comments

This is a great post.

Amazing.

However, I cannot understand a word in it :_)

On a less mathematically mundane level (in other words real ‘Maat’) :

http://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.

I think you missed the best one: E^Pi-Pi !

Ach! 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?

Brilliant 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}]