## Archive for January 1st, 2009

### The number 2009

January 1, 2009

Inspired in part by the legendary Hardy-Ramanujan anecdote as well as the post “What is interesting about the number 2009?” at  Walking Randomly, I offer a few arithmetic and demographic curiosities.

2009 is odd, which in particular guarantees it can be written as a difference of squares, most obviously as $1005^2 - 1004^2$, but perhaps more interestingly as $45^2 - 4^2$.

Being a difference of squares is equivalent to being a sum of consecutive odd numbers (in the first instance “2009”, in the second instance “9+11+13+…+89″);  some other curious sums that lead to our new friend 2009 include

$\frac{1}{15} + \frac{2}{15} + \frac{3}{15} + \cdots + \frac{245}{15}$

$\frac{1}{305} + \frac{4}{305} + \frac{9}{305} + \cdots + \frac{ 122^2}{305}$

$\frac{1}{1004} + \frac{2}{1004} + \frac{3}{1004}+\cdots +\frac{2008}{1004}$

$\frac{1}{1005}+\frac{2}{1005}+\frac{3}{1005}+\cdots+\frac{2009}{1005}$

Hmmm.  There are our friends 1004 and 1005 again!  Interesting….  [See below for conceptual continuity]

2009 can be written as $7^2 \cdot 41$, which among other things guarantees that all groups of order 2009 are solvable.

Based on data published by Statistics Finland, the 2009th largest city in the world has a population of 207,000 people [a three-way tie between São Leopoldo Brazil,  Ashdod Israel, and Lutsk Ukraine].

Forbes famously publishes lists of the top n individuals, companies, etc… in terms of wealth, corporate size, market share, etc…  Forbes’ list of the world’s billionaires has 1125 entries, so we can’t identify the 2009th wealthiest individual.  However, we can use their data to extrapolate and estimate the net worth of the 2009th richest person.

The Forbes data has a very good fit to a power law distribution (in the graph, the blue rhombi squares represent individual data points, the black line the power law of best fit).  This model predicts that the 2009th wealthiest individual will have a net worth of $279.27 (2009)^{-0.775} \approx 0.770$ billion dollars, or \$770 million.

Their list of the world’s largest companies, alas, is the “Forbes Global 2000″, and as one might guess, has only 2000 corporations listed.  Unfortunately, Forbes publishes the top 2000 companies as measured by their own internal (unpublished) index, a weighted average of some sort based on sales, profit, assets, market value, etc….  Without knowing how their index is calculated, one can’t easily extrapolate to determine the size of a hypothetical 2009th largest company in the world.  Presumably it is similar to Mitsubishi Gas Chemical, the company ranked 2000th by Forbes in their list.

If these notes inspire you to find more instances or properties of 2009, add them to the comments at “What is interesting about the number 2009?” at  Walking Randomly.

May 2009 bring all of our readers a wonderful new year!

Added 1/1/09: I knew there had to be a rhyme and reason to the recurrence of 1004 and 1005 in both the difference of squares and the sum of fractions deal.  Here’s the scoop:

If x is odd, then$x = \left( \frac{x+1}{2} \right)^2 - \left( \frac{x-1}{2} \right)^2$.

Furthermore, $1 + 2 + \cdots + (x-1) = \frac{ (x-1)x}{2}$, so in fact

$\frac{1}{(x-1)/2} + \frac{2}{(x-1)/2} + \cdots + \frac{x-1}{(x-1)/2} = x$,

and similarly since $1+2+\cdots +x = \frac{x(x+1)}{2}$, we also have

$\frac{1}{(x+1)/2} + \frac{2}{(x+1)/2} + \cdots + \frac{x}{(x+1)/2} = x$.